Criss-cross

methods

A

fresh

view

on

Pivot

algorithms

Report

K.

Fukuda

T.

Terlaky

Technische

Universiteit

Delft

Delft

University

of

Technology

Faculteit

der

Technische

Wiskunde

en

Informatica

Faculty

of

Technical

Mathematics

and

Informatics

ISSN

Copyright

c

by

the

Faculty

of

Technical

Mathematics

and

Informatics

Delft

The

Netherlands

No

part

of

this

Journal

may

be

reproduced

in

any

form

by

print

photoprint

micro?lm

or

any

other

means

without

permission

from

the

Faculty

of

Technical

Mathematics

and

Informatics

Delft

University

of

Technology

The

Netherlands

Copies

of

these

reports

may

be

obtained

from

the

bureau

of

the

Faculty

of

Technical

Mathematics

and

Informatics

Julianalaan

BL

Delft

phone

A

selection

of

these

reports

is

available

in

PostScript

form

at

the

Faculty

's

anonymous

ftpsite

They

are

located

in

the

directory

pub/publications/tech-reports

at

ftp.twi.tudelft.nl

DELFT

UNIVERSITY

OF

TECHNOLOGY

REPORT

Nr

-LCB-

Criss-Cross

Methods

A

Fresh

View

on

Pivot

Algorithms

K.

Fukuda

and

T.

Terlaky

ISSN

-LCB-

Reports

of

the

Faculty

of

Technical

Mathematics

and

Informatics

Nr

-LCB-

Delft

April

i

Komei

Fukuda

Department

of

Mathematics

Swiss

Federal

Institute

of

Technology

Lausanne

and

Institute

for

Operations

Research

Swiss

Federal

Institute

of

Technology

Z?urich

Switzerland

Tam?as

Terlaky

Faculty

of

Technical

Mathematics

and

Informatics

Delft

University

of

Technology

P.O.

Box

GA

Delft

The

Netherlands

Copyright

c

by

Faculty

of

Technical

Mathematics

and

Informatics

Delft

The

Netherlands

No

part

of

this

Journal

may

be

reproduced

in

any

form

by

print

photoprint

microfilm

or

any

other

means

without

written

permission

from

Faculty

of

Technical

Mathematics

and

Informatics

Delft

University

of

Technology

The

Netherlands

ii

Abstract

This

paper

surveys

the

origins

and

implications

of

-LRB-

finite

-RRB-

criss-cross

methods

in

linear

programming

and

related

problems

Most

pivot

algorithms

like

Dantzig

's

celebrated

Simplex

method

need

a

feasible

basis

to

start

with

when

the

basis

is

changed

feasibility

is

preserved

until

optimality

or

unboundedness

is

detected

To

obtain

a

feasible

basis

a

so-called

first

phase

problem

must

be

solved

On

the

contrary

a

criss-cross

method

is

a

pivot

algorithm

that

solves

the

linear

programming

problem

in

one

phase

without

preserving

possible

feasibility

of

the

intermediate

solutions

Getting

free

from

the

necessity

of

preserving

feasibility

and

monotonicity

needs

a

fresh

view

on

algorithmic

concepts

it

raises

new

questions

and

opens

new

perspectives

After

a

brief

introduction

and

historical

review

a

finite

criss-cross

algorithm

is

presented

Via

alternative

finiteness

proofs

the

flexibility

of

the

method

is

explored

The

finiteness

of

this

simple

one-phase

procedures

provides

us

with

a

most

simple

algorithmic

proof

of

the

strong

duality

theorem

of

LP

as

well

A

recent

result

on

the

existence

of

a

short

pivot

path

to

an

optimal

basis

is

given

indicating

a

certain

research

direction

for

designing

polynomial

pivot

algorithms

if

they

ever

exist

Some

extensions

to

convex

quadratic

programming

linear

complementarity

fractional

linear

programming

problems

and

to

oriented

matroid

programming

are

discussed

Notes

on

existentially

polytime

-LRB-

EP

-RRB-

theorems

for

linear

complementarity

problems

are

included

Some

conclusions

and

notes

on

possible

future

research

directions

close

the

paper

Introduction

Consider

the

following

primal

and

dual

linear

programming

-LRB-

LP

-RRB-

problem

pair

in

standard

form

-LRB-

P

-RRB-

minfcTx

Ax

b

x

g

-LRB-

D

-RRB-

maxfbTy

ATy

cg

where

A

is

a

given

m?n

real

matrix

c

and

b

-LRB-

x

and

y

-RRB-

are

n

and

m

dimensional

given

-LRB-

variable

-RRB-

real

vectors

respectively

and

AT

denotes

the

transpose

of

the

matrix

A

As

usual

the

first

linear

programming

problem

-LRB-

P

-RRB-

is

called

the

primal

problem

and

the

second

problem

-LRB-

D

-RRB-

is

called

the

dual

problem

This

pair

of

linear

programming

problems

will

be

frequently

referred

to

as

LP

-LRB-

A

b

c

-RRB-

Linear

programming

has

been

one

of

the

most

turbulent

areas

of

applied

mathematics

in

the

last

fifty

years

Despite

the

flux

of

many

recent

approaches

Dantzig

's

simplex

method

-LSB-

-RSB-

still

remains

to

be

one

of

the

most

efficient

algorithms

for

a

great

majority

of

practical

problems

-LSB-

-RSB-

It

is

generally

observed

that

for

most

cases

the

required

number

of

pivot

steps

is

a

linear

-LRB-

and

at

most

quadratic

-RRB-

function

of

the

number

of

variables

This

observation

about

the

practical

efficiency

of

the

simplex

method

-LSB-

-RSB-

is

theoretically

justified

by

proving

its

polynomial

behavior

in

the

expected

number

of

pivot

steps

-LSB-

-RSB-

for

various

probabilistic

models

On

the

other

hand

for

most

variants

-LRB-

specified

by

additional

pivot

rules

-RRB-

of

the

simplex

method

there

are

examples

for

which

the

number

of

pivot

steps

is

exponential

For

such

exponential

examples

see

e.g.

-LSB-

-RSB-

To

find

a

polynomial

pivot

algorithm

for

linear

programming

i.e.

one

for

which

the

number

of

pivot

operations

is

bounded

by

a

polynomial

function

of

the

input

size

or

to

prove

that

such

pivot

algorithms

do

not

exist

seems

to

be

very

hard

In

fact

to

clarify

if

there

exists

such

a

polynomial

time

simplex

algorithm

and

the

closely

related

Hirsch

conjecture

-LSB-

-RSB-

are

among

the

most

challenging

open

problems

in

linear

programming

and

in

polyhedral

theory

The

simplex

method

successfully

combines

greedy

concepts

such

as

preserving

feasibility

and

forcing

monotonicity

of

the

objective

value

It

utilizes

fundamental

geometrical

and

combinatorial

properties

of

linear

programming

problems

represented

by

basic

solutions

A

further

reason

for

the

practical

efficiency

of

the

simplex

method

is

in

the

fact

that

it

is

very

flexible

i.e.

there

are

plenty

of

possibilities

to

select

the

pivot

element

during

the

procedure

even

in

case

of

nondegeneracy

Therefore

many

simplex

variants

have

been

developed

during

the

last

fifty

years

In

the

recent

years

however

the

major

research

efforts

in

linear

programming

have

A

pivot

algorithm

might

be

a

polynomial

simplex

algorithm

or

more

generally

any

finite

pivot

method

that

traverses

through

possibly

both

primal

and

dual

infeasible

basic

solutions

To

date

no

exponential

example

is

known

for

Zadeh

's

-LSB-

-RSB-

pivot

selection

rule

A

recent

survey

on

pivot

rules

is

given

by

Terlaky

and

Zhang

-LSB-

-RSB-

been

shifted

first

to

the

ellipsoid

method

by

Khachian

-LSB-

-RSB-

and

later

even

more

intensively

to

interior

point

methods

initiated

by

the

seminal

paper

of

Karmarkar

-LSB-

-RSB-

New

studies

on

pivot-based

algorithms

are

not

receiving

much

attention

even

among

researchers

in

the

field

of

linear

programming

Moreover

a

large

part

of

the

work

was

presented

in

terms

of

oriented

matroid

programming

and

frequently

without

explicitly

specializing

the

new

pivot

methods

for

the

linear

programming

case

thus

remained

unknown

for

researchers

who

are

not

familiar

with

the

oriented

matroid

terminology

Fundamentals

Before

discussing

criss-cross

methods

first

we

consider

problems

-LRB-

P

-RRB-

and

-LRB-

D

-RRB-

again

As

usual

a

vector

x

Rn

is

called

primal

feasible

if

it

is

nonnegative

and

Ax

b

i.e.

satisfies

all

the

constraints

of

problem

-LRB-

P

-RRB-

a

vector

y

Rm

is

called

dual

feasible

if

s

c

ATy

i.e.

it

satisfies

all

the

constraints

in

-LRB-

D

-RRB-

The

weak

duality

theorem

provides

a

sufficient

condition

for

optimality

Proposition

-LRB-

Weak

duality

-RRB-

If

x

Rn

is

primal

feasible

and

y

Rm

is

dual

feasible

then

cTx

bTy

where

the

equality

is

satisfied

iff

xT

s

PROOF

Since

x

and

y

satisfy

the

primal

and

dual

constraints

we

have

cTx

yTAx

sTx

yTAx

bTy

Clearly

cTx

bTy

iff

xT

s

Corollary

-LRB-

Complementarity

-RRB-

If

x

Rn

is

primal

feasible

y

Rm

is

dual

feasible

and

xT

s

then

x

and

y

are

primal

and

dual

optimal

respectively

The

strong

duality

theorem

says

that

if

LP

-LRB-

A

b

c

-RRB-

is

both

primal

and

dual

feasible

then

there

are

primal

and

dual

feasible

solutions

such

that

cTx

bTy

-LRB-

or

equivalently

xTs

-RRB-

implying

both

are

optimal

Consequently

this

sufficient

condition

of

optimality

for

a

dual

pair

of

feasible

solutions

is

necessary

too

The

least-index

For

recent

results

on

interior

point

methods

the

reader

can

consult

the

books

of

Roos

Terlaky

and

Vial

-LSB-

-RSB-

Wright

-LSB-

-RSB-

Kojima

et.al

-LSB-

-RSB-

and

the

collection

of

survey

papers

in

-LSB-

-RSB-

criss-cross

method

described

in

Section

will

yield

a

very

simple

algorithmic

proof

of

this

fundamental

result

The

complementarity

condition

represents

a

strong

combinatorial

character

of

linear

programming

problems

that

is

featured

by

pivot

algorithms

In

the

view

of

the

above

discussions

the

set

of

optimal

solutions

can

be

characterized

as

the

set

of

solutions

of

the

system

Ax

b

ATy

s

c

xTs

-LRB-

-RRB-

x

s

Since

all

pivot

methods

generate

basic

solutions

their

intermediate

solutions

satisfy

both

equality

constraints

and

the

complementarity

condition

Simplex

pivot

methods

always

require

and

preserve

either

primal

or

dual

feasibility

of

the

generated

basic

solutions

A

primal

-LRB-

dual

respectively

-RRB-

simplex

method

is

initiated

by

a

primal

-LRB-

dual

-RRB-

feasible

basic

solution

If

neither

optimality

nor

unboundedness

is

detected

then

a

new

basis

and

the

related

basic

solution

is

chosen

in

such

a

way

that

a

dual

-LRB-

primal

-RRB-

infeasible

variable

enters

-LRB-

leaves

-RRB-

the

basis

and

-LRB-

-RRB-

the

new

basis

is

again

primal

-LRB-

dual

-RRB-

feasible

-LRB-

-RRB-

the

new

basis

differs

exactly

by

one

element

from

the

old

one

i.e.

the

new

basis

is

a

neighbor

of

the

old

one

Feasibility

of

the

basic

solution

is

preserved

throughout

and

the

primal

-LRB-

dual

-RRB-

objective

function

value

is

monotonically

decreasing

-LRB-

increasing

-RRB-

at

each

basis

exchange

To

produce

a

primal

or

dual

feasible

basic

solution

is

a

nontrivial

task

It

requires

the

solution

of

another

LP

problem

the

so-called

first

phase

problem

If

no

feasibility

-LRB-

nonnegativity

-RRB-

condition

is

forced

to

hold

or

forced

to

be

preserved

then

we

can

talk

about

criss-cross

methods

A

finite

criss-cross

method

goes

through

some

-LRB-

possibly

both

primal

and

dual

infeasible

-RRB-

basic

solutions

until

either

primal

infeasibility

dual

infeasibility

or

optimality

is

detected

This

procedure

can

be

started

with

any

basic

solution

and

solves

the

linear

programming

problem

in

one

phase

in

a

finite

number

of

pivot

steps

The

least-index

criss-cross

method

-LRB-

to

be

presented

in

Section

-RRB-

is

perhaps

the

simplest

such

algorithm

and

gives

rise

to

a

very

short

algorithmic

proof

of

the

LP

strong

duality

theorem

-LRB-

see

Theorem

-RRB-

The

theory

of

linear

inequalities

admits

also

a

simple

constructive

build-up

based

on

similar

techniques

-LSB-

-RSB-

Contrary

to

pivot

methods

where

complementarity

is

maintained

throughout

the

solution

process

intermediate

solutions

generated

by

interior

point

methods

typically

satisfy

all

equality

and

nonnegativity

conditions

but

complementarity

Complementarity

is

attained

just

at

termination

Thus

interior

point

methods

do

not

make

explicit

use

of

the

combinatorial

features

of

linear

programming

Figure

In

this

paper

pivot

methods

refer

to

those

algorithms

which

maintain

only

one

basis

at

a

time

There

are

pivot

algorithms

of

different

type

e.g.

the

algorithm

-LSB-

-RSB-

maintains

two

different

bases

at

a

time

Ax

b

ATy

s

c

x

s

xisi

Dual

Simplex

Primal

Simplex

ff

Criss-Cross

Method

Interior

Point

Methods

Fig.

Algorithms

illustrates

the

requirements

of

the

different

methods

Another

fundamental

property

in

linear

programming

is

that

the

primal

and

dual

feasible

directions

are

in

orthogonal

subspaces

i.e.

the

subspaces

fx

Ax

g

and

fs

s

AT

yg

are

orthogonal

This

orthogonality

property

in

a

slightly

extended

form

is

fully

utilized

not

only

in

pivot

methods

but

in

interior

point

methods

as

well

Certain

sign

properties

of

orthogonal

subspaces

furnish

the

base

for

a

combinatorial

abstraction

-LSB-

-RSB-

and

for

deeper

understanding

of

origins

and

perspectives

of

combinatorial

pivot

rules

The

main

purposes

of

this

paper

are

to

present

the

underlying

combinatorics

of

finite

criss-cross

pivot

methods

to

discuss

larger

classes

of

problems

which

can

be

solved

by

such

methods

and

to

highlight

some

new

questions

and

results

emerging

from

this

fresh

view-point

The

paper

is

organized

as

follows

The

origins

of

our

finite

criss-cross

method

including

a

historical

account

about

its

birth

are

given

in

Section

Then

at

the

end

of

this

section

we

present

some

standard

notation

which

we

use

throughout

the

paper

In

Section

termination

criteria

are

discussed

and

the

concept

of

admissible

pivots

is

introduced

In

Section

we

present

the

least-index

criss-cross

method

for

linear

programming

problems

Based

on

the

finiteness

result

a

simple

constructive

proof

of

the

strong

duality

theorem

of

LP

is

given

Some

discussions

on

the

flexibility

of

finite

criss-cross

methods

alternative

finiteness

proofs

and

notes

on

possible

finite

variants

and

on

worst

case

behavior

are

included

as

well

In

Section

recent

results

on

the

shortest

pivot

sequence

to

an

optimal

basis

is

discussed

Section

is

essentially

devoted

to

the

case

of

sufficient

linear

complementarity

problems

-LRB-

LCP

-RRB-

to

which

both

the

LP

strong

duality

theorem

and

the

least-index

criss-cross

method

are

extended

Basic

relations

to

convex

quadratic

programming

problems

are

briefly

discussed

as

well

In

Section

we

consider

the

generalization

of

sufficient

LCPs

to

oriented

matroids

and

present

a

purely

combinatorial

form

of

the

LCP

duality

theorems

Finally

in

Section

some

conclusions

follow

and

further

research

problems

are

given

Historical

Notes

Origins

From

the

early

days

of

LP

research

people

have

been

looking

for

an

algorithm

that

avoids

the

two

phase

procedure

needed

in

the

simplex

method

when

solving

the

general

LP

problem

Such

a

method

was

assumed

to

rely

on

the

intrinsic

symmetry

behind

the

primal

and

dual

problems

-LRB-

self-dual

-RRB-

and

it

should

be

able

to

start

with

any

basic

solution

There

were

several

attempts

made

to

relax

the

feasibility

requirement

in

the

simplex

method

It

is

important

to

mention

Dantzig

's

-LSB-

-RSB-

parametric

self-dual

simplex

algorithm

This

algorithm

can

be

interpreted

as

Lemke

's

-LSB-

-RSB-

algorithm

for

the

corresponding

linear

complementarity

problem

-LSB-

-RSB-

In

the

sixties

people

realized

that

pivot

sequences

through

possibly

infeasible

basic

solutions

might

result

in

significantly

shorter

paths

to

the

optimum

Moreover

a

self-dual

one

phase

procedure

was

expected

to

make

linear

programming

more

easily

accessible

for

broader

public

Probably

these

advantages

stimulated

the

birth

of

Zionts

-LSB-

-RSB-

criss-cross

method

-LRB-

after

which

the

finite

criss-cross

methods

discussed

in

this

paper

were

named

too

-RRB-

Remark

-LRB-

Zionts

criss-cross

method

-RRB-

Assuming

that

the

reader

is

familiar

with

both

the

primal

and

dual

simplex

methods

Zionts

criss-cross

method

can

easily

be

explained

-LCB-

It

can

be

initialized

by

any

possibly

both

primal

and

dual

infeasible

basis

If

the

basis

is

not

optimal

then

there

are

some

primal

or

dual

infeasible

variables

One

might

choose

any

of

these

It

is

advised

to

choose

primal

and

dual

infeasible

variables

alternately

if

possible

-LCB-

If

the

selected

variable

is

dual

infeasible

then

it

enters

the

basis

and

the

leaving

variable

is

chosen

among

the

primal

feasible

variables

in

such

a

way

that

primal

feasibility

of

these

variables

is

preserved

If

no

such

basis

exchange

is

possible

another

infeasible

variable

is

selected

-LCB-

If

the

selected

variable

is

primal

infeasible

then

it

leaves

the

basis

and

the

entering

variable

is

chosen

among

the

dual

feasible

variables

in

such

a

way

that

dual

feasibility

of

these

variables

is

preserved

If

no

such

basis

exchange

is

possible

another

infeasible

variable

is

selected

If

the

current

basis

is

infeasible

but

of

the

infeasible

variables

allows

a

pivot

fulfilling

the

above

requirements

then

it

is

proved

that

the

problem

LP

-LRB-

A

b

c

-RRB-

has

no

optimal

solution

Once

a

primal

or

dual

feasible

solution

is

reached

then

Zionts

method

reduces

to

the

primal

or

dual

simplex

method

respectively

One

attractive

character

of

Zionts

criss-cross

method

is

primal-dual

symmetry

-LRB-

selfduality

-RRB-

and

this

alone

differentiates

itself

from

the

simplex

method

However

it

is

not

clear

how

one

can

design

a

finite

version

-LRB-

i.e.

a

finite

pivot

rule

-RRB-

of

this

method

Consequently

it

did

not

get

enough

attention

and

theoretical

support

Nevertheless

this

is

the

first

root

of

the

finite

criss-cross

methods

studied

in

this

paper

The

need

for

a

simple

one

phase

self-dual

method

stayed

pending

The

other

root

of

finite

criss-cross

methods

was

the

intellectual

effort

to

find

finite

variants

-LRB-

other

than

the

lexicographic

rule

-LSB-

-RSB-

-RRB-

of

the

simplex

method

and

its

specialized

variants

These

efforts

were

also

stimulated

by

studying

the

combinatorial

structures

behind

linear

programming

From

the

early

seventies

in

several

branches

of

the

optimization

theory

finitely

convergent

algorithms

were

published

and

in

particular

consistent

labelling

and

pivot

selection

rules

based

on

least

index

ordering

established

a

solid

foundation

for

finite

pivot

algorithms

The

birth

of

oriented

matroids

and

oriented

matroid

programming

-LSB-

-RSB-

gave

an

essential

impulse

to

this

research

It

became

clear

that

although

the

simplex

method

has

rich

combinatorial

structures

some

essential

results

like

the

finiteness

of

Bland

's

minimal

index

rule

-LSB-

-RSB-

can

not

be

proved

in

the

oriented

matroid

context

In

fact

Edmonds

and

Fukuda

-LSB-

-RSB-

showed

that

it

might

cycle

in

the

oriented

matroid

case

due

to

the

possibility

of

nondegenerate

cycling

which

is

impossible

in

linear

programming

Interesting

recursive

pivot

algorithms

were

discovered

and

mainly

published

in

the

oriented

matroid

context

Among

them

are

Bland

's

recursive

algorithm

-LSB-

-RSB-

the

Edmonds-Fukuda

algorithm

-LSB-

-RSB-

-LRB-

the

interested

reader

can

find

variants

and

generalizations

in

e.g.

-LSB-

-RSB-

-RRB-

All

these

methods

in

the

linear

case

are

variants

of

the

simplex

method

i.e.

they

preserve

the

feasibility

of

the

basis

In

contrast

in

the

case

of

oriented

matroid

programming

only

Todd

's

finite

lexicographic

method

-LSB-

-RSB-

preserves

feasibility

of

the

basis

and

therefore

yields

a

finite

simplex

algorithm

for

oriented

matroids

For

a

thorough

survey

on

pivot

algorithms

the

reader

is

referred

to

-LSB-

-RSB-

All

these

research

results

made

the

time

ripe

for

the

discovery

of

finite

criss-cross

methods

It

is

remarkable

that

almost

at

the

same

time

in

different

parts

of

the

world

-LRB-

China

Hungary

USA

-RRB-

essentially

the

same

result

was

obtained

independently

by

approaching

the

problem

from

quite

different

directions

Because

in

some

cases

Zionts

method

reduces

to

the

simplex

method

lexicographic

perturbation

or

minimal

index

resolution

is

a

must

when

one

thinks

about

finiteness

However

these

do

not

seem

to

be

sufficient

to

prove

finiteness

in

the

general

case

when

the

initial

basis

is

both

primal

and

dual

infeasible

See

Tucker

's

-LSB-

-RSB-

consistent

labelling

technique

in

the

Ford-Fulkerson

flow

algorithm

See

Murty

's

-LSB-

-RSB-

Bard-type

scheme

for

the

P-matrix

linear

complementarity

problem

and

Bland

's

-LSB-

-RSB-

celebrated

least

index

rule

in

linear

programming

Fathers

The

first

finite

criss-cross

algorithm

which

we

call

the

least-index

criss-cross

method

was

discovered

independently

by

Chang

-LSB-

-RSB-

Terlaky

-LSB-

-RSB-

and

Wang

-LSB-

-RSB-

further

a

strongly

related

general

recursion

by

Jensen

-LSB-

-RSB-

In

his

unpublished

report

-LSB-

-RSB-

Chang

presented

in

this

algorithm

for

positive

semidefinite

linear

complementarity

problems

as

a

generalization

of

Murty

's

-LSB-

-RSB-

Bard-type

scheme

He

was

studying

the

role

of

least

index

resolution

of

degeneracy

problems

in

Lemke

's

algorithm

-LSB-

-RSB-

and

in

variants

of

the

principal

pivoting

method

-LSB-

-RSB-

as

one

solves

linear

complementarity

problems

If

one

specializes

the

algorithm

on

page

of

his

long

report

to

the

linear

programming

case

precisely

the

finite

least-index

criss-cross

method

is

obtained

Unfortunately

this

result

remained

unknown

until

the

early

nineties

-LSB-

-RSB-

At

the

end

of

Terlaky

first

presented

the

least-index

criss-cross

algorithm

for

LP

-LSB-

-RSB-

and

then

immediately

generalized

to

the

oriented

matroid

case

-LSB-

-RSB-

Since

then

the

algorithm

was

known

by

the

name

finite

-LRB-

or

convergent

-RRB-

crisscross

method

This

name

was

given

due

to

the

motivation

from

Zionts

-LSB-

-RSB-

ideas

These

papers

inspired

a

good

deal

of

research

on

criss-cross

methods

in

the

last

decade

Wang

's

motivation

was

quite

different

He

was

working

on

oriented

matroids

and

tried

to

resolve

the

problems

arising

from

possible

cycling

of

Bland

's

least

index

simplex

algorithm

in

oriented

matroids

In

Wang

-LSB-

-RSB-

presented

his

version

for

oriented

matroid

programming

under

the

name

a

finite

conformal

elimination

free

algorithm

which

again

turned

out

to

be

the

same

finite

criss-cross

method

Bland

and

his

then

Ph.D.

student

Jensen

worked

on

the

recursive

line

Their

research

resulted

in

a

general

recursive

scheme

-LSB-

-RSB-

Jensen

's

generalized

relaxed

recursive

scheme

is

actually

a

general

model

for

finite

pivot

algorithms

and

it

is

easy

to

see

that

the

least-index

criss-cross

method

can

be

interpreted

as

a

concrete

realization

of

this

general

scheme

Matrix

Notation

First

we

present

some

basic

notations

and

definitions

for

linear

systems

For

finite

sets

L

and

E

an

L?E

matrix

is

an

array

of

doubly

indexed

numbers

or

variables

A

-LRB-

aij

i

L

j

E

-RRB-

where

each

member

of

L

is

called

a

row

index

each

member

of

E

is

called

a

column

index

and

each

aij

is

called

a

component

For

R

L

and

S

E

the

R

S

matrix

-LRB-

ars

r

R

s

S

-RRB-

is

called

a

submatrix

of

A

and

will

be

denoted

by

ARS

We

use

simplified

notations

like

AR

for

ARE

A?S

for

ALS

Ai

for

AfigE

and

A?j

for

ALfjg

Also

for

a

positive

integer

m

we

use

expressions

such

as

m?E

matrix

to

mean

an

L?E

matrix

with

the

usual

index

set

L

mg

Thus

our

matrix

notation

is

simply

an

extension

of

the

usual

matrix

notation

and

this

extension

enables

us

to

describe

pivot

algorithms

in

a

simplest

and

rigorous

way

Let

m

jLj

and

assume

rank

-LRB-

A

-RRB-

m.

Consider

the

homogeneous

system

of

linear

equalities

A

x

-LRB-

-RRB-

where

x

is

an

unknown

vector

in

RE

The

system

can

be

written

alternately

as

X

aijxj

L

-LRB-

-RRB-

A

subset

B

of

E

is

said

to

be

a

basis

if

jBj

m

and

the

square

submatrix

A?B

is

nonsingular

The

complement

E

nB

of

a

basis

B

is

called

a

nonbasis

and

denoted

by

N

When

B

is

a

basis

we

denote

by

-LRB-

A?B

-RRB-

the

B

m

matrix

U

such

that

the

product

UA?B

is

the

B

B

identity

matrix

I

-LRB-

B

-RRB-

i.e.

U

is

the

left

inverse

Since

we

consider

matrix

rows

and

columns

to

be

indexed

by

sets

for

two

matrices

to

be

multipliable

the

column

index

set

of

the

first

matrix

must

be

equal

to

the

row

index

set

of

the

second

matrix

Let

B

be

any

basis

The

dictionary

D

D

-LRB-

B

-RRB-

of

the

basis

B

is

the

B

N

matrix

UA?N

-LRB-

A?B

-RRB-

N

It

is

important

to

note

that

the

dictionary

represents

the

system

of

linear

equations

equivalent

to

-LRB-

-RRB-

given

by

xB

DxN

-LRB-

-RRB-

which

can

also

be

written

as

xi

X

dijxj

B

-LRB-

-RRB-

Here

the

dij

's

denote

the

components

of

D.

For

any

specific

nonbasic

element

q

N

the

unique

solution

x

to

the

system

-LRB-

-RRB-

with

xq

and

xj

for

all

j

N

nfqg

is

called

the

basic

solution

-LRB-

associated

with

B

and

q

-RRB-

and

denoted

by

x

-LRB-

B

q

-RRB-

Once

the

dictionary

is

given

the

basic

solution

can

easily

be

read

x

-LRB-

B

q

-RRB-

B

D?q

and

x

-LRB-

B

q

-RRB-

Nnfqg

For

those

who

are

not

familiar

with

the

notion

of

dictionary

-LRB-

first

introduced

in

-LSB-

-RSB-

and

elaborated

in

-LSB-

-RSB-

-RRB-

we

note

that

the

LP

tableau

matrix

-LRB-

see

e.g.

-LSB-

-RSB-

-RRB-

T

T

-LRB-

B

-RRB-

is

related

to

the

dictionary

D

D

-LRB-

B

-RRB-

by

T

-LSB-

I

-LRB-

B

-RRB-

D

-RSB-

For

any

p

B

and

q

N

with

dpq

the

set

B

p

q

is

again

a

basis

and

the

replacement

of

B

by

B

p

q

is

called

the

pivot

operation

on

-LRB-

p

q

-RRB-

Here

p

and

q

are

a

short

form

of

single-element

deletion

nfpg

and

addition

-LSB-

fqg

respectively

Such

linear

systems

arise

in

linear

programming

as

one

introduces

a

new

variable

to

homogenize

the

right-hand-side

The

formal

definition

will

be

given

in

Section

We

define

two

vector

subspaces

of

RE

associated

with

the

matrix

A

N

-LRB-

A

-RRB-

fx

RE

Ax

g

-LRB-

-RRB-

R

-LRB-

A

-RRB-

fs

RE

s

ATy

y

RLg

-LRB-

-RRB-

where

N

-LRB-

A

-RRB-

is

the

space

of

A

and

R

-LRB-

A

-RRB-

is

the

row

space

of

A

When

a

basis

B

for

A

x

is

given

these

two

spaces

are

given

symmetrically

as

x

N

-LRB-

A

-RRB-

-LRB-

-RRB-

xB

DxN

-LRB-

-RRB-

s

R

-LRB-

A

-RRB-

-LRB-

-RRB-

sN

DT

sB

-LRB-

-RRB-

This

orthogonality

is

illustrated

on

Figure

Consequently

any

dictionary

repre

x

s

sB

DT

sB

xN

DxN

Fig.

Orthogonality

of

x

N

-LRB-

A

-RRB-

and

s

R

-LRB-

A

-RRB-

sents

a

dual

pair

of

orthogonal

subspaces

in

a

very

compact

form

Often

we

have

to

prove

certain

properties

of

orthogonal

subspaces

and

this

duality

relations

-LRB-

-RRB-

and

-LRB-

-RRB-

give

us

a

way

to

avoid

proving

two

statements

separately

which

are

dual

to

each

other

Remark

-LRB-

Dualization

-RRB-

We

use

the

convenient

term

dualization

to

mean

exchanging

the

dictionary

D

with

DT

Via

dualization

one

can

induce

a

correct

statement

on

the

dual

space

R

-LRB-

A

-RRB-

by

dualyzing

and

interpreting

a

proven

statement

on

the

primal

space

N

-LRB-

A

-RRB-

Although

there

is

some

asymmetry

in

the

representation

of

the

two

spaces

N

-LRB-

A

-RRB-

and

R

-LRB-

A

-RRB-

we

concern

only

the

properties

of

these

spaces

as

vector

subspaces

and

thus

it

is

of

no

importance

Clearly

the

vector

x

-LRB-

B

q

-RRB-

defined

as

x

-LRB-

B

q

-RRB-

B

D?q

x

-LRB-

B

q

-RRB-

q

and

x

-LRB-

B

q

-RRB-

N?q

is

in

N

-LRB-

A

-RRB-

Given

a

basis

B

and

a

vector

sB

there

is

a

unique

vector

s

R

-LRB-

A

-RRB-

of

the

dual

system

satisfying

sN

AT?N

-LRB-

A

B

-RRB-

T

sB

DT

sB

and

it

is

associated

with

the

dual

basis

N

We

define

the

dual

basic

solution

for

p

B

and

denote

it

by

s

-LRB-

N

p

-RRB-

if

s

-LRB-

N

p

-RRB-

p

s

-LRB-

N

p

-RRB-

B?p

and

s

-LRB-

N

p

-RRB-

N

-LRB-

DT

-RRB-

p

-LRB-

D?p

-RRB-

T

i.e.

the

dual

basic

vector

is

represented

in

the

pth

row

of

D.

Due

to

the

orthogonality

of

the

row

and

spaces

the

above

defined

vectors

x

-LRB-

B

q

-RRB-

and

s

-LRB-

N

p

-RRB-

are

orthogonal

One

also

realizes

that

x

-LRB-

B

q

-RRB-

p

s

-LRB-

N

p

-RRB-

q

dpq

For

space

efficiency

here

and

in

the

subsequent

figures

we

will

draw

all

vectors

as

if

they

were

row

vectors

Termination

and

admissible

pivots

Terminal

dictionaries

Considering

the

notation

introduced

in

Section

for

the

primal-dual

LP

problems

-LRB-

P

-RRB-

and

-LRB-

D

-RRB-

we

define

the

matrix

A

B

cT

A

b

CA

-LRB-

-RRB-

representing

the

orthogonal

linear

subspaces

related

to

-LRB-

P

-RRB-

and

-LRB-

D

-RRB-

respectively

Clearly

the

assumption

rank

-LRB-

A

-RRB-

m

implies

rank

-LRB-

A

-RRB-

m

We

define

two

index

sets

E

and

E

as

ng

and

n

f

gg

respectively

where

f

is

associated

with

the

unit

vector

-LRB-

i.e.

closely

related

to

the

objective

vector

c

of

the

LP

problem

-RRB-

and

g

is

associated

with

the

last

column

vector

i.e.

to

the

right-hand-side

vector

b.

Both

f

and

g

are

considered

special

symbols

not

in

E

and

the

matrix

A

is

an

-LRB-

m

-RRB-

E

matrix

Now

it

is

easy

to

see

that

the

primal

problem

-LRB-

P

-RRB-

and

the

dual

problem

-LRB-

D

-RRB-

of

LP

-LRB-

A

b

c

-RRB-

are

equivalent

to

the

following

formulation

which

we

use

in

the

sequel

-LRB-

P

-RRB-

minf

xf

x

N

-LRB-

A

-RRB-

xg

and

xj

Eg

-LRB-

-RRB-

-LRB-

D

-RRB-

maxf

sg

s

R

-LRB-

A

-RRB-

sf

and

sj

Eg

-LRB-

-RRB-

In

this

formulation

the

primal

problem

and

the

dual

problem

are

completely

symmetrical

In

particular

this

suggests

that

any

LP

problem

can

be

written

of

form

-LRB-

V

f

g

-RRB-

where

V

is

a

given

vector

subspace

of

RE

Its

dual

is

-LRB-

V

g

f

-RRB-

which

is

obtained

by

taking

the

orthogonal

subspace

and

interchanging

the

objective

f

and

the

right-hand-side

g

This

form

represents

a

very

strong

relation

between

the

duality

of

linear

programming

and

the

orthogonal

duality

of

vector

subspaces

Comparing

with

the

original

formulation

the

variable

xf

is

set

to

the

negative

of

the

original

objective

function

cTx

while

the

dual

objective

variable

sg

represents

the

original

dual

objective

function

bTy

Thus

the

weak

duality

condition

Corollary

reads

Proposition

-LRB-

Symmetric

form

of

weak

duality

-RRB-

For

any

pair

-LRB-

x

s

-RRB-

of

primaldual

feasible

solutions

xf

sg

nX

j

xjsj

-LRB-

-RRB-

and

they

are

both

optimal

if

the

equality

holds

above

nX

j

xjsj

-LRB-

-RRB-

A

basis

B

with

f

B

g

is

said

to

be

a

basis

of

the

linear

programming

problem

or

an

LP

basis

for

short

The

notation

B

B

f

will

also

be

used

One

says

that

a

basis

is

feasible

if

x

-LRB-

B

g

-RRB-

B

D

-LRB-

B

-RRB-

Bfgg

and

dual

feasible

if

for

N

E

B

N

N?g

one

has

s

-LRB-

N

f

-RRB-

N

DffgN

One

sees

that

by

definition

the

vectors

x

-LRB-

B

g

-RRB-

and

s

-LRB-

N

f

-RRB-

are

feasible

solutions

for

the

problems

-LRB-

P

-RRB-

and

-LRB-

D

-RRB-

respectively

A

basis

is

said

to

be

optimal

if

it

is

both

primal

and

dual

feasible

A

basis

B

is

said

to

be

primal

inconsistent

if

there

exists

p

B

such

that

dpg

and

DpN

and

dual

inconsistent

if

there

exists

q

N

such

that

dfq

and

DBq

We

will

call

a

basis

-LRB-

a

dictionary

-RRB-

terminal

if

it

is

either

optimal

primal

or

dual

inconsistent

The

sign

structure

of

optimal

primal

and

dual

inconsistent

dictionaries

are

illustrated

in

Figure

Here

and

for

the

subsequent

figures

we

indicate

the

positive

nonnegative

negative

nonpositive

and

zero

components

by

respectively

q

Optimal

Primal

inconsistent

Dual

inconsistent

p

Fig.

Terminal

tableau

structures

Proposition

Let

an

LP

basis

B

be

given

-LRB-

i

-RRB-

If

B

is

optimal

then

the

associated

basic

solutions

are

primal

and

dual

optimal

solutions

-LRB-

ii

-RRB-

If

B

is

primal

inconsistent

then

the

primal

problem

is

infeasible

-LRB-

iii

-RRB-

If

B

is

dual

inconsistent

then

the

dual

problem

is

infeasible

PROOF

-LRB-

i

-RRB-

If

the

basis

is

optimal

then

the

vectors

x

-LRB-

B

g

-RRB-

and

s

-LRB-

N

f

-RRB-

are

feasible

Further

by

construction

x

-LRB-

B

g

-RRB-

has

zero

components

at

N

and

s

-LRB-

N

f

-RRB-

has

zero

components

at

B

thus

they

are

complementary

and

hence

due

to

Proposition

optimal

-LRB-

ii

-RRB-

Let

p

be

an

element

in

B

such

that

dpg

and

DfpgN

The

row

p

represents

an

equation

xp

dpg

DfpgNxN

where

dpg

and

DfpgN

If

x

is

a

feasible

solution

xN

and

this

forces

the

nonnegative

variable

xp

to

be

negative

Thus

the

primal

problem

is

infeasible

-LRB-

iii

-RRB-

By

dualization

this

case

is

reduced

to

the

previous

case

To

see

how

to

prove

this

directly

let

q

be

an

element

in

N

such

that

dfq

and

DBfqg

By

the

relation

-LRB-

-RRB-

the

column

q

represents

an

equation

for

the

dual

problem

sq

dfq

DTBfqgsB

If

s

is

dual

feasible

solution

sB

and

this

forces

the

nonnegative

variable

sq

to

be

negative

Thus

the

dual

problem

is

infeasible

Admissible

pivots

A

variable

index

i

E

is

said

to

be

primal

-LRB-

dual

-RRB-

feasible

at

an

LP

basis

if

the

associated

value

in

the

current

primal

-LRB-

dual

-RRB-

basic

solution

is

nonnegative

i.e.

either

i

N

or

i

B

and

dig

-LRB-

either

i

B

or

i

N

and

df

i

-RRB-

A

natural

and

simple

requirement

that

one

can

ask

from

a

pivot

selection

is

the

following

when

a

primal

-LRB-

dual

-RRB-

infeasible

variable

is

selected

for

a

pivot

then

after

the

pivot

both

pivot

variables

should

become

primal

-LRB-

dual

-RRB-

feasible

Such

pivots

locally

aim

to

improve

the

infeasibility

status

of

the

solution

and

are

described

here

For

p

B

f

and

q

N

g

with

dpq

a

pivot

on

-LRB-

p

q

-RRB-

is

said

to

be

admissible

if

either

-LRB-

I

-RRB-

dpg

and

dpq

or

-LRB-

II

-RRB-

dfq

and

dpq

See

Figure

The

p

q

f

q

p

type

I

f

g

g

type

II

Fig.

Two

types

of

admissible

pivots

for

LP

reader

can

easily

verify

the

following

Proposition

After

a

pivot

of

type

I

both

of

p

and

q

become

primal

feasible

After

a

pivot

of

type

II

both

of

p

and

q

become

dual

feasible

Finite

Criss-Cross

Methods

Algorithm

Now

we

are

ready

to

describe

the

least-index

criss-cross

method

perhaps

the

simplest

finite

criss-cross

method

This

criss-cross

method

is

a

purely

combinatorial

pivoting

method

for

solving

LP

problems

it

uses

admissible

pivots

and

traverses

through

different

-LRB-

possibly

both

primal

and

dual

infeasible

-RRB-

bases

until

the

associated

basic

solution

is

optimal

or

an

evidence

of

primal

or

dual

infeasibility

is

found

It

assumes

that

the

variables

are

linearly

ordered

Least-Index

Criss-Cross

Method

Initialization

let

B

be

an

initial

basis

with

f

B

g

while

true

do

begin

if

dig

i

B

and

dfj

j

B

then

-LRB-

I

-RRB-

stop

the

current

solution

solves

LP

-LRB-

A

b

c

-RRB-

else

p

minfi

B

dig

g

q

minfj

N

dfj

g

r

minfp

qg

if

r

p

then

if

dpN

then

-LRB-

II

-RRB-

stop

LP

-LRB-

A

b

c

-RRB-

has

no

primal

feasible

solution

else

let

q

minfj

N

dpj

g

perform

pivot

B

B

p

q

endif

endif

if

r

q

then

if

dBq

then

-LRB-

III

-RRB-

stop

LP

-LRB-

A

b

c

-RRB-

has

no

dual

feasible

solution

else

let

p

minfi

N

diq

g

perform

pivot

B

B

p

q

endif

endif

endif

end

To

ease

the

understanding

we

include

Figure

that

shows

the

scheme

of

the

leastindex

criss-cross

method

Observe

the

simplicity

of

the

algorithm

-LCB-

It

can

be

initiated

with

any

basis

-LCB-

No

two

phases

are

needed

-LCB-

No

ratio

test

is

used

to

preserve

feasibility

only

the

signs

of

components

in

a

dictionary

and

a

prefixed

ordering

of

variables

determine

the

pivot

selection

We

have

proved

in

Proposition

that

at

termination

-LRB-

I

-RRB-

the

vectors

x

-LRB-

B

g

-RRB-

and

s

-LRB-

N

f

-RRB-

are

optimal

solutions

for

the

primal

and

the

dual

problem

of

LP

-LRB-

A

b

c

-RRB-

respectively

Further

primal

and

dual

infeasibility

at

the

cases

-LRB-

II

-RRB-

and

-LRB-

III

-RRB-

follows

from

the

same

proposition

The

only

thing

remained

to

be

proved

is

that

the

algorithm

reaches

one

of

these

termination

stages

in

a

finite

number

of

steps

Before

doing

so

let

us

consider

an

example

to

demonstrate

some

appealing

features

of

the

least-index

criss-cross

method

Example

We

consider

the

following

primal

and

dual

problems

g

z

q

q

p

q

p

Stop

optimal

Stop

primal

infeasible

Stop

dual

infeasible

pivot

pivot

p

q

choose

first

q

choose

first

infeasible

positive

in

row

p

dictionary

D

f

choose

first

negative

in

column

q

q

oe

Fig.

Scheme

of

the

least-index

criss-cross

method

Primal

problem

min

x

Dual

Problem

max

In

this

example

we

take

the

ordering

of

the

variables

as

they

are

listed

Again

f

and

g

will

indicates

the

objective

and

the

right-hand-side

element

respectively

optimum

two

steps

five

steps

simplex

path

start

objective

criss-cross

path

Fig.

Criss-cross

and

dual

simplex

paths

on

the

dual

problem

of

Example

The

reader

can

easily

verify

that

the

basis

fg

is

dual

feasible

The

dual

simplex

method

requires

six

steps

to

walk

around

anti-clockwise

from

a

feasible

vertex

to

a

feasible

vertex

until

optimum

is

reached

Observe

that

although

the

initial

basis

is

primal

degenerate

the

dual

simplex

path

is

uniquely

determined

The

step

from

fg

to

fg

is

not

a

dual

simplex

step

When

the

least-index

criss-cross

method

is

applied

it

jumps

from

the

initial

basis

fg

to

the

both

primal

and

dual

infeasible

basis

fg

and

from

there

to

the

optimal

basis

fg

If

another

initial

basis

e.g.

the

both

primal

and

dual

infeasible

basis

fg

is

chosen

then

here

the

primal

infeasible

indices

are

and

is

the

only

dual

infeasible

index

thus

the

primal

infeasible

leaves

and

according

to

the

pivot

rule

enters

the

basis

as

it

is

easy

to

verify

Thus

the

optimal

basis

fg

is

reached

by

just

one

pivot

in

this

case

Playing

a

bit

further

with

this

example

one

could

introduce

an

arbitrary

number

of

constraints

in

the

dual

problem

-LRB-

variables

in

the

primal

-RRB-

that

are

rotations

of

constraint

No.

up

till

constraints

No.

is

reached

to

produce

a

new

dual

feasible

vertex

-LRB-

basis

-RRB-

by

each

new

constraints

on

the

way

from

the

given

initial

dual

feasible

solution

to

the

optimum

This

way

one

can

force

the

dual

simplex

method

to

visit

all

these

vertices

Thus

if

the

number

of

constraints

will

be

n

then

n

steps

will

be

needed

to

solve

this

problem

These

new

constraints

will

not

have

any

influence

to

the

pivot

sequence

of

the

least-index

criss-cross

method

if

these

constraints

are

put

behind

constraint

No.

in

the

ordered

list

Finally

let

us

choose

another

ordering

of

the

variables

e.g.

Then

starting

from

the

both

primal

and

dual

infeasible

basis

fg

we

jump

to

the

primal

feasible

dual

infeasible

basis

fg

and

from

there

the

optimal

basis

fg

is

reached

Of

course

we

made

up

this

example

so

that

the

least-index

criss-cross

method

behaves

nicely

One

can

make

examples

for

which

this

method

takes

a

long

-LRB-

exponential

-RRB-

pivot

sequence

to

an

optimal

solution

Nevertheless

one

can

prove

that

there

is

always

a

very

short

sequence

of

admissible

pivots

to

an

optimal

basis

for

any

LP

as

shown

in

Section

This

provide

us

with

a

strong

hope

that

there

is

a

polynomial

time

admissible

pivot

algorithm

for

LP

problems

Finiteness

Directly

or

indirectly

most

-LRB-

if

not

all

-RRB-

finiteness

proofs

of

the

least-index

criss-cross

method

rely

on

some

fundamental

proposition

on

almost

terminal

dictionaries

First

we

present

this

proposition

with

two

different

proofs

A

dictionary

is

called

almost

terminal

-LRB-

with

respect

to

a

constraint

index

k

E

-RRB-

if

it

is

not

terminal

but

by

discarding

a

single

row

or

column

-LRB-

indexed

by

k

-RRB-

it

becomes

terminal

Observe

that

we

have

four

structurally

different

almost

terminal

dictionaries

although

we

have

only

three

different

terminal

ones

due

to

the

simple

fact

that

an

almost

terminal

dictionary

might

be

either

primal

or

dual

infeasible

in

a

single

element

and

by

discarding

this

element

in

both

cases

an

optimal

dictionary

follows

Let

us

fix

an

element

say

k

then

the

four

types

of

almost

terminal

dictionaries

are

depicted

on

Figure

k

k

k

k

A

A

B

B

q

p

Fig.

Almost

terminal

dictionaries

Proposition

Consider

a

linear

programming

problem

LP

-LRB-

A

b

c

-RRB-

and

fix

an

element

k

E

Then

at

most

one

type

of

the

four

almost

terminal

dictionaries

A

B

A

and

B

with

respect

to

k

may

exist

PROOF

We

have

six

pairs

of

cases

to

consider

We

denote

by

-LRB-

A

A

-RRB-

to

mean

the

case

that

both

types

A

and

A

exist

simultaneously

etc.

In

this

proof

a

dictionary

and

the

associated

basis

for

case

A

or

B

will

be

denoted

by

D

B

N

and

for

case

A

or

B

by

N

The

case

-LRB-

A

A

-RRB-

is

the

most

involved

thus

we

begin

first

with

the

easier

cases

-LRB-

B

B

-RRB-

The

vectors

s

-LRB-

N

p

-RRB-

and

x

-LRB-

q

-RRB-

are

in

the

orthogonal

subspaces

R

-LRB-

A

-RRB-

and

N

-LRB-

A

-RRB-

respectively

However

if

one

calculates

their

dot

product

observes

that

the

product

is

positive

For

ease

of

understanding

the

sign

structure

of

these

vectors

is

illustrated

on

the

following

figure

k

f

g

s

-LRB-

N

p

-RRB-

x

-LRB-

q

-RRB-

Fig.

Case

-LRB-

B

B

-RRB-

In

both

vectors

all

the

components

but

k

are

nonnegative

with

the

k-component

strictly

negative

in

both

This

contradicts

to

the

orthogonality

of

the

and

row

spaces

and

shows

that

cases

-LRB-

B

B

-RRB-

can

not

exist

simultaneously

-LRB-

A

B

-RRB-

The

vectors

s

-LRB-

N

f

-RRB-

and

x

-LRB-

q

-RRB-

are

in

the

orthogonal

subspaces

R

-LRB-

A

-RRB-

and

N

-LRB-

A

-RRB-

respectively

However

if

one

calculates

their

dot

product

observes

that

the

product

is

positive

For

ease

of

understanding

the

sign

structure

of

these

vectors

is

is

illustrated

on

the

following

figure

k

f

g

s

-LRB-

N

f

-RRB-

x

-LRB-

q

-RRB-

Fig.

Case

-LRB-

A

B

-RRB-

In

both

vectors

all

but

the

g

and

k

components

are

nonnegative

The

g

component

is

zero

in

x

-LRB-

q

-RRB-

and

the

k-component

is

strictly

negative

in

both

thus

give

a

positive

product

Finally

the

f

coordinates

in

both

vectors

are

positive

We

have

obtained

that

the

dot

product

of

these

two

vectors

is

positive

contradicting

the

fact

that

they

should

be

orthogonal

Thus

the

types

A

and

B

can

not

exist

simultaneously

-LRB-

B

A

-RRB-

By

dualization

this

case

reduces

to

the

case

-LRB-

A

B

-RRB-

and

thus

this

case

can

not

occur

-LRB-

A

A

-RRB-

This

is

a

slightly

more

involved

case

The

vectors

s

-LRB-

N

f

-RRB-

and

s

-LRB-

N

f

-RRB-

are

in

the

row

space

R

-LRB-

A

-RRB-

and

the

vectors

x

-LRB-

B

g

-RRB-

and

x

-LRB-

g

-RRB-

are

in

the

space

N

-LRB-

A

-RRB-

For

ease

of

understanding

the

sign

structure

of

these

vectors

is

illustrated

on

the

following

figure

The

vectors

-LSB-

s

-LRB-

N

f

-RRB-

s

-LRB-

N

f

-RRB-

-RSB-

and

-LSB-

x

-LRB-

g

-RRB-

x

-LRB-

B

g

-RRB-

-RSB-

are

orthogonal

as

well

However

if

one

calculates

their

dot

product

observes

that

the

product

is

positive

s

-LRB-

N

f

-RRB-

k

g

x

-LRB-

B

g

-RRB-

s

-LRB-

N

f

-RRB-

f

x

-LRB-

g

-RRB-

Fig.

Case

-LRB-

A

A

-RRB-

because

-LRB-

-RRB-

the

f

component

is

zero

in

the

first

vector

-LRB-

-RRB-

the

g

component

is

zero

in

the

second

vector

-LRB-

-RRB-

the

k

component

is

negative

in

both

vectors

-LRB-

-RRB-

the

rest

of

the

components

give

a

nonnegative

product

This

lead

to

contradiction

to

the

fact

that

they

should

be

orthogonal

Thus

types

A

and

A

can

not

exist

simultaneously

-LRB-

A

B

-RRB-

This

case

is

similar

to

the

case

-LRB-

A

B

-RRB-

-LRB-

A

B

-RRB-

This

case

is

dual

to

-LRB-

A

B

-RRB-

and

also

similar

to

the

case

-LRB-

B

A

-RRB-

The

key

property

used

in

this

proof

is

clearly

the

orthogonality

of

the

two

spaces

R

-LRB-

A

-RRB-

and

N

-LRB-

A

-RRB-

Below

we

give

another

proof

of

Proposition

which

makes

use

of

the

notion

of

redundancy

of

constraints

appeared

first

in

-LSB-

-RSB-

We

need

one

definition

An

inequality

or

equality

constraint

in

an

LP

problem

is

said

to

be

redundant

if

the

restriction

is

implied

by

other

constraints

Thus

any

redundant

constraint

can

be

removed

from

the

problem

without

changing

the

region

of

feasibility

PROOF

-LRB-

Second

proof

of

Proposition

-RRB-

Let

us

first

observe

the

following

basic

facts

-LRB-

a

-RRB-

If

there

exists

a

basis

of

type

A

then

-LCB-

there

is

a

primal

feasible

solution

x

with

xk

and

-LCB-

the

constraint

sk

is

nonredundant

for

-LRB-

D

-RRB-

-LRB-

a

-RRB-

If

there

exists

a

basis

of

type

A

then

-LCB-

there

is

a

dual

feasible

solution

s

with

sk

and

-LCB-

the

constraint

xk

is

nonredundant

for

-LRB-

P

-RRB-

-LRB-

b

-RRB-

If

there

exists

a

basis

of

type

B

then

-LCB-

the

constraint

xk

is

redundant

for

-LRB-

P

-RRB-

and

-LCB-

no

primal

feasible

solution

x

with

xk

exists

-LRB-

b

-RRB-

If

there

exists

a

basis

of

type

B

then

-LCB-

the

constraint

sk

is

redundant

for

-LRB-

D

-RRB-

and

-LCB-

no

dual

feasible

solution

s

with

sk

exists

All

claims

in

-LRB-

a

-RRB-

and

-LRB-

a

-RRB-

are

straightforward

from

the

associated

basic

solutions

To

see

-LRB-

b

-RRB-

we

look

at

the

pth

equation

in

a

type

B

dictionary

xp

dpg

P

dpjxj

dpkxk

Since

dpg

dpk

and

dpj

for

all

j

N?k

the

nonnegativity

of

the

variables

in

this

equation

excluding

xk

forces

xk

to

be

strictly

positive

and

thus

-LRB-

b

-RRB-

By

dualization

-LRB-

or

direct

interpretation

-RRB-

-LRB-

b

-RRB-

follows

too

The

claims

above

immediately

eliminate

the

possibilities

of

simultaneous

existence

of

all

pairs

of

almost

terminal

dictionaries

except

for

the

case

-LRB-

A

A

-RRB-

as

easily

verified

with

the

following

table

Type

A

Type

A

primal

feasible

x

with

xk

dual

feasible

s

with

sk

sk

is

nonredundant

for

-LRB-

D

-RRB-

xk

is

nonredundant

for

-LRB-

P

-RRB-

Type

B

Type

B

xk

is

redundant

for

-LRB-

P

-RRB-

sk

is

redundant

for

-LRB-

D

-RRB-

primal

feasible

x

with

xk

dual

feasible

s

with

sk

To

prove

that

the

remaining

case

-LRB-

A

A

-RRB-

can

not

hold

assume

we

have

a

dictionary

D

of

type

A

and

a

dictionary

of

type

A

Let

x

x

-LRB-

B

g

-RRB-

and

x

-LRB-

g

-RRB-

By

the

sign

patterns

of

these

dictionaries

it

is

easy

to

see

-LRB-

-RRB-

both

x

and

are

optimal

for

the

LP

problem

LP

obtained

from

the

primal

LP

by

reversing

the

inequality

in

xk

-LRB-

i.e.

xk

-RRB-

Moreover

observing

the

equation

represented

by

the

fth

row

of

D

since

dfk

and

dfj

for

all

j

B

k

every

feasible

solution

to

LP

with

a

negative

kth

component

has

an

objective

value

strictly

less

than

the

optimal

value

dfg

Thus

no

optimal

solution

to

LP

can

have

a

negative

kth

component

Since

we

have

a

contradiction

Now

we

are

ready

to

prove

the

finiteness

of

the

least-index

criss-cross

method

Theorem

The

least-index

criss-cross

method

is

finite

PROOF

Since

the

number

of

bases

is

finite

it

is

enough

to

show

that

each

basis

might

be

visited

at

most

once

by

the

algorithm

The

proof

is

by

contradiction

Assume

that

a

basis

is

visited

twice

Then

the

algorithm

makes

a

cycle

i.e.

starting

from

this

basis

after

a

certain

number

of

steps

the

same

basis

is

reached

again

Without

loss

of

generality

we

may

also

assume

that

this

cycling

example

is

minimal

e.g.

all

the

variables

entered

or

left

the

basis

at

least

once

during

the

cycle

The

highest

indexed

variable

n

has

entered

and

left

the

basis

as

well

We

will

examine

the

situations

when

n

entered

and

when

n

left

the

basis

There

are

two

situations

when

n

as

a

nonbasic

element

might

enter

the

basis

Case

A

is

when

n

is

the

first

-LRB-

only

-RRB-

infeasible

in

our

list

and

Case

B

is

when

a

basis

variable

say

p

which

is

before

n

in

the

list

is

primal

infeasible

and

the

component

related

to

n

gives

the

only

admissible

pivot

in

row

p

The

two

situations

when

n

as

a

basis

element

might

leave

the

basis

are

Case

A

when

all

the

elements

preceding

n

in

the

list

are

feasible

but

n

is

infeasible

and

Case

B

when

a

nonbasis

variable

say

q

which

is

before

n

in

the

list

is

dual

infeasible

and

the

component

related

to

n

provides

the

first

-LRB-

only

-RRB-

admissible

pivot

in

column

q.

Clearly

the

described

cases

A

B

A

and

B

coincide

with

the

four

almost

terminal

tableaus

of

Figure

By

Proposition

of

the

four

possible

combinations

-LRB-

A

A

-RRB-

-LRB-

A

B

-RRB-

-LRB-

B

A

-RRB-

and

-LRB-

B

B

-RRB-

when

n

enters

and

leaves

the

basis

is

possible

thus

cycling

can

not

occur

One

of

the

most

important

consequences

of

the

finiteness

of

the

least-index

crisscross

method

is

the

strong

duality

theorem

of

linear

programming

We

believe

this

is

one

of

the

simplest

algorithmic

proofs

of

this

fundamental

result

Theorem

-LSB-

Strong

duality

-RSB-

Exactly

one

of

the

following

two

cases

occurs

for

any

linear

programming

problem

LP

-LRB-

A

b

c

-RRB-

-LCB-

At

least

one

of

the

primal

problem

-LRB-

P

-RRB-

and

the

dual

problem

-LRB-

D

-RRB-

is

infeasible

-LCB-

Both

problems

have

an

optimal

solution

and

the

optimal

objective

values

are

equal

PROOF

By

using

Gaussian

elimination

one

finds

either

the

inconsistency

of

Ax

b

-LRB-

thus

identifies

primal

infeasibility

-RRB-

or

an

LP

basis

in

finite

steps

possibly

discarding

some

redundant

equations

For

the

latter

case

apply

the

least-index

criss-cross

method

with

this

initial

LP

basis

By

Theorem

it

terminates

with

one

of

the

three

terminal

dictionaries

of

Proposition

-LRB-

see

also

Figure

-RRB-

hence

either

an

optimal

solution

pair

is

obtained

or

infeasibility

is

detected

either

from

the

primal

or

from

the

dual

side

The

reader

might

want

to

try

to

prove

the

strong

duality

theorem

by

using

Proposition

and

by

induction

on

n.

Although

it

is

recursive

in

nature

the

proof

is

even

simpler

than

the

proof

by

the

criss-cross

method

and

gives

us

an

idea

to

design

various

criss-cross

type

algorithms

In

fact

the

least-index

criss-cross

method

can

be

considered

as

a

simple

sequential

method

to

unravel

this

recursion

It

is

worthwhile

to

mention

another

simple

proof

of

the

strong

duality

theorem

due

to

Edmonds

-LSB-

-RSB-

It

involves

a

new

way

of

solving

an

LP

problem

by

considering

a

sequence

of

feasibility

problems

with

an

extra

inequality

expressing

the

objective

function

bounded

by

certain

constant

By

a

clever

update

of

the

constant

the

algorithm

finds

the

optimal

value

by

solving

finitely

many

feasibility

problems

He

also

proposes

a

combinatorial

pivot

algorithm

to

solve

feasibility

problems

whose

correctness

can

be

proved

by

a

simpler

version

of

Proposition

The

following

corollaries

are

based

on

different

interpretations

of

the

least-index

criss-cross

method

and

this

way

alternative

proofs

can

be

derived

The

first

corollary

is

based

on

a

recursive

interpretation

of

the

algorithm

This

recursive

interpretation

and

the

finiteness

proof

based

on

it

can

be

derived

from

the

results

due

to

Bland

and

Jensen

-LSB-

-RSB-

and

can

be

found

in

V?aliaho

's

paper

-LSB-

-RSB-

Corollary

-LRB-

Recursive

interpretation

-RRB-

As

performing

the

least-index

crisscross

method

at

each

pivot

one

can

make

a

note

of

the

larger

of

the

two

indices

r

maxfp

qg

that

entered

or

left

the

basis

In

this

list

an

index

must

be

followed

by

another

larger

one

before

the

same

index

occurs

anew

PROOF

At

a

given

basis

by

using

the

predefined

ordering

of

the

indices

in

the

dictionary

we

identify

first

the

largest

index

r

for

which

all

the

variables

i

r

are

both

primal

and

dual

feasible

If

r

n

a

terminal

dictionary

is

reached

the

problem

is

optimally

solved

Else

r

is

primal

or

dual

infeasible

Let

t

be

the

largest

index

for

which

drj

for

all

j

t

or

dir

for

all

i

t

respectively

If

t

n

then

a

terminal

dictionary

is

reached

primal

or

dual

infeasibility

is

detected

Else

let

k

maxfr

tg

Then

we

have

a

terminal

dictionary

for

the

subproblem

involving

the

variables

i

k

and

an

almost

terminal

dictionary

for

the

subproblem

involving

the

variables

i

k

The

new

pivot

involves

the

index

k

as

the

larger

one

of

the

two

pivot

indices

This

happens

at

one

of

the

four

situations

illustrated

at

Figure

if

we

look

at

the

subproblem

involving

only

the

indices

i

k

When

the

same

index

k

occurs

anew

in

the

list

then

an

index

higher

than

k

must

have

been

involved

in

a

pivot

want

else

at

the

new

occurrence

of

the

index

k

another

dictionary

from

Figure

for

the

subproblem

i

k

would

occur

what

is

not

possible

as

we

have

proved

by

Proposition

The

recursive

interpretation

is

becoming

apparent

when

one

notes

that

it

is

based

on

the

observation

that

the

size

of

the

solved

subproblem

-LRB-

the

subproblem

for

which

a

terminal

dictionary

is

obtained

-RRB-

is

monotonically

increasing

The

third

finiteness

proof

the

second

corollary

is

based

on

the

proof

technique

developed

by

Edmonds

and

Fukuda

-LSB-

-RSB-

and

adapted

by

Fukuda

and

Matsui

-LSB-

-RSB-

to

the

case

of

the

leastindex

criss-cross

method

Corollary

-LRB-

Lexicographically

increasing

list

-RRB-

Let

u

Rn

be

a

binary

vector

set

initially

to

be

the

zero

vector

In

applying

the

algorithm

let

r

maxfp

qg

be

the

larger

of

the

two

indices

p

that

entered

or

q

that

left

the

basis

At

each

pivot

update

u

as

follows

let

ur

and

ui

r

The

remaining

components

of

u

stay

unchanged

Then

at

each

step

of

the

least-index

criss-cross

method

the

vector

u

strictly

increases

lexicographically

thus

the

method

terminates

in

a

finite

number

of

steps

PROOF

By

Corollary

as

one

makes

a

pivot

at

the

index

r

then

a

pivot

with

a

larger

-LRB-

say

k

-RRB-

index

involved

must

follow

before

the

same

index

r

occur

anew

That

means

that

at

the

pivot

with

the

index

k

r

all

the

coordinates

ui

i

k

are

set

to

zero

thus

ur

becomes

zero

as

well

Therefore

it

can

not

happen

that

a

coordinate

already

equal

to

one

is

set

again

to

one

what

also

implies

that

the

vector

u

strictly

increases

lexicographically

Since

the

procedure

is

initiated

by

the

zero

vector

and

the

all

one

vector

is

lexicographically

maximal

among

the

binary

vectors

the

finite

termination

of

the

method

follows

In

all

three

interpretations

above

the

key

tool

in

proving

finiteness

is

that

at

most

one

of

the

four

exclusive

cases

A

B

A

and

B

exposed

by

Proposition

may

occur

The

finiteness

of

the

least-index

criss-cross

method

have

been

proved

in

many

different

ways

-LRB-

see

e.g.

-LSB-

-RSB-

-RRB-

One

of

the

simplest

proofs

is

based

on

the

fact

that

the

number

of

possible

bases

is

finite

and

by

showing

that

each

basis

might

be

visited

at

most

once

thus

the

finite

termination

of

the

algorithm

follows

Such

a

proof

was

given

by

Terlaky

-LSB-

-LCB-

-RSB-

and

by

Wang

-LSB-

-RSB-

and

was

also

exposed

by

Theorem

and

the

first

proof

of

Proposition

Another

simple

proof

without

using

the

duality

of

vector

subspaces

was

given

in

-LSB-

-RSB-

Flexibility

Once

an

ordering

of

the

index

set

is

chosen

the

pivot

sequence

of

the

least-index

criss-cross

method

is

uniquely

determined

This

way

the

least-index

criss-cross

method

seems

to

be

completely

rigid

Having

a

closer

look

at

the

finiteness

proofs

one

get

a

deeper

insight

which

properties

are

really

needed

to

guarantee

finiteness

The

hidden

flexibility

allowing

still

a

finiteness

proof

of

a

criss-cross

method

can

most

easily

be

seen

from

the

last

convergence

proof

There

one

easily

can

analyze

what

kind

of

reordering

of

the

index

list

is

allowed

while

the

strict

lexicographic

increase

of

the

indicator

vector

u

can

be

preserved

At

any

stage

of

the

algorithm

one

may

reshuffle

all

the

zero

indices

at

the

head

of

the

vector

u

preceding

the

first

ui

component

Similarly

all

the

zero

indices

at

the

tail

of

u

behind

the

last

ui

coordinate

can

be

reordered

Finally

all

the

zero

indices

of

u

between

two

subsequent

ui

coordinates

can

be

reshuffled

too

At

all

these

reorderings

only

the

zero

coordinates

are

reshuffled

and

the

relative

position

of

the

zero

components

comparing

to

the

components

equal

to

one

remained

unchanged

These

reorderings

do

not

change

the

vector

u

do

not

change

the

position

of

the

components

equal

to

one

in

u

as

long

as

they

keep

the

value

one

A

possible

effect

of

this

reshuffling

is

illustrated

by

the

next

example

Example

This

example

indicates

that

clever

possibly

subsequent

reorderings

of

the

variables

might

significantly

speed

up

the

solution

process

Consider

the

problem

presented

in

Example

Let

us

initiate

the

criss-cross

method

by

the

basis

fg

basis

indicator

u

in

the

order

fg

fg

fg

fg

fg

fg

fg

fg

Table

A

criss-cross

path

basis

indicator

u

in

the

order

fg

fg

reorder

fg

fg

Table

Allowed

reordering

of

the

variables

The

generated

basis

sequence

and

the

associated

sequence

of

the

indicator

vectors

u

is

presented

in

Table

This

way

seven

pivot

steps

were

needed

to

solve

the

problem

Now

let

us

play

a

bit

with

the

flexibility

explained

above

Table

indicates

a

possible

sequence

of

pivots

with

an

allowed

reordering

of

the

variables

after

the

first

pivot

It

should

be

noted

that

one

can

exploit

the

variable

reordering

even

further

The

idea

is

to

forget

some

pivot

history

coded

in

the

u

vector

if

one

can

gain

more

An

extreme

case

is

when

it

is

better

to

reorder

all

the

variables

by

forgetting

all

the

history

and

starting

all

over

from

scratch

Such

an

idea

was

presented

in

-LSB-

-RSB-

Both

the

recursive

interpretation

and

the

flexibility

of

pivot

selection

in

the

leastindex

criss-cross

method

make

it

possible

to

develop

other

finite

variants

Such

finite

criss-cross

methods

which

do

not

rely

on

a

fixed

minimal

index

ordering

can

be

developed

based

on

the

finite

simplex

rules

presented

by

Zhang

-LSB-

-RSB-

Using

either

of

the

above

presented

two

Corollaries

or

the

reader

easily

verify

the

finiteness

of

the

following

two

criss-cross

rules

-LSB-

-RSB-

-LCB-

First

in

last

out

-LRB-

FILO

-RRB-

-LCB-

First

choose

a

primal

or

dual

infeasible

variable

that

has

changed

its

basis/nonbasis

status

most

recently

Then

choose

a

variable

in

the

selected

row

or

column

so

that

the

pivot

entry

fulfills

the

sign

requirement

of

the

admissible

pivot

selection

and

which

has

changed

its

basis/nonbasis

status

most

recently

When

more

than

one

candidates

occur

with

the

same

pivot

age

then

one

break

tie

as

you

like

-LRB-

e.g.

randomly

-RRB-

This

rule

can

easily

be

realized

by

assigning

an

age

vector

u

to

the

vector

of

the

variables

and

using

a

pivot

counter

k.

Initially

we

set

k

and

u

We

increase

k

by

one

at

each

pivot

and

we

set

the

pivot

coordinates

of

u

equal

to

k

Then

the

pivot

selections

are

made

by

choosing

the

variable

with

the

highest

possible

ui

value

satisfying

the

sign

requirements

-LCB-

Most

often

selected

variable

-LCB-

First

choose

a

primal

or

dual

infeasible

variable

that

has

changed

its

basis/nonbasis

status

most

frequently

Then

choose

a

variable

in

the

selected

row

or

column

so

that

the

pivot

entry

fulfills

the

sign

requirement

of

the

admissible

pivot

selection

and

which

has

changed

its

basis/nonbasis

status

most

frequently

When

more

than

one

candidates

occur

with

the

same

pivot

age

then

one

break

tie

as

you

like

-LRB-

e.g.

randomly

-RRB-

The

most

often

selected

rule

can

easily

be

realized

by

assigning

another

age

vector

u

to

the

vector

of

the

variables

Initially

we

set

u

At

each

pivot

we

increase

the

pivot-variable

components

of

u

by

one

Then

the

pivot

selections

are

made

by

choosing

the

variable

with

the

highest

possible

ui

value

satisfying

the

sign

requirement

Exponential

and

average

behavior

The

worst

case

exponential

behavior

of

the

least-index

criss-cross

method

was

studied

by

Roos

-LSB-

-RSB-

Roos

exponential

example

is

a

variant

of

the

Klee-Minty

-LSB-

-RSB-

cube

In

this

example

the

starting

solution

is

the

origin

defined

by

a

feasible

basis

the

variables

are

ordered

so

that

the

finite

criss-cross

method

follows

a

simplex

path

i.e.

without

making

any

ratio

test

feasibility

of

the

starting

basis

is

preserved

Another

exponential

example

was

presented

by

Fukuda

and

Namiki

-LSB-

-RSB-

for

linear

complementarity

problems

-LRB-

criss-cross

methods

for

linear

complementarity

problems

are

discussed

in

Section

-RRB-

Contrary

to

the

clear

result

on

the

worst

case

behavior

to

date

not

much

is

known

about

the

expected

or

average

number

of

pivot

steps

required

by

criss-cross

methods

to

solve

LP

problems

The

only

notable

result

was

obtained

by

Fukuda

and

Namiki

-LSB-

-RSB-

They

averaged

the

length

of

all

possible

pivot

sequences

on

an

exponential

example

for

linear

complementarity

problems

by

considering

all

possible

orderings

of

the

variables

They

proved

that

on

the

problem

where

the

longest

criss-cross

path

can

be

exponential

the

average

length

of

all

possible

criss-cross

paths

is

linear

in

the

dimension

of

the

problem

space

Best-case

Analysis

of

Admissible

Pivot

Methods

As

it

is

the

case

for

many

simplex

algorithms

the

least-index

criss-cross

method

discussed

in

Section

is

not

a

polynomial-time

algorithm

see

-LSB-

-RSB-

A

question

we

would

like

to

ask

is

as

to

whether

there

exists

a

polynomial

criss-cross

method

Unfortunately

we

have

no

reasonable

ways

to

answer

this

question

at

this

moment

However

a

recent

work

of

Fukuda-L

uthi-Namiki

-LSB-

-RSB-

shows

the

existence

of

a

short

sequence

of

admissible

pivots

-LRB-

introduced

in

Section

-RRB-

of

length

at

most

m

to

an

optimal

basis

from

any

given

basis

In

this

section

we

shall

present

this

result

-LRB-

in

a

slightly

weaker

form

-RRB-

which

indicates

a

potential

direction

for

us

to

look

for

polynomial

time

pivot

algorithms

It

is

noteworthy

that

we

do

not

know

of

any

such

result

for

feasibility

preserving

algorithms

and

in

fact

the

maximum

length

of

shortest

feasibility-preserving

pivot

sequences

between

two

feasible

bases

is

not

known

to

be

bounded

by

a

polynomial

in

the

size

of

the

associated

LP

Short

pivot

sequences

Throughout

we

assume

that

an

LP

basis

is

given

Finding

one

LP

basis

-LRB-

or

proving

that

exists

-RRB-

can

easily

be

done

by

Gaussian

elimination

The

goal

of

a

pivot

algorithm

is

to

find

a

terminal

basis

by

a

sequence

of

pivot

operations

starting

with

a

given

LP

basis

Recall

that

for

p

B

f

and

q

N

g

with

dpq

a

pivot

on

-LRB-

p

q

-RRB-

is

said

to

be

admissible

if

either

-LRB-

I

-RRB-

dpg

and

dpq

or

-LRB-

II

-RRB-

dfq

and

dpq

-LRB-

see

Figure

-RRB-

Admissible

pivots

are

very

natural

elementary

operations

to

be

used

in

any

pivot

algorithms

by

the

following

reasons

Proposition

Let

B

be

any

LP

basis

Then

the

following

statements

hold

-LRB-

a

-RRB-

If

B

is

not

terminal

then

there

exists

an

admissible

pivot

at

B.

-LRB-

b

-RRB-

If

the

LP

has

an

optimal

solution

then

there

exists

an

admissible

pivot

at

B

if

and

only

if

B

is

not

optimal

PROOF

The

statement

-LRB-

a

-RRB-

is

clear

with

the

definitions

To

prove

-LRB-

b

-RRB-

assume

that

the

LP

has

an

optimal

solution

This

together

with

the

LP

duality

theorem

Theorem

implies

that

neither

a

primal

inconsistent

nor

a

dual

inconsistent

basis

exists

Thus

if

a

basis

B

is

not

optimal

there

exists

an

admissible

pivot

The

other

implication

is

trivial

We

define

an

admissible

pivot

method

for

LP

as

a

pivot

method

that

only

uses

admissible

pivots

It

is

not

difficult

to

verify

the

following

Remark

The

primal

simplex

method

the

dual

simplex

method

Bland

's

recursive

pivot

method

-LSB-

-RSB-

Edmonds-Fukuda

method

-LSB-

-RSB-

and

the

least-index

criss-cross

method

-LSB-

-RSB-

are

all

admissible

pivot

methods

More

precisely

the

primal

simplex

method

and

Edmonds-Fukuda

method

use

admissible

pivots

of

type

II

only

the

dual

simplex

method

uses

admissible

pivots

of

type

I

only

and

both

Bland

's

recursive

method

and

the

criss-cross

methods

discussed

in

Section

use

admissible

pivots

of

both

type

I

and

type

II

Now

a

simple

fundamental

question

arises

Question

What

is

the

length

of

the

shortest

sequence

of

admissible

pivots

from

a

given

LP

basis

B

to

any

fixed

optimal

basis

B

It

turns

out

that

to

this

question

one

has

a

rather

disappointing

answer

In

some

cases

no

such

sequence

exists

and

thus

the

shortest

length

is

undefined

as

shown

in

the

following

example

Example

Consider

the

LP

illustrated

in

Figure

In

this

example

E

f

gg

and

the

set

of

feasible

solutions

is

the

shaded

triangle

region

There

are

two

optimal

bases

B

fg

and

B

fg

At

the

dictionary

for

the

basis

B

fg

there

is

only

one

admissible

pivot

that

is

at

-LRB-

-RRB-

Clearly

by

this

pivot

one

reaches

the

basis

B

Thus

the

basis

B

is

not

reachable

from

B

by

any

admissible

pivots

f

max

B

B

B

f

g

f

g

B

B

Fig.

B

unreachable

from

B

by

admissible

pivots

The

question

becomes

interesting

once

we

assume

nondegeneracy

We

say

that

an

LP

basis

B

is

nondegenerate

if

dig

for

all

basic

variable

i

B?f

An

LP

basis

B

is

said

to

be

dual-nondegenerate

if

the

associated

nonbasis

is

nondegenerate

for

the

dual

problem

or

equivalently

dfj

for

all

nonbasic

variable

j

N?g

Finally

an

LP

basis

is

fully

nondegenerate

if

it

is

both

nondegenerate

and

dual-nondegenerate

One

can

easily

prove

the

following

Lemma

If

LP

has

a

fully

nondegenerate

optimal

basis

then

it

is

the

unique

optimal

basis

Our

main

theorem

in

this

section

is

the

following

which

is

a

consequence

of

the

uniqueness

lemma

above

Theorem

If

an

LP

has

a

fully

nondegenerate

optimal

basis

B

then

there

exists

a

sequence

of

jB?nBj

admissible

pivots

from

any

LP

basis

B.

PROOF

Let

B

be

a

fully

nondegenerate

optimal

basis

for

a

given

LP

and

let

B

be

any

LP

basis

different

from

B

Set

BnB

B

B

NnN

B?nB

and

N

N

Consider

the

submatrix

of

the

dictionary

D

D

-LRB-

B

-RRB-

The

matrix

represents

a

smaller

linear

optimization

problem

with

variables

-LSB-

see

Figure

D

g

f

Fig.

A

Partition

of

the

dictionary

D

D

-LRB-

B

-RRB-

It

is

easy

to

see

that

has

a

fully

nondegenerate

optimal

basis

namely

and

by

Lemma

it

is

a

unique

optimal

basis

This

implies

that

the

basis

can

not

be

terminal

This

together

with

Proposition

-LRB-

a

-RRB-

implies

that

there

exists

an

admissible

pivot

in

This

transformation

makes

the

resulting

basis

closer

to

the

feasible

basis

B

by

one

Hence

by

applying

the

same

argument

we

can

arrive

at

B

by

a

sequence

of

admissible

transformations

involving

exactly

jB?nBj

pivots

One

might

suspect

that

this

theorem

is

a

disguised

form

of

a

simple

well-known

lemma

in

linear

algebra

any

two

bases

B

and

of

a

matrix

A

can

be

connected

by

a

sequence

of

pivot

operations

As

it

was

illustrated

by

Example

the

validity

of

our

theorem

critically

depends

on

the

full

nondegeneracy

assumption

which

has

no

counterpart

in

this

lemma

Therefore

we

do

not

see

any

easy

way

to

relate

these

two

statements

with

similar

appearances

In

order

to

apply

the

theorem

to

general

degenerate

LP

's

we

can

use

a

standard

technique

of

lexicographic

-LRB-

symbolic

-RRB-

perturbation

which

makes

all

LP

basis

fully

nondegenerate

see

-LSB-

-RSB-

for

details

Extensions

Finite

criss-cross

methods

can

be

generalized

to

solve

larger

classes

of

optimization

problems

These

problems

include

convex

quadratic

optimization

-LSB-

-RSB-

linear

complementarity

problems

-LRB-

LCP

-RRB-

with

sufficient

matrices

-LSB-

-RSB-

and

fractional

linear

optimization

-LSB-

-RSB-

In

this

section

we

concentrate

our

discussion

on

the

LCP

case

In

Section

we

shall

present

an

LCP

duality

theorem

due

to

-LSB-

-RSB-

and

a

finite

least-index

criss-cross

method

for

sufficient

linear

complementarity

problems

The

duality

theorem

is

a

direct

consequence

of

the

finiteness

of

the

criss-cross

method

It

can

be

considered

as

an

extension

of

the

LP

strong

duality

theorem

Theorem

Furthermore

the

LCP

criss-cross

method

is

a

natural

extension

of

the

LP

criss-cross

method

presented

in

the

previous

section

In

order

to

explain

the

solvability

of

convex

quadratic

optimization

problems

by

the

LCP

criss-cross

method

it

is

important

to

note

that

every

positive

semidefinite

matrix

is

sufficient

Since

any

convex

quadratic

programming

-LRB-

QP

-RRB-

problem

can

be

reduced

to

LCPs

with

positive

semidefinite

matrices

the

-LRB-

least-index

-RRB-

crisscross

method

for

sufficient

LCPs

can

be

applied

to

convex

quadratic

optimization

problems

For

those

unfamiliar

with

this

reduction

we

shall

present

it

in

Section

Sufficient

Linear

Complementarity

Problems

Linear

complementarity

problems

are

defined

as

LCP

-LRB-

M

b

-RRB-

Mx

s

b

x

s

xisi

n

where

the

given

n

n

matrix

M

is

sufficient

b

Rn

is

given

and

x

s

Rn

The

least-index

criss

cross

method

solves

LCPs

with

sufficient

matrices

by

traversing

infeasible

complementary

bases

until

the

associated

basic

solution

is

nonnegative

or

an

evidence

of

non-existence

of

a

solution

is

obtained

We

begin

with

some

definitions

An

n

n

matrix

M

is

called

column

sufficient

if

x

Rn

such

that

xi

-LRB-

Mx

-RRB-

i

for

all

i

ng

xj

-LRB-

Mx

-RRB-

j

for

some

j

ng

-LRB-

-RRB-

A

matrix

M

is

called

row

sufficient

if

its

transpose

is

column

sufficient

A

matrix

M

is

called

sufficient

if

it

is

both

row

and

column

sufficient

The

notion

of

the

sufficiency

was

first

introduced

by

Cottle

Pang

and

Venkateswaran

-LSB-

-RSB-

The

class

of

sufficient

matrices

is

identical

to

the

class

of

P

matrices

introduced

by

Kojima

e.al

-LSB-

-RSB-

This

result

was

proved

by

V?aliaho

-LSB-

-RSB-

For

a

given

n

n

matrix

M

and

an

n-vector

b

we

consider

the

linear

system

V

-LRB-

M

b

-RRB-

fx

RE

-LSB-

I

M

b

-RSB-

x

g

where

E

n

n

gg

is

the

index

set

of

the

columns

of

-LSB-

I

A

b

-RSB-

For

each

i

E

g

we

define

the

complement

of

i

denoted

by

i

is

i

n

i

if

i

n

and

i

i?n

if

n

i

A

subset

S

of

E

g

is

called

complementary

if

fi

ig

S

for

all

i

E

g

i.e.

S

S

Also

a

vector

x

RE

is

called

complementary

if

xi

xi

for

all

i

E

g

A

vector

x

in

RE

is

called

strictly

sign

reversing

-LRB-

s.s.r.

-RRB-

if

xi

xi

for

all

i

E?g

and

xj

xj

for

some

j

E?g

A

vector

x

in

RE

is

called

strictly

sign

preserving

-LRB-

s.s.p.

-RRB-

if

xi

xi

for

all

i

E

g

and

xj

xj

for

some

j

E

g.

Observe

that

a

matrix

M

is

column

sufficient

if

and

only

if

V

-LRB-

M

b

-RRB-

contains

no

s.s.r.

vector

x

with

xg

-LRB-

-RRB-

and

row

sufficient

if

and

only

if

V

-LRB-

M

b

-RRB-

contains

no

s.s.p.

vector

-LRB-

-RRB-

where

V

-LRB-

M

b

-RRB-

is

the

orthogonal

complement

of

V

-LRB-

M

b

-RRB-

These

conditions

on

vector

subspaces

allow

us

to

make

statements

on

matrices

in

terms

of

sign

-LRB-

i.e.

combinatorial

-RRB-

statements

on

the

associated

vector

subspaces

We

shall

see

these

sign

conditions

in

an

abstract

setting

of

oriented

matroids

in

Section

where

we

present

the

essence

of

combinatorics

behind

LP

QP

and

LCP

Now

we

are

ready

to

rewrite

the

LCP

as

follows

We

call

this

form

the

primal

LCP

PLCP

-LRB-

M

b

-RRB-

find

x

RE

satisfying

-LRB-

-RRB-

x

V

-LRB-

M

b

-RRB-

-LRB-

-RRB-

xi

xi

for

all

i

E

g

-LRB-

-RRB-

xg

and

xi

for

all

i

E

g

-LRB-

-RRB-

The

dual

problem

of

PLCP

-LRB-

M

b

-RRB-

is

defined

as

DLCP

-LRB-

M

b

-RRB-

find

y

RE

satisfying

-LRB-

-RRB-

y

V

-LRB-

M

b

-RRB-

-LRB-

-RRB-

yi

yi

for

all

i

E

g

-LRB-

-RRB-

yg

and

yi

for

all

i

E

g

-LRB-

-RRB-

We

now

state

a

slightly

weaker

form

of

the

LCP

duality

theorem

by

Fukuda

and

Terlaky

-LSB-

-RSB-

Theorem

-LRB-

-LSB-

-RSB-

-RRB-

For

any

sufficient

n?n

matrix

M

and

n-vector

b

exactly

one

of

the

following

statements

holds

-LRB-

-RRB-

There

exists

a

solution

x

to

PLCP

-LRB-

M

b

-RRB-

-LRB-

-RRB-

There

exists

a

solution

y

to

DLCP

-LRB-

M

b

-RRB-

This

theorem

implies

that

if

M

is

sufficient

and

rational

the

non-existence

of

a

solution

to

PLCP

-LRB-

M

b

-RRB-

is

well

characterized

That

is

there

is

a

polynomially

bounded

evidence

of

the

non-existence

of

a

solution

because

there

exists

a

basic

solution

corresponding

to

the

evidence

And

thus

the

decision

problem

of

LCP

with

sufficient

matrices

is

in

both

NP

and

co-N

P

although

the

recognition

of

sufficiency

seems

to

be

a

difficult

problem

We

shall

discuss

how

to

solve

any

sufficient

LCPs

without

checking

the

sufficiency

explicitly

in

Section

In

this

section

we

will

describe

a

finite

method

for

the

LCP

with

sufficient

matrices

This

can

be

seen

as

an

algorithm

to

find

a

solution

to

exactly

one

of

-LRB-

-RRB-

and

-LRB-

-RRB-

of

the

LCP

duality

theorem

The

algorithm

which

we

call

the

least-index

criss-cross

method

is

a

natural

extension

of

the

least-index

criss-cross

method

of

Section

for

linear

programming

The

finiteness

of

this

algorithm

immediately

implies

the

LCP

duality

theorem

Theorem

Let

us

fix

some

notations

We

assume

that

the

n

complementary

pairs

of

indices

are

linearly

ordered

This

order

induces

a

linear

order

on

any

complementary

subset

of

indices

For

example

if

B

then

the

order

is

OE

OE

OE

min

B

and

maxB

An

LCP

basis

is

a

basis

B

of

the

homogeneous

system

-LSB-

I

M

b

-RSB-

x

which

does

not

contain

g

and

complementary

i.e.

contains

exactly

one

of

i

and

i

for

i

n

The

problem

PLCP

-LRB-

M

b

-RRB-

has

a

trivial

LCP

basis

ng

An

LCP

basis

B

is

said

to

be

feasible

if

dig

for

all

i

B

and

inconsistent

if

there

exists

r

B

such

that

drg

and

drj

for

all

j

N

g

Here

D

fdij

i

B

j

Ng

denotes

the

B

times

N

E

n

B

dictionary

Clearly

if

B

is

feasible

the

associated

basic

solution

x

-LRB-

B

g

-RRB-

solves

the

PLCP

-LRB-

M

b

-RRB-

If

B

is

inconsistent

because

of

its

rth

row

the

dual

basic

solution

s

-LRB-

N

r

-RRB-

is

nonnegative

and

the

scaled

vector

s

-LRB-

N

r

-RRB-

drg

solves

DLCP

-LRB-

M

b

-RRB-

The

least-index

criss-cross

method

starts

with

the

initial

basis

ng

and

performs

pivot

operations

until

it

finds

either

a

feasible

or

an

inconsistent

basis

Least-Index

Criss-Cross

Method

for

LCP

Initialization

let

B

ng

be

an

initial

complementary

basis

while

true

do

begin

if

dig

for

all

i

B

then

-LRB-

I

-RRB-

stop

the

current

basic

solution

solves

PLCP

-LRB-

M

b

-RRB-

else

r

minfi

B

dig

g

if

drj

for

all

j

B

then

-LRB-

II

-RRB-

stop

DLCP

-LRB-

M

b

-RRB-

has

a

feasible

solution

s

-LRB-

N

r

-RRB-

drg

else

t

minfj

B

drj

g

p

maxfr

tg

q

minfr

tg

if

dpp

then

perform

diagonal

pivot

B

B

p

fpg

else

perform

exchange

pivots

B

B

fp

qg

fp

qg

endif

endif

endif

end

Figure

shows

the

flow

of

the

least-index

criss-cross

method

The

sufficiency

property

of

the

matrix

M

imply

that

if

dpp

then

it

is

positive

and

if

dpp

then

dpqdqp

Finiteness

and

flexibility

Recall

that

Proposition

played

the

most

important

role

in

proving

the

finiteness

of

the

criss-cross

method

for

LP

problems

The

critical

message

there

was

that

there

exists

at

most

one

almost

terminal

dictionary

when

a

variable

index

k

is

fixed

For

the

proof

of

finiteness

of

the

LCP

criss-cross

method

one

can

use

a

straightforward

generalization

of

Proposition

We

refer

to

-LSB-

-RSB-

for

a

proof

of

this

proposition

Proposition

Consider

a

sufficient

LCP

-LRB-

M

b

-RRB-

and

fix

an

element

k

E

Then

at

most

one

of

the

four

almost

terminal

dictionaries

A

B

A

and

B

defined

in

Figure

with

respect

to

k

may

exist

Just

as

in

Proposition

the

finiteness

of

the

LCP

criss-cross

method

can

be

explained

by

monotonic

improvements

of

some

binary

vector

associated

with

the

algorithm

Thus

the

same

flexibility

of

the

algorithm

without

sacrificing

finiteness

can

be

claimed

The

reader

is

referred

to

-LSB-

-RSB-

for

details

EP-Theorems

The

LCP

duality

theorem

Theorem

is

an

existence

theorem

in

which

each

object

claimed

to

exist

has

a

short

evidence

for

its

existence

In

fact

each

object

itself

can

be

chosen

in

such

a

way

that

its

size

is

polynomially

bounded

in

the

size

complementary

dictionary

r

top

most

r

stop

t

t

left

most

r

stop

p

maxfr

tg

q

minfr

tg

q

p

q

p

q

q

ffl

p

ffl

p

ffl

diagonal

pivot

exchange

pivot

complementary

dictionary

Fig.

Illustration

of

the

least-index

criss-cross

method

for

LCP

of

the

input

However

this

problem

is

not

of

the

most

preferable

type

in

the

sense

that

the

assumption

of

the

input

matrix

being

sufficient

does

not

seem

to

admit

a

polynomial-time

verification

Consequently

the

direct

proof

of

the

theorem

and

as

such

the

criss-cross

method

has

a

considerable

disadvantage

that

one

must

verify

the

sufficiency

of

the

input

matrix

M

before

running

the

algorithm

k

type

-LRB-

A

-RRB-

k

r

r

k

type

-LRB-

B

-RRB-

k

type

-LRB-

A

-RRB-

k

t

t

k

type

-LRB-

B

-RRB-

Fig.

Four

types

of

dictionaries

-LCB-

Only

one

type

can

exist

The

notion

of

EP

-LRB-

existentially

polytime

-RRB-

theorem

due

to

-LSB-

-RSB-

encourages

us

to

write

every

existence

theorem

in

a

particular

form

i.e.

EP

form

Its

essence

is

to

get

rid

of

nontrivial

assumptions

while

all

outcomes

of

the

theorem

can

be

verified

by

a

short

evidence

In

this

section

following

the

paper

-LSB-

-RSB-

we

present

how

the

LCP

duality

theorem

can

be

written

in

EP

form

This

rewriting

appeared

to

be

critical

in

modifying

the

criss-cross

method

so

that

it

can

take

any

LCP

input

and

terminate

in

either

finding

a

primal

solution

a

dual

solution

or

a

short

evidence

that

the

input

matrix

is

not

sufficient

LCP

Duality

Theorem

in

EP

form

An

EP

-LRB-

existentially

polytime

-RRB-

theorem

-LSB-

-RSB-

is

a

theorem

of

the

form

-LSB-

-LRB-

x

-RRB-

-LRB-

x

-RRB-

Fk

-LRB-

x

-RRB-

-RSB-

where

Fi

-LRB-

x

-RRB-

is

a

predicate

formula

which

has

the

form

Fi

-LRB-

x

-RRB-

-LSB-

such

that

jjyijj

jjxjjni

and

fi

-LRB-

x

yi

-RRB-

-RSB-

Here

ni

Z

jjzjj

denotes

the

encoding

length

of

z

and

fi

-LRB-

x

yi

-RRB-

is

a

predicate

for

which

there

is

a

polytime

algorithm

The

most

important

feature

of

EP

theorem

is

that

it

naturally

motivates

us

to

look

for

a

polynomial

algorithm

which

produces

or

or

or

yk

Such

an

algorithm

is

called

a

constructing

polytime

algorithm

abbreviated

by

CPA

The

automatic

conjecture

that

there

exists

an

CPA

for

an

instance

of

EP

theorem

arises

as

follows

A

special

EP

theorem

in

which

k

and

or

is

exclusive

is

called

a

good

characterization

of

EP

form

Thus

an

EP

theorem

with

a

good

characterization

might

admit

a

CPA

as

if

a

decision

problem

with

a

good

characterization

might

admit

a

polynomial

algorithm

-LRB-

this

is

well-known

conjecture

that

NP

co-N

P

P

-RRB-

On

the

other

hand

EP

theorem

-LSB-

-LRB-

x

-RRB-

or

-LRB-

x

-RRB-

or

or

Fk

-LRB-

x

-RRB-

-RSB-

can

be

translated

to

a

good

characterization

of

EP

form

by

setting

F

-LRB-

x

-RRB-

-LRB-

x

-RRB-

or

-LRB-

x

-RRB-

or

or

Fk

-LRB-

x

-RRB-

and

F

-LRB-

x

-RRB-

false

for

any

x.

Thus

the

automatic

conjecture

that

there

is

a

CPA

for

an

instance

of

EP

theorem

arises

It

is

also

noticed

in

-LSB-

-RSB-

that

this

conjecture

may

be

stronger

than

the

conjecture

that

NP

co-NP

P

In

order

to

write

the

LCP

duality

theorem

in

EP

form

first

we

rewrite

Theorem

as

follows

Theorem

Let

M

be

any

n

n

matrix

and

b

be

any

n-vector

Then

at

least

one

of

the

following

three

statements

holds

-LRB-

-RRB-

There

exists

a

solution

x

to

PLCP

-LRB-

M

b

-RRB-

-LRB-

-RRB-

There

exists

a

solution

y

to

DLCP

-LRB-

M

b

-RRB-

-LRB-

-RRB-

The

matrix

M

is

not

sufficient

This

theorem

is

not

yet

in

EP

form

As

far

as

-LRB-

-RRB-

or

-LRB-

-RRB-

is

concerned

it

is

clear

that

there

exists

a

corresponding

basic

solution

whose

sizes

is

polynomially

bounded

The

main

problem

is

in

the

statement

-LRB-

-RRB-

To

modify

this

theorem

to

an

EP

form

we

must

show

that

there

exists

an

evidence

for

M

not

satisfying

the

sufficiency

whose

size

is

polynomially

bounded

by

the

size

of

the

input

We

give

below

three

lemmas

which

imply

this

claim

Let

V

Rn

be

any

linear

subspace

The

support

oe

-LRB-

x

-RRB-

fi

xi

g

of

a

vector

x

Rn

is

the

set

of

its

nonzero

coordinates

A

vector

x

Rn

is

called

an

elementary

vector

-LSB-

-RSB-

of

the

subspace

V

if

its

support

is

minimal

i.e.

there

exist

no

vector

x

Rn

such

that

oe

-LRB-

x

-RRB-

ae

oe

-LRB-

x

-RRB-

Elementary

vectors

will

be

referred

to

in

the

sequel

as

circuits

of

V

For

later

use

note

that

for

any

given

basis

B

and

dictionary

D

the

vectors

x

-LRB-

B

j

-RRB-

and

s

-LRB-

N

i

-RRB-

for

all

i

B

and

j

N

are

elementary

vectors

-LRB-

circuits

-RRB-

in

the

subspaces

V

and

V

respectively

A

circuit

x

of

V

-LRB-

V

-RRB-

will

be

called

a

fundamental

circuit

if

xg

Let

x

and

xk

be

vectors

in

V

The

vector

x

is

conformally

decomposed

into

xk

if

xi

xki

xi

=-RRB-

xki

xi

=-RRB-

xki

and

xi

=-RRB-

xki

for

each

i

n

The

first

lemma

is

the

conformal

decomposition

lemma

Lemma

Let

V

be

a

linear

subspace

of

Rn

Then

any

vector

x

V

can

conformally

be

decomposed

into

circuits

ck

of

V

By

using

the

above

we

obtain

the

next

lemma

which

shows

if

M

is

not

sufficient

then

the

evidence

can

be

represented

by

a

circuit

or

a

sum

of

two

circuits

Lemma

Let

V

fx

RE?gj

-LSB-

I

M

-RSB-

x

g

If

M

is

not

column

sufficient

-LRB-

row

sufficient

respectively

-RRB-

then

there

exists

a

s.s.r.

-LRB-

a

s.s.p.

-RRB-

circuit

or

a

s.s.r.

-LRB-

an

s.s.p.

-RRB-

vector

x

in

V

-LRB-

y

in

V

-RRB-

which

is

conformally

decomposable

into

two

complementary

circuits

PROOF

Suppose

M

is

not

column

sufficient

Then

V

contains

an

s.s.r.

vector

say

x.

From

Lemma

we

have

that

x

can

conformally

be

decomposed

into

circuits

say

x

ck

If

k

we

are

done

Assume

k

Let

K

fi

xi

xi

g

and

let

j

K

Then

there

exists

a

circuit

c

ckg

such

that

cj

If

cj

cj

then

c

is

the

required

one

Otherwise

there

exists

in

ckg

such

that

Because

ckg

is

a

conformal

decomposition

of

x

c

is

a

s.s.r.

vector

in

V

The

proof

for

the

case

that

M

is

not

row

sufficient

is

essentially

the

same

This

completes

the

proof

Lemma

Let

M

be

an

n?n

matrix

which

is

not

sufficient

Then

there

exists

an

evidence

for

M

not

satisfying

the

sufficiency

whose

size

is

polynomially

bounded

by

the

size

of

the

input

M

PROOF

From

Lemma

the

evidence

for

M

not

satisfying

the

sufficiency

can

be

represented

by

a

circuit

or

by

a

sum

of

two

circuits

Every

circuit

is

expressed

by

a

fundamental

circuit

of

some

dictionary

The

size

of

the

coefficients

of

the

dictionary

are

polynomially

bounded

by

that

of

input

matrix

M

Thus

an

evidence

is

polynomially

bounded

by

the

size

of

the

input

matrix

M

This

completes

the

proof

Finally

we

can

strengthen

Theorem

to

the

following

LCP

duality

theorem

in

EP

form

Theorem

Let

M

be

any

n

n

rational

matrix

and

b

be

any

rational

n-vector

Then

at

least

one

of

the

following

three

statements

holds

-LRB-

-RRB-

There

exists

a

solution

to

PLCP

-LRB-

M

b

-RRB-

whose

size

is

polynomially

bounded

by

the

sizes

of

M

and

b.

-LRB-

-RRB-

There

exists

a

solution

to

DLCP

-LRB-

M

b

-RRB-

whose

size

is

polynomially

bounded

by

the

sizes

of

M

and

b.

-LRB-

-RRB-

There

exists

an

evidence

for

the

matrix

M

not

satisfying

the

sufficiency

whose

size

is

polynomially

bounded

by

the

size

of

M

This

theorem

is

now

in

EP

form

in

which

there

are

no

assumptions

on

the

input

that

are

hard

to

check

and

every

conclusion

of

the

theorem

can

be

verified

with

a

small

certificate

It

is

possible

to

modify

the

LCP

criss-cross

method

so

that

the

modified

algorithm

proves

Theorem

directly

by

outputing

a

small

evidence

of

either

-LRB-

-RRB-

-LRB-

-RRB-

or

-LRB-

-RRB-

To

present

such

an

algorithm

is

beyond

the

scope

of

the

present

paper

see

-LSB-

-RSB-

for

a

precise

description

Here

we

note

only

that

the

criss-cross

method

can

fail

on

two

ways

if

the

matrix

M

is

not

sufficient

Either

the

method

finds

a

component

at

a

pivot

position

that

can

not

occur

at

a

sufficient

LCP

or

cycling

occurs

To

identify

cycling

and

to

derive

a

certificate

of

non-sufficiency

of

M

in

this

case

needs

the

indicator

vector

u

introduced

at

Corollary

in

the

LP

case

and

the

storage

of

the

bases

when

a

coordinate

of

the

vector

u

becomes

Cycling

is

identified

when

u

would

lexicographically

decrease

and

then

two

mutually

exclusive

basis

of

Proposition

is

found

The

modified

LCP

criss-cross

method

behaves

significantly

better

than

the

original

one

method

since

it

solves

all

sufficient

LCPs

but

yet

it

does

not

require

the

input

matrix

to

be

sufficient

which

appears

to

be

hard

to

verify

Since

the

criss-cross

method

is

not

a

polynomial

algorithm

for

LP

problems

the

LCP

criss-cross

method

is

not

a

polynomial

algorithm

either

However

the

theory

of

the

EP

theorem

suggests

the

existence

of

a

polynomial

time

algorithm

for

constructing

evidence

of

the

LCP

duality

theorem

in

EP

form

To

find

such

an

algorithm

is

one

of

the

most

challenging

open

problems

Reduction

of

Convex

Quadratic

Programming

to

LCP

In

this

section

we

show

how

one

can

formulate

any

convex

quadratic

optimization

problem

as

a

special

sufficient

LCP

problem

Also

we

discuss

the

specialization

of

the

LCP

duality

theorem

to

this

class

of

problems

Let

A

be

a

real

m

n

matrix

let

G

and

H

be

symmetric

positive

semidefinite

matrices

of

order

n

and

m

and

let

b

and

c

be

real

m

and

n-vectors

respectively

The

following

optimization

problems

QP

and

QD

are

known

to

be

a

dual

pair

of

convex

quadratic

optimization

problems

-LRB-

QP

-RRB-

max

cTx

xT

G

x

yTH

y

subject

to

A

x

H

y

b

x

-LRB-

QD

-RRB-

min

bTy

xT

G

x

yTH

y

subject

to

G

x

AT

y

c

y

-LRB-

-RRB-

When

both

G

and

H

are

totally

zero

matrices

these

problems

are

simply

a

dual

pair

of

linear

optimization

problems

The

following

weak

duality

theorem

is

a

direct

extension

of

the

LP

weak

duality

theorem

and

leads

naturally

to

an

LCP

formulation

Proposition

-LRB-

QP

Weak

Duality

-RRB-

For

any

feasible

solution

-LRB-

-RRB-

to

QP

and

any

feasible

solution

-LRB-

-RRB-

to

QD

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

T

-LRB-

-RRB-

T

-LRB-

-RRB-

where

the

equality

holds

if

and

only

if

-LRB-

a

-RRB-

i

mg

and

-LRB-

b

-RRB-

i

-RRB-

-LRB-

b

-RRB-

j

ng

and

-LRB-

AT

c

-RRB-

j

-RRB-

and

-LRB-

c

-RRB-

-LRB-

-RRB-

T

G

-LRB-

-RRB-

-LRB-

-RRB-

T

H

-LRB-

-RRB-

Just

like

in

the

LP

duality

case

one

direct

consequence

of

the

QP

weak

duality

is

that

the

relation

-LRB-

-RRB-

holds

with

equality

above

then

both

the

primal

and

the

dual

feasible

solutions

are

optimal

LCP

Formulation

of

QP

Consider

the

LCP

which

simply

expresses

the

conditions

-LRB-

a

-RRB-

-LRB-

b

-RRB-

and

-LRB-

c

-RRB-

above

by

setting

the

two

sets

-LRB-

-RRB-

and

-LRB-

-RRB-

of

solutions

identical

-LRB-

LCP

-RRB-

u

v

H

A

AT

G

y

x

b

c

y

x

u

v

y

x

T

u

v

-LRB-

-RRB-

It

is

easily

seen

that

if

this

LCP

has

a

solution

-LRB-

x

y

-RRB-

then

by

the

weak

duality

and

the

reduction

it

is

optimal

for

both

QP

and

QD

The

beauty

of

the

convex

QP

problems

is

a

sort

of

converse

statement

if

both

QP

and

QD

have

feasible

solutions

then

the

LCP

has

a

solution

-LRB-

which

must

be

optimal

for

both

QP

and

QD

by

the

first

statement

-RRB-

In

order

to

obtain

this

nice

result

observe

the

matrix

H

A

AT

G

is

positive

semidef

inite

This

implies

that

the

LCP

duality

theorem

Theorem

can

be

applied

to

this

LCP

By

interpreting

the

theorem

in

the

convex

QP

setting

one

obtains

the

second

statement

above

known

as

the

duality

theorem

of

QP

as

a

corollary

Theorem

-LRB-

QP

Duality

Theorem

-LSB-

-RSB-

-RRB-

The

following

two

statements

hold

-LRB-

a

-RRB-

If

both

QP

and

QD

are

feasible

then

each

has

an

optimal

solution

and

the

optimal

values

are

equal

moreover

the

optimal

solutions

can

be

taken

to

be

the

same

-LRB-

b

-RRB-

If

QP

-LRB-

QD

-RRB-

is

infeasible

then

its

dual

problem

QD

-LRB-

QP

-RRB-

is

either

infeasible

or

unbounded

Combinatorial

Abstractions

Oriented

Matroids

Many

of

the

extensions

of

the

least-index

criss-cross

method

and

the

LP

duality

theory

have

been

studied

originally

in

the

combinatorial

framework

of

oriented

matroids

There

are

in

fact

good

reasons

why

the

theory

of

oriented

matroids

was

helpful

in

discovering

quite

new

and

refreshing

arguments

and

algorithms

for

the

classical

linear

and

quadratic

optimization

theory

In

this

abstract

setting

the

duality

theorems

in

LP

QP

and

LCP

have

natural

combinatorial

counterparts

which

must

be

proved

using

only

combinatorial

conditions

-LRB-

axioms

-RRB-

In

particular

we

can

not

use

an

argument

such

as

an

algorithm

is

finite

because

the

associated

objective

value

increases

monotonically

and

therefore

we

are

forced

to

use

completely

different

ideas

to

design

finite

algorithms

Consequently

the

notion

of

oriented

matroids

provides

us

with

a

perfect

abstract

framework

to

seek

for

purely

combinatorial

proofs

and

algorithms

In

this

section

we

shall

briefly

discuss

the

essence

of

this

combinatorial

framework

and

present

the

LCP

duality

theorems

in

this

setting

Let

E

be

a

finite

set

A

sign

vector

X

on

E

is

a

vector

in

f

gE

The

opposite

X

of

a

sign

vector

X

is

defined

in

the

obvious

way

-LRB-

-RRB-

-LRB-

-RRB-

Let

X

and

Y

be

sign

vectors

on

E

An

element

j

E

is

said

to

separate

X

and

Y

if

Xj

and

Yj

are

nonzero

and

opposite

in

signs

The

set

of

elements

separating

X

and

Y

is

denoted

by

S

-LRB-

X

Y

-RRB-

The

composition

of

X

and

Y

denoted

by

X

ffi

Y

is

the

sign

vector

on

E

defined

by

-LRB-

X

ffi

Y

-RRB-

j

Xj

if

Xj

Yj

otherwise

-LRB-

j

E

-RRB-

The

support

oe

-LRB-

X

-RRB-

of

X

is

the

set

fj

E

Xj

g

An

oriented

matroid

on

E

is

a

pair

M

-LRB-

E

F

-RRB-

where

E

is

a

finite

set

and

F

is

a

set

of

sign

vectors

on

E

satisfying

the

following

axioms

-LRB-

-RRB-

F

-LRB-

where

denotes

the

zero

vector

-RRB-

-LRB-

-RRB-

X

F

=-RRB-

X

F

-LRB-

-RRB-

X

X

F

=-RRB-

X

ffi

X

F

-LRB-

-RRB-

X

X

F

and

p

S

-LRB-

X

X

-RRB-

=-RRB-

there

exists

Z

F

such

that

Zp

and

Zj

-LRB-

X

ffi

X

-RRB-

j

for

all

j

S

-LRB-

X

X

-RRB-

We

call

each

member

of

F

a

vector

of

the

oriented

matroid

Each

vector

with

minimal

nonempty

support

is

called

a

circuit

One

can

define

oriented

matroids

in

terms

of

circuits

see

e.g.

-LSB-

-RSB-

Note

that

the

notion

of

oriented

matroids

was

introduced

independently

by

BlandLas

Vergnas

-LSB-

-RSB-

and

Folkman-Lawrence

-LSB-

-RSB-

The

definition

of

oriented

matroids

above

is

due

to

Edmonds

Fukuda

and

Mandel

-LSB-

-RSB-

For

a

vector

x

RE

we

denote

the

sign

incidence

vector

of

x

as

ffi

-LRB-

x

-RRB-

i.e.

ffi

-LRB-

x

-RRB-

is

the

sign

vector

on

E

with

ffi

-LRB-

x

-RRB-

j

sign

-LRB-

xj

-RRB-

for

each

j

E

One

can

easily

check

that

for

any

linear

subspace

V

of

RE

the

set

F

ffi

-LRB-

V

-RRB-

of

all

sign

vectors

coming

from

V

satisfies

the

axioms

-LRB-

-RRB-

-LRB-

-RRB-

Such

an

oriented

matroid

is

called

representable

or

linear

There

are

several

interesting

constructions

of

non-representable

oriented

matroids

see

-LSB-

-RSB-

We

shall

give

below

some

basic

properties

of

oriented

matroids

without

proofs

For

those

unfamiliar

with

oriented

matroids

good

references

are

-LSB-

-RSB-

It

will

be

convenient

to

define

the

product

ff

-LRB-

or

simply

ff

-RRB-

of

two

signs

ff

f

g

in

the

natural

way

if

both

are

nonzero

and

of

same

sign

if

both

are

nonzero

and

of

opposite

sign

and

otherwise

Two

sign

vectors

X

and

Y

are

called

orthogonal

-LRB-

denoted

by

X

Y

-RRB-

if

the

existence

of

an

element

p

E

with

Xp

Yp

implies

the

existence

of

an

element

q

E

with

Xq

Yq

Xp

Yp

The

dual

M

-LRB-

E

F

-RRB-

of

an

oriented

matroid

M

is

defined

by

F

fY

f

gE

X

Y

for

all

X

Fg

The

dual

M

is

also

an

oriented

matroid

and

F

F

-LRB-

-RRB-

Furthermore

if

F

ffi

-LRB-

V

-RRB-

for

a

linear

subspace

V

of

RE

then

F

ffi

-LRB-

V

-RRB-

A

vector

circuit

of

M

may

be

called

covector

cocircuit

of

M

There

are

two

basic

operations

on

oriented

matroids

which

will

be

used

frequently

in

this

section

For

a

subset

S

of

E

let

F

nS

fXnS

X

F

and

Xj

for

all

j

Sg

-LRB-

-RRB-

F

S

fXnS

X

Fg

-LRB-

-RRB-

where

X

nS

denotes

the

subvector

-LRB-

Xj

j

EnS

-RRB-

of

X

It

is

easy

to

see

from

the

definition

both

-LRB-

E

F

nS

-RRB-

and

-LRB-

E

F

S

-RRB-

are

oriented

matroids

which

are

denoted

by

MnS

and

M

S

respectively

When

S

is

a

singleton

set

feg

Mnfeg

and

M

feg

will

be

denoted

by

Mne

and

M

e

respectively

The

following

relations

hold

for

any

oriented

matroid

M

-LRB-

E

F

-RRB-

-LRB-

F

nS

-RRB-

F

S

-LRB-

-RRB-

-LRB-

F

S

-RRB-

F?nS

-LRB-

-RRB-

Oriented

Matroid

Complementarity

Problem

For

a

nonnegative

integer

n

let

denote

a

set

of

elements

partitioned

into

prescribed

n

pairs

for

each

e

there

is

a

unique

element

e

in

called

the

complement

of

e

such

that

e

e

and

e

e.

For

a

subset

S

of

let

S

fe

e

Sg

A

subset

S

is

called

complementary

if

S

S

We

use

to

denote

-LSB-

fgg

for

a

given

and

a

distinguished

element

g

which

is

assumed

to

be

not

in

One

canonical

choice

of

is

where

i

i

n

if

i

n

and

i

i

n

if

n

i

For

a

given

oriented

matroid

M

-LRB-

F

-RRB-

with

n

the

complementarity

problem

-LRB-

OMCP

-RRB-

is

to

find

X

F

satisfying

-LRB-

-RRB-

Xe

Xe

for

all

e

-LRB-

-RRB-

X

Xg

-LRB-

-RRB-

Here

we

use

the

inequality

in

the

obvious

way

and

we

shall

do

so

in

the

sequel

The

conditions

-LRB-

-RRB-

-LRB-

-RRB-

and

-LRB-

-RRB-

are

straightforward

abstractions

of

-LRB-

-RRB-

-LRB-

-RRB-

and

-LRB-

-RRB-

respectively

Again

we

simply

make

the

automatic

abstraction

of

the

definitions

in

the

LCP

case

A

sign

vector

X

on

or

on

is

called

complementary

if

the

condition

-LRB-

-RRB-

is

satisfied

called

strictly

sign-preserving

-LRB-

s.s.p.

-RRB-

if

Xe

Xe

for

all

e

and

-LRB-

-RRB-

Xp

Xp

for

some

p

-LRB-

-RRB-

and

called

strictly

sign-reversing

-LRB-

s.s.r.

-RRB-

if

Xe

Xe

for

all

e

and

-LRB-

-RRB-

Xp

Xp

for

some

p

-LRB-

-RRB-

We

call

an

oriented

matroid

-LRB-

F

-RRB-

sufficient

if

F

contains

no

s.s.r.

vector

and

its

dual

-LRB-

F

-RRB-

contains

no

s.s.p.

vector

By

the

characterizations

of

sufficient

matrices

in

-LRB-

-RRB-

and

-LRB-

-RRB-

the

sufficiency

definition

of

oriented

matroids

is

simply

an

extension

of

sufficiency

of

the

associated

matrices

We

say

M

-LRB-

E

-LSB-

fgg

F

-RRB-

is

a

single-element

extension

of

an

oriented

matroid

-LRB-

E

F

-RRB-

by

a

new

element

g

E

if

Mng

The

following

theorem

is

a

combinatorial

generalization

of

the

LCP

duality

theorem

Theorem

Theorem

-LRB-

OMCP

Duality

Theorem

-RRB-

Let

M

-LRB-

F

-RRB-

be

any

single-element

extension

of

a

sufficient

oriented

matroid

by

g

Then

exactly

one

of

the

following

statements

holds

-LRB-

a

-RRB-

There

exists

a

nonnegative

complementary

vector

X

in

F

with

Xg

-LRB-

b

-RRB-

There

exists

a

nonnegative

complementary

vector

Y

in

F

with

Yg

A

very

important

tool

in

oriented

matroids

is

the

notion

of

minors

Minors

-LRB-

or

subproblems

-RRB-

of

OMCPs

were

introduced

by

Todd

-LSB-

-RSB-

Let

M

-LRB-

F

-RRB-

be

an

oriented

matroid

with

n

For

each

complementary

subset

S

of

let

M

-LRB-

S

-RRB-

-LRB-

-LRB-

S

-LSB-

S

-RRB-

F

-LRB-

S

-RRB-

-RRB-

denote

the

oriented

matroid

MnS

S

which

is

called

a

complementary

minor

of

M.

Theorem

can

be

proved

by

using

minors

with

induction

on

n

and

the

fact

that

the

sufficiency

assumption

is

inherited

by

the

minor

subproblems

Since

sufficient

oriented

matroids

serve

as

combinatorial

abstractions

of

sufficient

linear

complementarity

problems

the

generalizability

of

finite

criss-cross

methods

to

these

problem

classes

was

expected

In

fact

one

can

design

a

finite

criss-cross

method

for

OMCP

as

an

algorithmic

version

of

the

inductive

proof

of

Theorem

see

-LSB-

-RSB-

It

is

not

surprising

that

the

simpler

linear

programming

problem

has

also

an

analogous

abstract

oriented

matroid

linear

programming

problem

The

reader

can

consult

with

the

following

papers

the

definition

and

basic

properties

of

oriented

matroid

linear

programming

-LSB-

-RSB-

and

solving

it

by

the

least-index

criss-cross

method

-LSB-

-RSB-

the

definition

and

basic

properties

of

oriented

matroid

quadratic

programming

-LSB-

-RSB-

and

solving

it

by

the

least-index

criss-cross

method

-LSB-

-RSB-

the

definition

and

basic

properties

of

semidefinite

and

sufficient

oriented

matroid

linear

complementarity

problems

-LSB-

-RSB-

and

solving

it

by

the

least-index

criss-cross

method

-LSB-

-RSB-

Conclusions

and

open

problems

The

goal

of

the

present

paper

was

to

provide

a

good

motivation

to

study

a

class

of

pivot

algorithms

which

is

more

general

than

the

class

of

simplex

algorithms

This

broader

view

on

pivot

methods

resulted

in

a

family

of

simple

finite

one-phase

methods

it

made

possible

to

give

simple

constructive

proofs

for

such

fundamental

results

as

the

strong

duality

theorems

of

LP

QP

and

LCP

and

it

allowed

us

to

explore

the

combinatorial

structures

of

linear

optimization

problems

Some

new

results

on

the

existence

of

a

short

admissible

pivot

path

were

presented

This

is

a

weaker

analogue

of

the

old

very

difficult

problem

the

so-called

d-step

conjecture

-LSB-

-RSB-

Some

related

open

problems

are

sketched

below

The

simplex

method

has

been

studied

extensively

by

many

researchers

since

it

was

invented

in

by

Dantzig

-LSB-

-RSB-

but

yet

it

resists

the

resolution

of

the

longstanding

open

problem

Open

Problem

Is

there

any

refinement

of

the

simplex

method

which

terminates

in

a

number

of

pivots

bounded

by

a

polynomial

in

m

and

n

A

refinement

of

the

simplex

method

is

a

simplex

algorithm

which

at

each

feasible

basis

uniquely

specifies

such

pivot

elements

which

satisfy

the

simplex

pivot

selection

criteria

Although

the

simplex

method

allows

extremely

efficient

practical

implementations

no

polynomial

refinement

of

the

simplex

method

is

known

to

date

In

fact

many

well

known

refinements

such

as

the

minimal

index

simplex

rule

the

largest

coefficient

simplex

rule

the

maximum

improvement

simplex

rule

and

the

shadow

vertex

algorithm

are

known

to

require

an

exponential

number

of

pivots

for

certain

LP

's

-LRB-

see

e.g.

-LSB-

-RSB-

-RRB-

Open

Problem

seems

to

be

too

hard

because

the

existence

question

Open

Problem

Is

there

a

simplex-pivot

path

to

an

optimal

basis

from

any

given

feasible

basis

whose

length

is

bounded

by

a

polynomial

of

m

and

n

appears

to

be

already

too

hard

Observe

that

a

solution

of

the

first

problem

gives

a

solution

to

the

latter

one

Further

the

affirmative

answer

to

this

latter

simpler

question

would

imply

the

solution

of

the

famous

diameter

problem

of

convex

polytopes

Open

Problem

Is

the

combinatorial

diameter

of

a

d-dimensional

polytope

with

m-facets

bounded

by

a

polynomial

in

m

and

d

Although

some

interesting

results

on

the

diameter

of

polytopes

have

been

obtained

recently

-LRB-

e.g.

-LSB-

-RSB-

-RRB-

we

must

admit

that

we

are

still

quite

far

from

the

solution

of

this

simpler

problem

In

Section

a

positive

answer

to

a

much

weaker

question

is

presented

We

have

seen

that

if

the

optimal

basis

is

fully

nondegenerate

a

short

admissible

pivot

path

from

any

basis

to

the

optimal

basis

exists

This

existence

result

raises

the

following

more

difficult

question

-LRB-

posed

by

H.-J

L?uthi

-RRB-

Open

Problem

For

any

given

LP

is

there

an

ordering

of

variables

for

which

the

least

index

criss-cross

method

finds

an

optimal

basis

from

any

given

basis

in

a

polynomial

number

of

pivots

A

much

simplified

question

which

seems

to

be

already

hard

is

Open

Problem

For

any

given

LP

a

given

optimal

basis

and

a

given

initial

basis

is

there

an

ordering

of

variables

for

which

the

least

index

criss-cross

method

terminates

in

a

polynomial

number

of

pivots

Although

the

least-index

criss-cross

method

presented

in

Section

admits

an

exponential

example

the

ultimate

goal

of

our

study

is

of

course

to

answer

the

most

basic

question

Open

Problem

Is

there

a

polynomial

criss-cross

method

Another

unresolved

problem

concerns

the

practicality

of

criss-cross

rules

Contrary

to

the

many

efficient

implementations

of

simplex

methods

to

date

no

practically

efficient

criss-cross

method

is

known

Thus

we

have

the

following

question

Open

Problem

Is

there

a

practically

efficient

finite

criss-cross

method

Interesting

experimental

results

on

some

modifications

of

the

least-index

criss-cross

method

were

presented

in

-LSB-

-RSB-

To

date

not

much

is

known

on

the

expected

number

of

pivot

steps

of

criss-cross

methods

This

is

in

sharp

contrast

to

the

extensive

literature

on

the

expected

number

of

steps

of

simplex

methods

-LRB-

see

e.g.

-LSB-

-RSB-

-RRB-

Open

Problem

What

is

the

expected

number

of

pivot

steps

needed

by

finite

criss-cross

methods

to

solve

LPs

We

have

presented

an

EP

version

of

the

LCP

least-index

criss-cross

method

in

Section

Since

the

criss-cross

method

is

not

a

polynomial

algorithm

for

LCP

the

following

problem

remains

to

be

open

Open

Problem

Is

there

a

polynomial

time

algorithm

for

constructing

evidence

of

the

LCP

duality

theorem

in

EP

form

Acknowledgement

The

authors

are

grateful

to

Tibor

Ill?es

Thomas

M.

Liebling

Hans-Jakob

L?uthi

Kees

Roos

and

Shuzhong

Zhang

for

useful

comments

regarding

the

content

of

the

paper

and

for

reading

and

correcting

an

earlier

version

The

first

author

thanks

Makoto

Namiki

and

Akihisa

Tamura

for

letting

him

use

a

part

of

the

joint

paper

-LSB-

-RSB-

References

-LSB-

-RSB-

I.

Adler

and

N.

Megiddo

A

simplex

algorithm

whose

average

number

of

steps

is

bounded

between

two

quadratic

functions

of

the

smaller

dimension

Journal

of

the

Association

of

Computing

Machinery

-LRB-

-RRB-

-LCB-

-LSB-

-RSB-

D.

Avis

and

V.

Chv?atal

Notes

on

Bland

's

rule

Mathematical

Programming

Study

-LRB-

-RRB-

-LCB-

-LSB-

-RSB-

D.

Avis

and

K.

Fukuda

A

pivoting

algorithm

for

convex

hulls

and

vertex

enumeration

of

arrangements

and

polyhedra

Research

Report

Department

of

Information

Sciences

Tokyo

Institute

of

Technology

Tokyo

Japan

-LRB-

-RRB-

-LSB-

-RSB-

J.

Beasley

-LRB-

ed

-RRB-

Advances

in

Linear

and

Integer

Programming

-LRB-

Oxford

University

Press

Oxford

England

-RRB-

-LSB-

-RSB-

R.E.

Bixby

The

simplex

method

it

keeps

getting

better

Lecture

at

the

International

Symposium

on

Mathematical

Programming

Amsterdam

The

Netherlands

-LRB-

-RRB-

-LSB-

-RSB-

A.

Bj?orner

M.

Las

Vergnas

B.

Sturmfels

N.

White

and

G.

Ziegler

Oriented

Matroids

-LRB-

Cambridge

University

Press

-RRB-

-LSB-

-RSB-

R.G.

Bland

New

finite

pivoting

rules

for

the

simplex

method

Mathematics

of

Operations

Research

-LRB-

-RRB-

-LCB-

-LSB-

-RSB-

R.G.

Bland

A

combinatorial

abstraction

of

linear

programming

Journal

of

Combinatorial

Theory

-LRB-

Ser

B

-RRB-

-LRB-

-RRB-

-LCB-

-LSB-

-RSB-

R.G.

Bland

and

M.

Las

Vergnas

Orientability

of

matroids

Journal

of

Combinatorial

Theory

Ser

B.

-LRB-

-RRB-

-LCB-

-LSB-

-RSB-

K.H.

Borgwardt

The

Simplex

Method

A

Probabilistic

Analysis

Algorithms

and

Combinatorics

Vol

-LRB-

Springer

Verlag

-RRB-

-LSB-

-RSB-

K.

Cameron

and

J.

Edmonds

Existentially

polytime

theorems

Proceedings

of

the

DIMACS

Workshop

in

Polyhedral

Combinatorics

-LRB-

-RRB-

-LSB-

-RSB-

Y.Y.

Chang

Least

index

resolution

of

degeneracy

in

linear

complementarity

problems

Technical

Report

-LCB-

Department

of

Operations

Research

Stanford

University

Stanford

California

USA

-LRB-

-RRB-

-LSB-

-RSB-

A.

Charnes

Optimality

and

degeneracy

in

linear

programming

Econometrica

-LRB-

-RRB-

-LCB-

-LSB-

-RSB-

H.-D

Chen

P.M.

Pardalos

M.

A.

Saunders

The

simplex

algorithm

with

a

new

primal

and

dual

pivot

rule

Operations

Research

Letters

-LRB-

-RRB-

-LCB-

-LSB-

-RSB-

V.

Chv?atal

Linear

Programming

-LRB-

W.H.

Freeman

and

Company

-RRB-

-LSB-

-RSB-

J.

Clausen

A

note

on

Edmonds-Fukuda

pivoting

rule

for

the

simplex

method

European

Journal

of

Operations

Research

-LRB-

-RRB-

-LCB-

-LSB-

-RSB-

R.W.

Cottle

Symmetric

dual

quadratic

programs

Quart

Appl

Math

-LRB-

-RRB-

-LSB-

-RSB-

R.W.

Cottle

and

G.B.

Dantzig

Complementary

pivot

theory

of

mathematical

programming

Linear

Algebra

and

Its

Applications

-LRB-

-RRB-

-LSB-

-RSB-

R.W.

Cottle

J.-S

Pang

and

V.

Venkateswaran

Sufficient

matrices

and

the

linear

complementarity

problem

Linear

Algebra

and

Its

Applications

-LRB-

-RRB-

-LCB-

-LSB-

-RSB-

G.B.

Dantzig

Programming

in

a

linear

structure

Comptroller

USAF

Washington

DC

-LRB-

February

-RRB-

-LSB-

-RSB-

G.B.

Dantzig

Linear

Programming

and

Extensions

-LRB-

Princeton

University

Press

Princeton

N.J.

-RRB-

-LSB-

-RSB-

G.B.

Dantzig

Linear

programming

The

story

about

how

it

began

In

History

of

Mathematical

Programming

eds

A.H.G.

Rinnoy

Kan

L.K.

Lenstra

and

A.

Schrijver

-LRB-

CWI

North-Holland

Amsterdam

-RRB-

-LCB-

-LSB-

-RSB-

G.B.

Dantzig

A.

Orden

and

P.

Wolfe

The

generalized

simplex

method

for

minimizing

a

linear

form

under

linear

inequality

restraints

Pacific

Journal

of

Mathematics

-LRB-

-RRB-

-LCB-

-LSB-

-RSB-

J.

Edmonds

Exact

pivoting

presentation

ECCO

VII

February

-LSB-

-RSB-

J.

Edmonds

A

Helly

method

for

linear

programming

presentation

Amsterdam

April

-LSB-

-RSB-

J.

Edmonds

and

K.

Fukuda

NP

easy

and

LP

theory

Cours

postgrade

en

Recherche

Op?erationnelle

Ecole

Polytechnique

F?ed?erale

de

Lausanne

Lausanne

May

-LSB-

-RSB-

J.

Folkman

and

J.

Lawrence

Oriented

matroids

Journal

of

Combinatorial

Theory

-LRB-

Ser

B

-RRB-

-LRB-

-RRB-

-LCB-

-LSB-

-RSB-

K.

Fukuda

Oriented

Matroid

Programming

Ph.D.

Thesis

-LRB-

Waterloo

University

Waterloo

Ontario

Canada

-RRB-

-LSB-

-RSB-

K.

Fukuda

H.-J

L?uthi

and

M.

Namiki

The

existence

of

a

short

sequence

of

admissible

pivots

to

an

optimal

basis

in

LP

and

LCP

Technical

Report

Institute

for

Operations

Research

ETH-Zentrum

Z?urich

Switzerland

-LRB-

-RRB-

-LSB-

-RSB-

K.

Fukuda

and

H.-J

L?uthi

Combinatorial

maximum

improvement

algorithm

for

LP

and

LCP

presented

at

Franco-Japanese

Days

Brest

France

-LRB-

-RRB-

-LSB-

-RSB-

K.

Fukuda

and

T.

Matsui

On

the

finiteness

of

the

criss-cross

method

European

Journal

of

Operations

Research

-LRB-

-RRB-

-LCB-

-LSB-

-RSB-

K.

Fukuda

and

M.

Namiki

On

extremal

behaviors

of

Murty

's

least

index

method

Mathematical

Programming

-LRB-

-RRB-

-LCB-

-LSB-

-RSB-

K.

Fukuda

and

M.

Namiki

Finding

all

common

basis

in

two

matroids

Discrete

Applied

Mathematics

-LRB-

-RRB-

-LCB-

-LSB-

-RSB-

K.

Fukuda

M.

Namiki

and

A.

Tamura

EP

theorems

and

linear

complementarity

problems

Technical

Report

Institute

for

Operations

Research

ETH

Zentrum

Z?urich

Switzerland

-LRB-

-RRB-

-LSB-

-RSB-

K.

Fukuda

and

T.

Terlaky

Linear

complementarity

and

oriented

matroids

Journal

of

the

Operational

Research

Society

of

Japan

-LRB-

-RRB-

-LCB-

-LSB-

-RSB-

D.

Goldfarb

Worst

case

complexity

of

the

shadow

vertex

simplex

algorithm

Technical

Report

Department

of

Industrial

Engineering

and

Operations

Research

Columbia

University

USA

-LRB-

-RRB-

-LSB-

-RSB-

M.

Haimovitz

The

simplex

method

is

very

good

-LCB-

on

the

expected

number

of

pivot

steps

and

related

properties

of

random

linear

programs

Technical

Report

Columbia

University

New

York

USA

-LRB-

-RRB-

-LSB-

-RSB-

D.

Den

Hertog

C.

Roos

and

T.

Terlaky

The

linear

complementarity

problem

sufficient

matrices

and

the

criss-cross

method

Linear

Algebra

and

Its

Applications

-LRB-

-RRB-

-LCB-

-LSB-

-RSB-

T.

Ill?es

A.

Szirmai

and

T.

Terlaky

A

finite

Criss-Cross

Method

for

Hyperbolic

Programming

Report

No.

-LCB-

Faculteit

der

Technische

Wiskunde

en

Informatica

Technische

Universiteit

Delft

The

Netherlands

-LRB-

-RRB-

To

appear

in

European

Journal

of

Operations

Research

-LSB-

-RSB-

D.

Jensen

Coloring

and

Duality

Combinatorial

Augmentation

Methods

Ph.D.

Thesis

-LRB-

School

of

OR

and

IE

Cornell

University

Ithaca

N.Y.

USA

-RRB-

-LSB-

-RSB-

G.

Kalai

The

diameter

of

graphs

of

convex

polytopes

and

f-vector

theory

In

Applied

Geometry

and

Discrete

Mathematics

The

Victor

Klee

Festschrift

eds

P.

Gritzman

and

B.

Strumfels

DIMACS

Series

in

Discrete

Mathematics

and

Theoretical

Computer

Science

-LRB-

American

Mathematical

Society

-RRB-

-LSB-

-RSB-

N.

Karmarkar

A

new

polynomial-time

algorithm

for

linear

programming

Combinatorica

-LRB-

-RRB-

-LCB-

-LSB-

-RSB-

L.G.

Khachian

Polynomial

algorithms

in

linear

programming

Zhurnal

Vichislitelnoj

Matematiki

i

Matematischeskoi

Fiziki

-LRB-

-RRB-

-LCB-

-LRB-

in

Russian

-RRB-

USSR

Computational

Mathematics

and

Mathematical

Physics

-LRB-

-RRB-

-LCB-

-LRB-

English

translation

-RRB-

-LSB-

-RSB-

E.

Klafszky

and

T.

Terlaky

Some

generalizations

of

the

criss-cross

method

for

quadratic

programming

Math

Oper

und

Stat

ser

Optimization

-LRB-

-RRB-

-LCB-

-LSB-

-RSB-

E.

Klafszky

and

T.

Terlaky

Some

generalizations

of

the

criss-cross

method

for

the

linear

complementarity

problem

of

oriented

matroids

Combinatorica

-LRB-

-RRB-

-LCB-

-LSB-

-RSB-

E.

Klafszky

and

T.

Terlaky

The

role

of

pivoting

in

proving

some

fundamental

theorems

of

linear

algebra

Linear

Algebra

and

Its

Applications

-LRB-

-RRB-

-LCB-

-LSB-

-RSB-

V.

Klee

and

G.J.

Minty

How

good

is

the

simplex

algorithm

In

Inequalities-III

ed

O.

Shisha

-LRB-

Academic

Press

New

York

-RRB-

-LCB-

-LSB-

-RSB-

V.

Klee

and

P.

Kleinschmidt

The

d

-LCB-

step

conjecture

and

its

relatives

Mathematics

of

Operations

Research

-LRB-

-RRB-

-LCB-

-LSB-

-RSB-

M.

Kojima

N.

Megiddo

T.

Noma

and

A.

Yoshise

A

Unified

Approach

to

Interior

Point

Algorithms

for

Linear

Complementarity

Problems

In

Lecture

Notes

in

Computer

Science

-LRB-

Springer

Verlag

-RRB-

-LSB-

-RSB-

C.E.

Lemke

J.T.

Jr.

Howson

Equilibrium

points

of

bimatrix

games

SIAM

Journal

-LRB-

-RRB-

-LCB-

-LSB-

-RSB-

C.E.

Lemke

Bimatrix

equilibrium

points

and

mathematical

programming

Management

Science

-LRB-

-RRB-

-LCB-

-LSB-

-RSB-

I.

Lustig

The

equivalence

of

Dantzig

's

self-dual

parametric

algorithm

for

linear

programs

to

Lemke

's

algorithm

for

linear

complementarity

problems

applied

to

linear

programming

Technical

Report

SOL

-LCB-

Department

of

Operations

Research

Stanford

University

Stanford

California

USA

-LRB-

-RRB-

-LSB-

-RSB-

A.

Mandel

Topology

of

Oriented

Matroids

Ph.D.

Thesis

-LRB-

Waterloo

University

Waterloo

Ontario

Canada

-RRB-

-LSB-

-RSB-

W.

Morris

Jr.

and

M.J.

Todd

Symmetry

and

positive

definiteness

in

oriented

matroids

Technical

Report

No.

Cornell

University

School

of

Operations

Research

and

Industrial

Engineering

Itacha

N.Y.

-LRB-

-RRB-

-LSB-

-RSB-

K.G.

Murty

A

note

on

a

Bard

type

scheme

for

solving

the

complementarity

problem

Opsearch

-LRB-

-LCB-

-RRB-

-LRB-

-RRB-

-LCB-

-LSB-

-RSB-

K.G.

Murty

Linear

and

Combinatorial

Programming

-LRB-

Krieger

Publishing

Company

Malabar

Florida

USA

-RRB-

-LSB-

-RSB-

K.G.

Murty

Computational

complexity

of

parametric

linear

programming

Mathematical

Programming

-LRB-

-RRB-

-LCB-

-LSB-

-RSB-

M.

Namiki

and

T.

Matsui

Some

modifications

of

the

criss-cross

method

Research

Report

Department

of

Information

Sciences

Tokyo

Institute

of

Technology

Tokyo

Japan

-LRB-

-RRB-

-LSB-

-RSB-

K.

Paparrizos

Pivoting

rules

directing

the

simplex

method

through

all

feasible

vertices

of

Klee-Minty

examples

Opsearch

-LRB-

-RRB-

-LRB-

-RRB-

-LCB-

-LSB-

-RSB-

R.T.

Rockafellar

Elementary

vectors

of

a

subspace

of

Rn

In

Combinatorial

Mathematics

and

Its

Applications

-LRB-

R.C.

Bose

and

T.A.

Dowling

eds

-RRB-

University

of

North

Carolina

Press

Chapel

Hill

North

Carolina

-LRB-

-RRB-

-LCB-

-LSB-

-RSB-

C.

Roos

An

exponential

example

for

Terlaky

's

pivoting

rule

for

the

criss-cross

simplex

method

Mathematical

Programming

-LRB-

-RRB-

-LCB-

-LSB-

-RSB-

C.

Roos

T.

Terlaky

and

J.-Ph

Vial

Theory

and

Algorithms

for

Linear

Optimization

An

Interior

Point

Approach

-LRB-

John

Wiley

Sons

New

York

USA

-RRB-

-LSB-

-RSB-

J.E.

Strum

Introduction

to

Linear

Programming

-LRB-

Holden

Day

California

USA

-RRB-

-LSB-

-RSB-

T.

Terlaky

Egy

uj

v?eges

criss-cross

m?odszer

line?aris

programoz?asi

feladatok

megold?as?ara

Alkalmazott

Matematikai

Lapok

-LRB-

-RRB-

-LRB-

In

Hungarian

English

title

A

new

finite

criss-cross

method

for

solving

linear

programming

problems

-RRB-

-LSB-

-RSB-

T.

Terlaky

A

convergent

criss-cross

method

Math

Oper

und

Stat

ser

Optimization

-LRB-

-RRB-

-LRB-

-RRB-

-LCB-

-LSB-

-RSB-

T.

Terlaky

A

finite

criss-cross

method

for

oriented

matroids

Journal

of

Combinatorial

Theory

-LRB-

Ser

B

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LCB-

-LSB-

-RSB-

T.

Terlaky

-LRB-

ed

-RRB-

Interior

Point

Methods

of

Mathematical

Programming

-LRB-

Kluwer

Academic

Publishers

Dordrecht

The

Netherlands

-RRB-

-LSB-

-RSB-

T.

Terlaky

and

S.

Zhang

Pivot

rules

for

linear

programming

a

survey

on

recent

theoretical

developments

Annals

of

Oper

Res

-LRB-

-RRB-

-LCB-

-LSB-

-RSB-

M.J.

Todd

Complementarity

in

oriented

matroids

SIAM

Journal

on

Algebraic

and

Discrete

Mathematics

-LRB-

-RRB-

-LCB-

-LSB-

-RSB-

M.J.

Todd

Linear

and

quadratic

programming

in

oriented

matroids

Journal

of

Combinatorial

Theory

-LRB-

Ser

B

-RRB-

-LRB-

-RRB-

-LCB-

-LSB-

-RSB-

M.J.

Todd

Polynomial

expected

behavior

of

a

pivoting

algorithm

for

linear

complementarity

and

linear

programming

problems

Mathematical

Programming

-LRB-

-RRB-

-LCB-

-LSB-

-RSB-

A.

Tucker

A

note

on

convergence

of

the

Ford-Fulkerson

flow

algorithm

Mathematics

of

Operations

Research

-LRB-

-RRB-

-LRB-

-RRB-

-LCB-

-LSB-

-RSB-

H.

V?aliaho

A

new

proof

of

the

finiteness

of

the

criss-cross

method

Math

Oper

und

Stat

ser

Optimization

-LRB-

-RRB-

-LCB-

-LSB-

-RSB-

H.

V?aliaho

P

matrices

are

just

sufficient

Linear

Algebra

and

Its

Applications

-LRB-

-RRB-

-LSB-

-RSB-

Zh

Wang

A

conformal

elimination

free

algorithm

for

oriented

matroid

programming

Chinese

Annals

of

Mathematics

-LRB-

-RRB-

-LRB-

-RRB-

-LSB-

-RSB-

Zh

Wang

A

modified

version

of

the

Edmonds-Fukuda

algorithm

for

LP

in

the

general

form

Asia-Pacific

Journal

of

Operations

Research

-LRB-

-RRB-

-LSB-

-RSB-

Zh

Wang

A

general

deterministic

pivot

method

for

oriented

matroid

programming

Chinese

Annals

of

Mathematics

B.

-LRB-

-RRB-

-LSB-

-RSB-

Zh

Wang

T.

Terlaky

A

general

scheme

for

solving

linear

complementarity

problems

in

the

setting

of

oriented

matroids

-LRB-

-RRB-

In

Combinatorics

and

Graph

Theory

eds

H.P.

Yap

T.H.

Ku

E.K.

Lloyd

and

Zh

Wang

-LRB-

-RRB-

-LCB-

Proceedings

of

the

Spring

School

and

International

Conference

on

Combinatorics

SSIC

China

World

Scientific

Singapore

-LSB-

-RSB-

S.J.

Wright

Primal-Dual

Interior

Point

Methods

SIAM

Publications

Philadelphia

USA

-LRB-

-RRB-

-LSB-

-RSB-

N.

Zadeh

What

is

the

worst

case

behavior

of

the

simplex

algorithm

Technical

Report

No.

Department

of

Operations

Research

Stanford

University

Stanford

California

USA

-LRB-

-RRB-

-LSB-

-RSB-

S.

Zhang

On

anti-cycling

pivoting

rules

for

the

simplex

method

Operations

Research

Letters

-LRB-

-RRB-

-LCB-

-LSB-

-RSB-

S.

Zhang

New

variants

of

finite

criss-cross

pivot

algorithms

for

linear

programming

Technical

Report

Econometric

Institute

Erasmus

University

Rotterdam

-LRB-

-RRB-

-LSB-

-RSB-

S.

Zionts

The

criss-cross

method

for

solving

linear

programming

problems

Management

Science

-LRB-

-RRB-

-LRB-

-RRB-

-LCB-

-LSB-

-RSB-

S.

Zionts

Some

empirical

test

of

the

criss-cross

method

Management

Science

-LRB-

-RRB-

cycling
linear complementarity problems
linear programming
oriented matroids
pivot rules
quadratic programming
recursion