existence of bounded positive solutions of semilinear elliptic equations abstract in this paper by using fixed point theory under quite general conditions on the nonlinear term we obtain an existence result on bounded positive solutions of certain semilinear elliptic equations in r n n 3

a simplex based algorithm to solve separated continuous linear programs we consider the separated continuous linear programming problem with linear data we characterize the form of its optimal solution and present an algorithm which solves it in a finite number of steps using an analog of the simplex method in the space of bounded measurable functions

existence of unbounded positive solutions of quasilinear elliptic equations in two dimensional exterior domains in this paper by using the fixed point theory we obtain a new existence result for bounded positive solutions of the quasilinear elliptic equations in two dimensional exterior domains

p matrices are just sufficient abstract the classes of sufficient matrices and of p matrices have recently arisen in connection with the linear complementarity problem it is known that p matrices are sufficient we show that conversely every sufficient matrix is a p matrix

sufficient matrices and the linear complementarity problem abstract we pose and answer two questions about solutions of the linear complementarity problem lcp the first question is concerned with the conditions on a matrix m which guarantee that for every vector q the solutions of the lcp q m are identical to the karush kuhn tucker points of the natural quadratic program associated with q m in answering this question we introduce the class of row sufficient matrices the transpose of such a matrix is what we call column sufficient the latter matrices turn out to furnish the answer to our second question which asks for the conditions on m under which the solution set of q m is convex for every q in addition to these two main results we discuss the connections of these twonew matrix classes with other well known matrix classes in linear complementarity theory

a simplex algorithm whose average number of steps is bounded between two quadratic functions of the smaller dimension it has been a challenge for mathematicians to confirm theoretically the extremely good performance of simplex type algorithms for linear programming in this paper the average number of steps performed by a simplex algorithm the so called self dual method is analyzed the algorithm is not started at the traditional point 1 l t but points of the form 1 e e 2 t with e sufficiently small are used the result is better in two respects than those of the previous analyses first it is shown that the expected number of steps is bounded between two quadratic functions c 1 min m n 2 and c 2 min m n 2 of the smaller dimension of the problem this should be compared with the previous two major results in the field borgwardt proves an upper bound of o n 4 m 1 n 1 under a model that implies that the zero vector satisfies all the constraints and also the algorithm under his consideration solves only problems from that particular subclass smale analyzes the self dual algorithm starting at 1 1 t he shows that for any fixed m there is a constant c m such the expected number of steps is less than c m ln n m m 1 megiddo has shown that under smale s model an upper bound c m exists thus for the first time a polynomial upper bound with no restrictions except for nondegeneracy on the problem is proved and for the first time a nontrivial lower bound of precisely the same order of magnitude is established both borgwardt and smale require the input vectors to be drawn from spherically symmetric distributions in the model in this paper invariance is required only under certain

a quasi polynomial bound for the diameter of graphs of polyhedra the diameter of the graph of a d dimensional polyhedron with n facets is at most nlog d 2 let p be a convex polyhedron the graph of p denoted by g p is an abstract graph whose vertices are the extreme points of p and two vertices u and v are adjacent if the interval v u is an extreme edge 1 dimensional face of p the diameter of the graph of p is denoted by p let d n be the maximal diameter of the graphs of d dimensional polyhedra p with n facets a facet is a d 1 dimensional face thus p is the set of solutions of n linear inequalities in d variables it is an old standing problem to determine the behavior of the function d n the value of d n is a lower bound for the number of iterations needed for dantzig s simplex algorithm for linear programming with any pivot rule in 1957 hirsch conjectured 2 that d n n d klee and walkup 6 showed that the hirsch conjecture is false for unbounded polyhedra they proved that for n 2d d n n d d 5 this is the best known lower bound for d n the statement of the hirsch conjecture for bounded polyhedra is still open for a recent survey on the hirsch conjecture and its relatives see 5 in 1967 barnette proved 1 3 that d n n3 an improved upper bound d n n2 was proved in 1970 by larman 7 barnette s and larman s bounds are linear in n but exponential in the dimension d in 1990 the first author 4 proved a subexponential bound d n 2 n d the purpose of this paper is to announce and to give a complete proof of a quasipolynomial upper bound for d n such a bound was proved by the first author in march 1991 the proof presented here is a substantial simplification that was subsequently found by the second author see 4 for the original proof and related results the existence of a polynomial or even linear upper bound for d n is still open recently the first author found a randomized pivot rule for linear programming which requires an expected n d or less arithmetic operations for every linear programming problem with d variables and n constraints 1991 mathematics subject classification primary 52a25 90c05 received by the editors july 1 1991 the first author was supported in part by a bsf grant by a gif grant the second author was supported by an afosr grant c 1992 american mathematical society 0273 0979 92 1 00 25 per page

structural properties of xpath fragments we study structural properties of each of the main sublanguages of navigational xpath w3c recommendation commonly used in practice first we characterize the expressive power of these language fragments in terms of both logics and tree patterns second we investigate closure properties focusing on the ability to perform basic boolean operations while remaining within the fragment we give a complete picture of the closure properties of these fragments treating xpath expressions both as functions of arbitrary nodes in a document tree and as functions that are applied only at the root of the tree finally we provide sound and complete axiom systems and normal forms for several of these fragments these results are useful for simplification of xpath expressions and optimization of xml queries

positive solutions of nonlinear differential equations with prescribed decay of the first derivative an existence and uniqueness result for bounded positive solutions x t of the equation u f t u u 0 t greater than or equal to t 0 greater than or equal to 0 is established by means of the banach contraction principle for such a solution it is shown that alpha t less than or equal to x t less than or equal to beta t t greater than or equal to t 0 where alpha beta are given nonnegative continuous functions which are integrable over t 0 infinity the result complements others known in the literature

an exponential example for terlaky s pivoting rule for the criss cross simplex method recently t terlaky has proposed a new pivoting rule for the criss cross simplex method for linear programming and he proved that his rule is convergent in this note we show that the required number of iterations may be exponential in the number of variables and constraints of the problem

simulating alternating tree automata by nondeterministic automata new results and new proofs of the theorems of rabin mcnaughton and safra abstract we give a proof that alternating tree automata can be simulated by nondeterministic tree automata which yields new complexity results and a unified proof of the theorems of rabin mcnaughton and safra we also give a simple axiomatic framework for uniformizing strategies

an easy way to teach interior point methods abstract in this paper the duality theory of linear optimization lo is built up based on ideas emerged from interior point methods ipms all we need is elementary calculus we will embed the lo problem and its dual in a self dual skew symmetric problem most duality results except the existence of a strictly complementary solution are trivial for this embedding problem the existence of the central path and its convergence to the analytic center of the optimal face will be proved the proof is based on an elementary careful analysis of a newton step we show also that if an almost optimal solution on the central path is known then a simple strongly polynomial rounding procedure provides an exact strictly complementary optimal solution the all one vector is feasible for the embedding problem and it is an interior point on the central path this way an elegant solution to the initialization of ipms is obtained as well this approach allows to apply any ipm to the embedding problem while complexity results obtained for feasible ipms are preserved

decaying positive solutions of quasilinear elliptic equations in exterior domains in r2 abstract the existence of decaying positive solutions of quasilinear elliptic equations of the form u x u x u x 2 0 is established in an exterior domain r 2 under suitable smoothness and growth conditions the main result is proved by means of a supersolution subsolution method given by noussair and swanson by using phase plane analysis of a system of lienard type a supersolution and a subsolution of the above equation are found out an extension of the main result to more general case is also attempted finally some examples are attached

pivot versus interior point methods pros and cons linear optimization lo is the fundamental problem of mathematical optimization it admits an enormous number of applications in economics engineering science and many other fields the three most significant classes of algorithms for solving lo problems are pivot ellipsoid and interior point methods because ellipsoid methods are not efficient in practice we will concentrate on the computationally successful simplex and primal dual interior point methods only and summarize the pros and cons of these algorithm classes

criss cross methods a fresh view on pivot algorithms criss cross methods are pivot algorithms that solve linear programming problems in one phase starting with any basic solution the first finite criss cross method was invented by chang terlaky and wang independently unlike the simplex method that follows a monotonic edge path on the feasible region the trace of a criss cross method is neither monotonic with respect to the objective function nor feasibility preserving the main purpose of this paper is to present mathematical ideas and proof techniques behind finite criss cross pivot methods a recent result on the existence of a short admissible pivot path to an optimal basis is given indicating shortest pivot paths from any basis might be indeed short for criss cross type algorithms the origins and the history of criss cross methods are also touched upon

on simplex method with most obtuse angle rule and cosine rule a class of simplex methods for solving linear programming lp problems with cosine pivot rule have been presented in some recent papers in this paper we show that the cosine rule used in this class is equivalent to the most obtuse angle pivot rule proposed by pan 1990 6 the relation between the direct method for lp and the most obtuse angle rule is discussed

deterministic generalized automata a generalized automaton ga is a finite automaton where the single transitions are defined on words rather than on single letters generalized automata were considered by k hashiguchi who proved that the problem of calculating the size of a minimal ga is decidable

sensitivity analysis in linear programming just be careful in this paper we review the topic of sensitivity analysis in linear programming we describe the problems that may occur when using standard software and advocate a framework for performing complete sensitivity analysis three approaches can be incorporated within it one using bases an approach using the optimal partition and one using optimal values we elucidate problems and solutions with an academic example and give results from an implementation of these approaches to a large practical linear programming model of an oil refinery this shows that the approaches are viable and useful in practice

deciding determinism of caterpillar expressions caterpillar expressions have been introduced by bruggemann klein and wood for applications in markup languages caterpillar expressions provide a convenient formalism for specifying the operation of tree walking automata on unranked trees here we give a formal definition of determinism of caterpillar expressions that is based on the language of instruction sequences defined by the expression we show that determinism of caterpillar expressions can be decided in polynomial time

positive solutions of second order differential equations we prove a nonoscillation result for second order nonlinear ordinary differential equations an application to the schrodinger equation is also given c 2004 elsevier ltd all rights reserved

a practical anti cycling procedure for linearly constrained optimization a procedure is described for preventing cycling in active set methods for linearly constrained optimization including the simplex method the key ideas are a limited acceptance of infeasibilities in all variables and maintenance of a working feasibility tolerance that increases over a long sequence of iterations the additional work per iteration is nominal and stalling cannot occur with exact arithmetic the method appears to be reliable based on computational results for the first 53 linear programming problems in thenetlib set

uniform random generation of decomposable structures using floating point arithmetic the em recursive method formalized by nijenhuis and wilf cite niwi78 and systematized by flajolet van cutsem and zimmermann cite flziva94 is extended here to floating point arithmetic the resulting adz method enables one to generate decomposable data structures both labelled or unlabelled uniformly at random in expected o n 1 epsilon time and space after a preprocessing phase of o n 2 epsilon time which reduces to o n 1 epsilon for context free grammars

on the power of tree walking automata tree walking automata twas recently received new attention in the fields of formal languages and databases to achieve a better understanding of their expressiveness we characterize them in terms of transitive closure logic formulas in normal form it is conjectured by engelfriet and hoogeboom that twas cannot define all regular tree languages or equivalently all of monadic second order logic we prove this conjecture for a restricted but powerful class of twas in particular we show that 1 bounded twas that is twas that are only allowed to traverse every edge of the input tree at most once in every direction cannot define all regular languages we then extend this result to a class of twas that can simulate first order logic fo and is capable of expressing properties not definable in fo extended with regular path expressions the latter logic being a valid abstraction of current query languages for xml and semistructured data

on the finiteness of the criss cross method abstract in this short paper we prove the finiteness of the criss cross method by showing a certain binary number of bounded digits associated with each iteration increases monotonically this new proof immediately suggests the possibility of relaxing the pivoting selection in the criss cross method without sacrificing the finiteness

tree walking automata do not recognize all regular languages tree walking automata are a natural sequential model for recognizing tree languages it is well known that every tree language recognized by a tree walking automaton is regular we show that the converse does not hold

the sagitta method for solving linear programs abstract a new non simplex active set method for solving linear programs is presented this method uses a global viewpoint of the problem it starts without any iteration point and its basic characteristic is that the first feasible point determined by the method is generally an optimal solution the computational results obtained for a range space implementation are highly encouraging

enumeration and random generation of accessible automata we present a bijection between the set a n of deterministic and accessible automata with n states on a k letters alphabet and some diagrams which can themselves be represented as partitions of a set of kn 1 elements into n non empty subsets this combinatorial construction shows that the asymptotic order of the cardinality of a n is related to the stirling number knn our bijective approach also yields an efficient random sampler for the uniform distribution of automata with n states its complexity is o n 3 2 using the framework of boltzmann samplers

the d step conjecture and its relatives the d step conjecture arose from an attempt to understand the computational complexity of edge following algorithms for linear programming such as the simplex algorithm it can be stated in terms of diameters of graphs of convex polytopes in terms of the existence of nonrevisiting paths in such graphs in terms of an exchange process for simplicial bases of a vector space and in terms of matrix pivot operations first formulated by w m hirsch in 1957 the conjecture remains unsettled though it has been proved in many special cases and counterexamples have been found for slightly stronger conjectures if the conjecture is false as we believe to be the case then finding a counterexample will be merely a small first step in the line of investigation related to the conjecture this report summarizes what is known about the d step conjecture and its relatives a considerable amount of new material is included but it does not seem to come close to settling the conjecture of special interest is the first example of a polytope that is not vertex decomposable showing that a certain natural approach to the conjecture will not work also significant are the quantitative relations among the lengths of paths associated with various forms of the conjecture

the linear complimentarity problem sufficient matrices and the criss cross method specially structured linear complementarity problems lcps and their solution by the criss cross method are examined the criss cross method is known to be finite for lcps with positive semidefinite bisymmetric matrices and with p matrices it is also a simple finite algorithm for oriented matroid programming problems recently cottle pang and venkateswaran identified the class of column row sufficient matrices they showed that sufficient matrices are a common generalization of p and psd matrices cottle also showed that the principal pivoting method with a clever modification can be applied to row sufficient lcps in this paper the finiteness of the criss cross method for sufficient lcps is proved further it is shown that a matrix is sufficient if and only if the criss cross method processes all the lcps defined by this matrix and all the lcps defined by the transpose of this matrix and any parameter vector

the s monotone index selection rules for pivot algorithms of linear programming abstract in this paper we introduce the concept of s monotone index selection rule for linear programming problems we show that several known anti cycling pivot rules like the minimal index last in first out and the most often selected variable pivot rules are s monotone index selection rules furthermore we show a possible way to define new s monotone pivot rules we prove that several known algorithms like the primal dual simplex mbu simplex algorithms and criss cross algorithm with s monotone pivot rules are finite methods we implemented primal simplex and primal mbu simplex algorithms in matlab using three s monotone index selection rules the minimal index the last in first out lifo and the most often selected variable mosv index selection rule numerical results demonstrate the viability of the above listed s monotone index selection rules in the framework of pivot algorithms

the finite criss cross method for hyperbolic programming in this paper the finite criss cross method is generalized to solve hyperbolic fractional linear programming problems just as in the case of linear or quadratic programming the criss cross method can be initialized with any not necessarily feasible basic solution it is known that if the feasible region of the problem is unbounded then some of the known algorithms fail to solve the problem our criss cross algorithm does not have such drawback finiteness of the procedure is proved under the usual mild assumptions some small numerical examples illustrate the main features of the algorithm and show that our method generates different iterates than other earlier published methods

parametric random generation of deterministic tree automata uniform random generators deliver a simple empirical means to estimate the average complexity of an algorithm we present a general rejection algorithm that generates sequential letter to letter transducers up to isomorphism we also propose an original parametric random generation algorithm to produce sequential letter to letter transducers with a fixed number of transitions we tailor this general scheme to randomly generate deterministic tree walking automata and deterministic top down tree automata we apply our implementation of the generator to the estimation of the average complexity of a deterministic tree walking automata to nondeterministic top down tree automata construction we also implemented

transitive closure logic nested tree walking automata and xpath we study fo mtc first order logic with monadic transitive closure a logical formalism in between fo and mso on trees we characterize the expressive power of fo mtc in terms of nested tree walking automata using the latter we show that fo mtc is strictly less expressive than mso solving an open problem we also present a temporal logic on trees that is expressively complete for fo mtc in the form of an extension of the xml document navigation language xpath with two operators the kleene star for taking the transitive closure of path expressions and a subtree relativisation operator allowing one to restrict attention to a specific subtree while evaluating a subexpression we show that the expressive power of this xpath dialect equals that of fo mtc for boolean unary and binary queries we also investigate the complexity of the automata model as well as the xpath dialect we show that query evaluation be done in polynomial time combined complexity but that emptiness or satisfiability is 2exptime complete

sampling different kinds of acyclic automata using markov chains we propose algorithms that use markov chain techniques to generate acyclic automata uniformly at random we first consider deterministic accessible and acyclic automata then focus on the class of minimal acyclic automata in each case we explain how to define random local transformations that describe an ergodic and symmetric markov chain the distribution of the automaton obtained after t random steps in this markov chain tends to the uniform distribution as t tends to infinity

polynomial expected behavior of a pivoting algorithm for linear complementarity and linear programming problems we show that a particular pivoting algorithm which we call the lexicographic lemke algorithm takes an expected number of steps that is bounded by a quadratic inn when applied to a random linear complementarity problem of dimensionn we present two probabilistic models both requiring some nondegeneracy and sign invariance properties the second distribution is concerned with linear complementarity problems that arise from linear programming in this case we give bounds that are quadratic in the smaller of the two dimensions of the linear programming problem and independent of the larger similar results have been obtained by adler and megiddo

existence for unbounded positive solutions of schrodinger equations in two dimensional exterior domains in this paper by using the fixed point theory under quite general conditions on the nonlinear term we obtain a existence result of unbounded positive solutions of schrodinger equations in two dimensional exterior domains

new variants of the criss cross method for linearly constrained convex quadratic programming abstract in this paper s zhang s eur j oper res 116 1999 607 new and more flexible criss cross type algorithms with lifo and most often selected variable pivot rules are generalized for linearly constrained convex primal dual quadratic programming problems these criss cross type algorithms are different from the one described in klafszky and terlaky math oper und stat ser optim 24 1992 127 even though the finiteness proof of these new criss cross type algorithms is similar to the original one for the algorithm of klafszky and terlaky in the sense that both these proofs are based on the orthogonality theorem more cases have to be considered due to the flexibility of pivot lifo most often selected variable rules which requires a deeper and more careful analysis when the primal dual problem is a linear programming problem no quadratic terms in the objective function the structure of the corresponding linear complementarity problem is simpler i e the matrix of the problem is skew symmetric for such problem pairs our proof of finiteness simplifies to the proof of illes and meszaros yugoslav j oper res 11 2001 17 and provides a new finiteness proof for s zhang s criss cross type algorithms

a phase 1 approach for the generalized simplex algorithm abstract a new simplex variant allowing basis deficiency has recently been proposed to attack the degeneracy 1 as a generalization of the simplex algorithm it uses a phase 1 procedure solving an auxiliary problem with piecewise linear sums of infeasibilities as its objective in this paper we develop another phase 1 approach that only introduces a single artificial variable unlike the former which needs a crash procedure to supply an initial basis the proposed phase 1 is able to get itself started from scratch with an artificial basis having a single column computational results with a set of standard test problems from netlib are also reported

new variants of finite criss cross pivot algorithms for linear programming abstract in this paper we generalize the so called first in last out pivot rule and the most often selected variable pivot rule for the simplex method as proposed in s zhang operations research letters 10 1991 189 192 to the criss cross pivot setting where neither the primal nor the dual feasibility is preserved the finiteness of the new criss cross pivot variants is proven

on extremal behaviors of murty s least index method in this small note we observe some extremal behaviors of murty s least index method for solving linear complementarity problems in particular we show that the expected number of steps for solving murty s exponential example with a random permutation of variable indices is exactly equal ton wheren is the size of the input square matrix

a note on the edmonds fukuda pivoting rule for simplex algorithms abstract the pivot selection rule of edmonds fukuda for simplex algorithms was described by jack edmonds in 1982 at the xith international symposium on mathematical programming also the essential characteristics of the rule namely that it prevents cycling in simplex algorithms was proved by edmonds contrary to other pivot selection rules however it is not obvious that the edmonds fukuda rule maintains feasibility of the solution during the solution process this was stated without proof by edmonds and no proof has hitherto appeared a proof is given in the present note also edmonds proof of the anti cycling property is stated finally the relation to the recursive rule ii method of bland is discussed