# On Properties of Geodesicη-Preinvex Functions

Authors: I. Ahmad; Akhlad Iqbal; Shahid Ali
Journal: Advances in Operations Research
 (2009)

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## Abstract

The present paper deals with the properties of geodesicη-preinvex functions and their relationships with η-invex functions and strictly geodesic η-preinvex functions. The geodesic η-pre-pseudo-invex and geodesic η-pre-quasi-invex functions on the geodesic invex set are introduced and some of their properties are discussed.


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## Body

## 1. Introduction

In the recent
years, several generalizations have been developed for the classical properties
of convexity. This concept in linear topological spaces relies on the
possibility of connecting any two points of the space by the line segment
between them. In 1981, Hanson [1] introduced the concept of invexity by
generalizing the difference (x−y) in the definition of convex function to any
function η(x,y).
Hanson's initial results inspired a great deal of subsequent work, which has
greatly expanded the role and applications of invexity in nonlinear
optimization and other branches of pure and applied sciences.Ben-Israel and
Mond [2] introduced a new generalization of convex sets and convex functions.
Craven [3] called them invex sets and preinvex functions, respectively.
Jeyakumar [4] studied the properties of preinvex functions and their
role in optimization and mathematical programming.In general, a
manifold is not a linear space, but the extension of concepts and techniques
from linear spaces to Riemannian manifold are natural. Rapcsák [5] and Udrişte
[6] considered a generalization of convexity, called geodesic convexity and
extended many results of convex analysis and optimization theory to Riemannian
manifolds. In this setting, the linear space has been replaced by Riemannian
manifold and the line segment by a geodesic. For more details, readers may see
[5, 6] and the references cited therein.The notion of
invex functions on Riemannian manifold was introduced in [7], however its
generalization has been investigated by Mititelu [8]. Barani and Pouryayevali
[9] introduced the geodesic invex set, geodesic invex function and geodesic
preinvex function on Riemannian manifold with respect to particular maps, and
studied the relations between them.In this paper,
we discuss various concepts, definitions and properties for the functions on
Riemannian manifold. The notion of invexity and its generalization on
Riemannian manifold is recalled in Section2. In Section 3, we discuss some
properties of geodesic η-preinvex functions and their relationships
with η-invex functions. The relationship between
strictly geodesic η-preinvex function and geodesic η-preinvex function is investigated. We also
prove that the composite function is η-invex on the geodesic invex set.The geodesicη-pre-pseudo-invex (p.p.i) and geodesic η-pre-quasi-invex (p.q.i) functions are
introduced and their properties are also discussed in Section 4. The results
obtained here generalize those which are present in the literature.


## 2. Preliminaries

In this
section, we recall some definitions and known results about Riemannian
manifolds, which will be used throughout the paper. For the standard material
on differential geometry, one can consult [10].Suppose that(M,g) is a complete n-dimensional Riemannian
manifold. A subset A of M is called totally convex, if A contains every geodesic γx,y of M,
whose end points x and y belong to A [6].Definition 2.1.
LetM be a complete n-dimensional Riemannian
manifold and A be a totally convex set in M.
A function f:A→R is said to be geodesic convex if for all
geodesic arcs γ:[x,y]→M and all t∈[0,1],
one has(2.1)f(γx,y(t))≤(1−t)f(y)+tf(x),or(2.2)f(γ(y))−f(γ(x))≥d(f(γ))x(y−x)if f is a differentiable function.LetM be an n-dimensional differentiable manifold
and TpM be the tangent space to M at p.
Also, assume that TM=⋃p∈MTpM is the tangent bundle of M.
For any ϵ>0, Iϵ=(−ϵ,ϵ)⊂R.Definition 2.2 (see [7, 8]).
A differentiable
curveα:Iϵ→M is called a differentiable application.
Letα(0)=p∈M.
Then tangent vector to the curve α at p is v=α′(0)∈Tα(0)M=TpM.
Assume thatN is another differentiable manifold and φ:M→N is a differentiable application.Definition 2.3 (see [8]).
The linear applicationdφp:TpM→Tφ(p)N defined by dφp(v)=β′(0),
where v=α′(0)∈TpM and β=φ(α) is called the differential of φ at the point p. But(2.3)β′(0)=φ′(α(0))α′(0)=φ′(p)v.So,(2.4)dφp(v)=φ′(p)v,v∈TpM. The length of a
piecewise C1 curve γ:[a,b]→M is defined by(2.5)L(γ):=∫ab∥γ′(t)∥γ(t)dt.For any two points p,q∈M,
we define(2.6)d(p,q):=inf{L(γ):γisapiecewiseC1curvejoiningptoq}.Then d is a distance which induces the original
topology on M.
We know that on every Riemannian manifold there exists exactly one covariant
derivation called Levi-Civita connection denoted by ∇XY for any vector fields X,Y∈M.Definition 2.4.
A geodesic is aC∞ smooth path γ whose tangent is parallel along the path γ, that is, γ satisfies the equation ∇dγ(t)/dtdγ(t)/dt=0.
Any path γ joining p and q in M such that L(γ)=d(p,q) is a geodesic and it is called a minimal
geodesic.
In other words,
aC∞ curve γ:I→M whose acceleration vector field vanishes
identically is called geodesic [6].We consider now
an applicationη:M×M→TM such that η(p,q)∈TqM for every q∈M and any p∈M.
For a differentiable function f:M→R,
Pini [7] defined invexity as follows.Definition 2.5.
The differentiable
functionf is said to be η-invex on M if for any x,y∈M,(2.7)f(x)−f(y)≥dfy(η(x,y)).
Iff is a differentiable map from the manifold M to the manifold N,
we will denote by dfx the differential of f at x.Mititelu [8]
generalized the above definition as follows.Definition 2.6.
The differentiable
functionf is said to be η-pseudoinvex on M if for any x,y∈M,(2.8)dfy(η(x,y))≥0⇒f(x)≥f(y).Definition 2.7.
The differentiable
functionf is said to be η-quasiinvex on M if for any x,y∈M,(2.9)f(x)≤f(y)⇒dfy(η(x,y))≤0.Remark 2.8.
Iff is a differentiable and η-invex function defined on M and dfy(η(x,y))≥0 for all x,y∈M, then f is η-pseudoinvex on M.In all these
definitions,(2.10)dfy(η(x,y))=[df(y)]η(x,y).If (M,g) is a Riemannian manifold and f is a differentiable map from M to N,
then(2.11)dfy(η(x,y))=gy(gradf(y),η(x,y)),where gradf(y) is the gradient of f at the point y.Barani and
Pouryayevali [9] defined the geodesic invex set and the invexity of a function f on an open geodesic invex subset of a
Riemannian manifold.Definition 2.9.
LetM be a Riemannian manifold and let η:M×M→TM be a function such that for every x,y∈M, η(x,y)∈TyM.
A nonempty subset S of M is said to be geodesic invex set with respect
to η if for every x,y∈S
there exists a unique geodesic γx,y:[0,1]→M such that(2.12)γx,y(0)=y,γx,y′(0)=η(x,y),γx,y(t)∈S,∀t∈[0,1].Definition 2.10.
LetM be a Riemannian manifold, let S be an open subset of M which is geodesic invex with respect to η:M×M→TM, and let f be a real differentiable function defined on S.
Then, f is said to be η-invex on S if for every x,y∈S, (2.13)f(x)−f(y)≥dfy(η(x,y)).  Differentiable
convex functions (on an open convex subset A) are invex. Also, the wider class
of geodetically convex functions on manifolds is included in the class of invex
functions, under some additional hypotheses on manifold [11]. In particular,
this is true if the manifold has the property that given any two points x,y there exists a unique geodesic γx,y(t) joining them. In this case a function which is
geodetically convex is also geodesic invex with respect to η(x,y)=γx,y′(0).


## 3. Some Properties of Geodesicη-Preinvex Functions

The definition
of preinvex function onRn is given in [12]. See also [13, 14] for the
properties of preinvex functions. Barani and Pouryayevali [9] extended this
notion to Riemannian manifolds.Definition 3.1 (see [9]).
LetM be a Riemannian manifold and let  S be an open subset of M which is geodesic invex with respect to η:M×M→TM.
Then, f:S→R is said to be geodesic η-preinvex on S if(3.1)f(γx,y(t))≤tf(x)+(1−t)f(y),for every x,y∈S,t∈[0,1].
If the above inequality is strict then, f is said to be strictly geodesic η-preinvex on S.Theorem 3.2.
LetM be a Riemannian manifold and let S be an open subset of M which is geodesic invex with respect to η:M×M→TM.
Let f:S→R be a geodesic η-preinvex function and let g:I→R be an increasing convex function such that
range (f)⊆I.
Then the composite function g(f) is geodesic η-preinvex on S.Proof.
Sincef is geodesic η-preinvex function, we have(3.2)f(γx,y(t))≤tf(x)+(1−t)f(y).Since g is an increasing and convex function, we get (3.3)g[f(γx,y(t))]≤g[tf(x)+(1−t)f(y)]≤tg(f(x))+(1−t)g(f(y)).  Hence, g(f) is geodesic η-preinvex on S.Theorem 3.3.
LetM be a Riemannian manifold and let S be an open subset of M which is geodesic invex with respect to η:M×M→TM.
Let f:S→R be a geodesic η-preinvex function and let g:I→R be a strictly increasing convex function such
that range (f)⊆I.
Then the composite function g(f) is strictly geodesic η-preinvex on S.Proof.
Sincef is geodesic η-preinvex function, we have(3.4)f(γx,y(t))≤tf(x)+(1−t)f(y).Since g is the strictly increasing and convex
function, we get (3.5)g[f(γx,y(t))]<g[tf(x)+(1−t)f(y)]≤tg(f(x))+(1−t)g(f(y)).   Or (3.6)g[f(γx,y(t))]<tg(f(x))+(1−t)g(f(y)). Hence, g(f) is strictly geodesic η-preinvex on S.Similarly, we
can prove the following results.Theorem 3.4.
LetM be a Riemannian manifold and let S be an open subset of M which is geodesic invex with respect to η:M×M→TM.
Let f:S→R be a geodesic η-preinvex function and let g:I→R be increasing and strictly convex function
such that range (f)⊆I.
Then the composite function g(f) is strictly geodesic η-preinvex on S.Theorem 3.5.
LetS be an open subset of M which is geodesic invex with respect to η:M×M→TM.
Suppose fi:S→R, i=1,2,…,p,
be geodesic η-preinvex. Then(3.7)f=∑i=1pλifi,∀λi∈R,λi≥0,i=1,2,…,p,is geodesic η-preinvex function on S.Now, we prove
the following proposition, which guarantees that a differentiable and geodesicη-preinvex function f is η-invex.Proposition 3.6.
LetM be a complete manifold and S⊆M which is geodesic invex with respect to η:M×M→TM.
Let f:S→R be a differentiable function and there exists
a sequence {tn} of positive real numbers such that tn→0 as n→∞ and(3.8)f(γ(tn))≤tnf(x)+(1−tn)f(y)for every x,y∈S,
then f is η-invex on S.Proof.
We have(3.9)f(γ(tn))−f(y)tn≤f(x)−f(y).Since f is differentiable on S,
taking the limit as n→∞ on both sides, we get(3.10)dfγx,y(0)(γx,y′(0))≤f(x)−f(y).Therefore,(3.11)dfy(η(x,y))≤f(x)−f(y).Hence, the result.It is to be
noted that the converse of above proposition is not true in general. However,
Barani and Pouryayevali [9] proved that a η-invex function on S is geodesic η-preinvex on S if η satisfies the condition C in [9].It is revealed
in the following proposition that like convex functions,η-invex functions are transformed into η-invex functions by a suitable class of
monotone functions.Proposition 3.7.
Letϕ:R→R be a monotone increasing differentiable convex
function. If f is η-invex on geodesic invex set S,
then the composite function ϕ(f) is η-invex.Proof.
Using the fact thatϕ(x+h)≥ϕ(x)+ϕ′(x)h for every x,h∈R,
we have (3.12)ϕ(f(x))≥ϕ(f(y)+dfy(η(x,y)))≥ϕ(f(y))+ϕ′(f(y))dfy(η(x,y))=ϕ(f(y))+d(ϕ(f))yη(x,y)⇒ϕ(f(x))−ϕ(f(y))≥d(ϕ(f))yη(x,y).   Hence ϕ(f) is η-invex on S.


## 4. Properties of Generalized Geodesicη-Preinvex Functions

In [15], Pini introduced the notion of η-pre-pseudo-invex and η-pre-quasi-invex functions on an invex set. We
extend these notions to geodesic η-pre-pseudo-invexity and geodesic η-pre-quasi-invexity on a geodesic invex set S by replacing the line segment with the
geodesic.Letf be a function defined on a geodesic invex
subset S of a Riemannian manifold M with respect to η:M×M→TM.Definition 4.1.
Functionf is said to be geodesic η-pre-pseudo-invex (p.p.i) on S if there exist a geodesic γx,y(t) and a strictly positive function b:S×S→R+ such that(4.1)f(x)<f(y)⇒f(γx,y(t))≤f(y)+t(t−1)b(x,y),for every t∈(0,1) and x,y∈S.Theorem 4.2.
LetM be a Riemannian manifold and S be an open subset of M which is geodesic invex with respect to η:M×M→TM.
If f is geodesic η-preinvex, then f is geodesic η-pre-pseudo-invex for the same geodesic.Proof.
Iff(x)<f(y) for every t∈(0,1) and f is geodesic η-preinvex, then (4.2)f(γx,y(t))≤tf(x)+(1−t)f(y)=f(y)+t(f(x)−f(y))<f(y)+t(f(x)−f(y))−t2(f(x)−f(y))=f(y)+t(t−1)(f(y)−f(x))=f(y)+t(t−1)b(x,y), where b(x,y)=f(y)−f(x)>0.Theorem 4.3.
Letf:S→R be a geodesic η-pre-pseudo-invex function on S and let g:I→R be strictly increasing convex function such
that range (f)⊆I.
Then, the composite function g(f) is geodesic η-pre-pseudo-invex on S.Proof.
Sincef is geodesic η-pre-pseudo-invex function on S,
we have(4.3)f(x)<f(y)⇒f(γx,y(t))≤f(y)+t(t−1)b(x,y),for every t∈(0,1) and x,y∈S,
where b(x,y) is strict positive function.
Sinceg is strictly increasing convex function, we get(4.4)g(f(x))<g(f(y))⇒g(f(γx,y(t)))<g[f(y)+t(t−1)b(x,y)]<g(f(y))+t(t−1)g(b(x,y)) for every t∈(0,1) and x,y∈S, where g(b(x,y)) is strict positive function. Which shows that g(f) is a geodesic η-pre-pseudo-invex function on S.Definition 4.4.
Functionf is said to be geodesic η-pre-quasi-invex (p.q.i) on S if(4.5)f(γx,y(t))≤max(f(x),f(y)),for all x,y∈S and for every t∈[0,1].Now, we
characterize geodesicη-pre-quasi-invex function in terms of its
lower level sets.Theorem 4.5.
LetS be a geodesic invex subset of M and f:S→R.
Then, f is geodesic η-pre-quasi-invex on S if and only if its lower level sets are
geodesic invex.Proof.
Suppose thatf is geodesic η-pre-quasi-invex function on S and C(α):={x:f(x)≤α} is the subset of S.
If C(α) is empty, the result is trivial. If C(α) is neither empty nor the whole set S,
take any two points x and y in C(α).
We have to show that the geodesic γx,y(t) is contained in C(α).
Since f is geodesic η-pre-quasi-invex function, we have(4.6)f(γx,y(t))≤max(f(x),f(y))≤α,for all x,y∈S and for every t∈[0,1].
Hence C(α) is geodesic invex.
Conversely,
suppose that for every real numberα
the set C(α) is geodesic invex. Take any two points x,y∈S and suppose that f(x)≤f(y).
Consider the lower level set C(f(y)).
Since C(α) is geodesic invex, the geodesic γx,y(t) is contained in C(f(y)).
Thus,(4.7)f(γx,y(t))≤f(y)=max(f(x),f(y))for every t∈[0,1].
The proof is complete.Proposition 4.6.
Letf be a geodesic η-pre-quasi-invex function on S.
Then,
(i)
every strict local minimum off is also a strict global minimum;(ii)
the set of all strict global minimum points is
geodesic invex set.Proof.
(i) Let y be a strict local minimum which is not global;
then there exists a point x*∈S,
such that, f(x*)<f(y).
Since, f is geodesic η-pre-quasi-invex, we have f(γx*,y(t))≤f(y),
which contradicts the hypothesis that y is a strict local minimum.
(ii) If f has no minimum value in S,
then the set of minimum points is empty and hence geodesic invex. If f has the minimum point α on S,
then the set of minimum points is S∩C(α),
which is geodesic invex.
The geodesicη-pre-quasi-invexity is preserved under
composition with nondecreasing function ϕ:R→R as can be seen below.Proposition 4.7.
Letf be a geodesic η-pre-quasi-invex function and let  ϕ:R→R be a nondecreasing function. Then, ϕ(f) is geodesic η-pre-quasi-invex.Proof.
Given thatf is a geodesic η-pre-quasi-invex function and ϕ is a nondecreasing function. Then, we have (4.8)(ϕ(f))(γx,y(t))≤ϕ(max(f(x),f(y)))=max(ϕ(f(x)),ϕ(f(y))),  which shows
that the composite function ϕ(f) is geodesic η-pre-quasi-invex.Proposition 4.8.
If the functionf is geodesic η-preinvex on S,
then f is geodesic η-pre-quasi-invex on S.Proof.
Letf be geodesic η-preinvex function on S.
Then for every x,y∈S and t∈[0,1],
it follows that (4.9)f(γx,y(t))≤tf(x)+(1−t)f(y)≤tmax{f(x),f(y)}+(1−t)max{f(x),f(y)}=max{f(x),f(y)},   which shows
that f is geodesic η-pre-quasi-invex on S.Proposition 4.9.
Iff:S→R is geodesic η-pre-pseudo-invex on S then f is geodesic η-pre-quasi-invex on S.Proof.
Letf(x)<f(y).
Since f is geodesic η-pre-pseudo-invex function on S, for all x,y∈S,  and for all t∈(0,1),
we have (4.10)f(γx,y(t))≤f(y)+t(t−1)b(x,y)<f(y)=max{f(x),f(y)}.   Hence, f is geodesic η-pre-quasi-invex on S.



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