1. Introduction

The study of Quasi-convex functions and optimizations, which play important roles in numerous fields including, economics, engineering, management science, operations research, industrial organization, computer vision, curve fitting, and various applied sciences, is several decades old. [1,11,19,22]. The notion of Quasi-convex functions and the characteristics convexity of its level set was first recognized by De Finetti in his work, “Sulle Straficazoni Convesse” in 1949, [9]. Since then, efforts have been focused on this class of functions because of its similar features with convex functions and its wider applications [11,14,13,21]. A quasi-convex optimization problem is a mathematical optimization problem in which the objective is to minimize a quasi-convex function over a convex set. Because every convex function is also quasi-convex, Quasi-convex programs therefore generalize convex programs [1].

The prefix ‘quasi” means “as if”. Thus, we expect quasi-convex functions to possess some special qualities that are similar to those of convex functions. However, while some properties of convex functions and optimizations have analogues equivalence of quasi-convexity some properties do not have. For instance, although the sub-level sets of both convex and quasi-convex functions are convex, quasi-convex functions differ from convex functions in the following ways among others; quasi-convex functions can be discontinuous in the interior of their domain, not every local minimum is a global minimum, local minimum that are not global cannot be strict minima. First order conditions are not sufficient to identify even local optima under quasi-convexity. [11,14].

Many theorems involving the characterizations of quasi-convex functions and optimizations in finite dimensional spaces appear in the literature. One of the most important properties of convex functions is that their level sets are convex. This property is also a fundamental geometric characterization of quasi-convex functions which sometimes is treated as their definition [10,11,14,19,21]. However, the most attractive characterizations of quasi-convex functions are those which involve gradients (a detailed account of the current state of research on the topic can be found in [5]). As to generalized derivatives of quasi-convex functions, very few results exist (see [5,12]). In [12], a study of quasi-convex functions is presented via Clarke's sub-differential, but the authors restricted themselves to the case of Lipschitz functions on a finite dimensional space only.

Interestingly, [21] and independently, [2-4] characterized the lower semi-continuous quasi-convex functions in terms of generalized (Clarke-Rockafellar) sub-differentials and directional derivatives in infinite dimensional spaces with the concept of quasi-monotone maps and prove that a lower semi-continuous function on an infinite dimensional space is quasi-convex if and only if its generalized sub-differential or its directional derivative is quasi-monotone. Although [2-4] and [21] studied the subdifferential characterizations of quasi-convex functions in infinite dimensional spaces, none of the studies covered their optimality conditions in infinite dimensional spaces. [20] did a study on a necessary optimality condition for lower semi-continuous quasi-convex functions on closed convex sets but did not cover the sufficient optimality condition of the problems. He didn’t adopt the variational inequality approach in his study of optimality conditions but rather adopted the normal cone approach in his minimization of the quasi-convex function. Variational inequalities have found many applications in optimization and in order fields of applied, especially in mechanics [15].

Although the condition is known to be necessary optimality condition for existence of a minimizer in quasi-convex programming for some sub-differentials, it is not a sufficient condition. We extend the study of [2-4] and [21] by using some variational inequalities approach to obtain a necessary and sufficient condition for to be either a local minimum or a global minimum.

2. Preliminaries

Let be a Banach space with norm its topological dual for the duality pairing and the value of at For each we define the closed line segment for some for and define and analogously and we denote an open ball centered at with radius by

Given a lower semi-continuous (l.s.c.) function the effective domain is defined by

For a multivalued operator the domain of T is

Definition 2.1. A function is said to be quasi-convex if for each

(1)

This is equivalent to the convexity of the level sets

(2)

is said to be strictly quasi-convex if the inequality (1) is strict when

Definition 2.2. [4] A differentiable function is called quasi-convex if for every

(3)

Definition 2.3 An operator that associates to any l.s.c. function and a point a subset of is a sub-differential if it satisfies the following properties:

(i) whenever is convex;

(ii) whenever is a local minimum of

(iii) , whenever is a real a real-valued convex continuous function which is -differentiable at where -differentiable at means that both and are non-empty. We say that is -differentiable at when is non-empty while are called the sub-gradients of at

The Clarke-Rockafellar general derivative of at in the direction is given by

where indicates the fact that and

and means that both and

The Clarke-Rockafellar subdifferential of is defined by

then

We extend the notion of quasi-convexity to less smooth function using the concept of generalized directional derivatives and sub-differential.

Definition 2.4. [4] A l.s.c function is called quasi-convex (with respect to Clarke-Rockerfeller Subdifferentials) if for any

(4)

Definition 2.5. A multivalued operator is said to be quasi-monotone if

3. Sub-Differential Characterizations of Quasi-Convex Functions

Our aim is to show that if quasi-convex if and only is quasi-monotone. We need the following lemma.

Lemma 3.1. Let with Then, exist and sequence and with

for every with

Proof. By Approximate mean value inequality theorem [3], we can find an and a sequence and verifying

(5)

Letting with it holds

(6)

for sufficiently large.

Theorem 3.2. (Quasi-convexity). is quasi-convex if and only if is quasi-monotone.

Proof. We show that if is not quasi-convex then, is not quasi-monotone.

Suppose that there exist some in with and According Lemma 3.1 applied with and there exists a sequence and such that

(7)

Let be such that and set , so that . Since is lower semi-continuous, we may pick very large with Apply Lemma 3.1 again with and to find sequences such that

(8)

In particular, and

(9)

hence, for sufficiently large. But showing that is not quasi-monotone.

Conversely, we suppose that is quasi-convex and show that is quasi-monotone. Let and with We need to verify that We fix and such that

for all

We fix Since we can find and such that From the quasi-convexity of we deduce that whence,

so that

Combining the inequalities and for any there exists such that

which shows that

4. Optimality Conditions and Variational Inequalities

Let be a multivalued operator, and Recall from [17,18] that, satisfies the variational inequality (10) if and only if

(10)

Let be a lower semi-continuous (l.s.c.) function and consider the minimization problem

(11)

Then, if is a convex open neighborhood of we have the following

Lemma 4.1. If satisfies (10), the following assertions hold.

(i) If then is a global minimum of

(ii) If then is a local minimum of

Proof. It suffices to prove (ii) Suppose by contradiction that is not a solution of (11), then there exist such that By Lemma 3.1, there exist and two sequences with

for any

Since is a convex open neighborhood of then Furthermore, for large enough

For we have

which contradicts (10). Thus, is a local minimum of .

Consider now the quasi-convex minimization problem (11) again,

(12)

where is l.s.c. and quasi-convex, then we have:

Theorem 4.2. If then the following assertions are equivalent:

(i) is an optimal solution of (12).

(ii) satisfies (10).

Proof. Suppose is a strict minimum of (12). Then for all such that we have By Lemma 3.1, there exist and two sequences with

for any where

For we have

Since is quasi-convex, by Theorem 3.2., is quasi-monotone. This implies that

Thus, satisfies the variational inequality (10)

Suppose that is not is a strict minimum of (12) and consider the function defined by

(13)

where it is obvious that is l.s.c., quasi-convex and is a strict local minimum of

Then, we have

Since depends only on the values of in the neighborhood of

When i.e. the interior of we obtain a more precise result

Lemma 4.3. If then satisfies the variational inequality (10) on the whole space and is an optimal solution of (12) with Moreover, is a global minimum of

Proof. Suppose that then

such that

where

Let and consider the linear mapping

By open mapping theorem [4, Pseudo 8], we

Since is quasi-convex, then is quasi-monotone. By Definition 2.1 of [16], the multivalued operator defined by

is quasi-monotone. And then,

for all satisfies (10).

5. Conclusions

We have studied optimizations under a weak condition of convexity, called quasi-convexity in infinite dimensional spaces. Although many theorems involving the characterizations of quasi-convex functions and optimizations in finite dimensional spaces appear in the literature, very few results exist on the characterizations of quasi-convex functions in infinite dimensional spaces which involve a generalized derivatives of quasi-convex functions. Although the condition is known to be necessary optimality condition for existence of a minimizer in quasi-convex programming for some sub-differentials, it is not a sufficient condition. This study is an extension of the study of [2-4] and [21] by using some variational inequalities approach instead of the normal cone approach to obtain a necessary and sufficient condition for to be either a local minimum or a global minimum.