Lobachevskii Journal of Mathematics http://ljm.ksu.ru ISSN 1818-9962 Vol. 24, 2006, 55–62

© S.Lahrech, A.Jaddar, A.Ouahab, and A.Mbarki

Abstract. Using the notion of radially Clarke-Rockafellar subdifferentiable functions (RCRS-functions), we characterize strictly pseudoconvex functions with respect to the Clarke-Rockafellar subdifferential in two different ways, and we study a maximization problem involving RCRS-strictly pseudoconvex functions over a convex set.

1. Introduction

Generalized convexity has proved to be a good tool in the study of some economic problems and in mathematical programming. Strict pseudoconvexity is a kind of generalized convexity that appeared recently as an important part of the class of pseudoconvex functions. The former class has been characterized by many authors (see for instance [1, 2, 4, 7, 10]). In this paper we will refine these results in section 2, using the Clarke Rockafellar subdifferential. While, in section 3 we give a necessary and sufficient condition for a point to be a maximum of a strictly pseudoconvex function over a convex set.

Let us recall some definitions and well known results in connection with what we shall do in the sequel. By $X$ we mean a Banach space and by $X^{\ast }$ its topological dual, while $\langle .,.\rangle$ is the duality pairing between $X$ and $X^{\ast }$. For $x$ and $y$ in $X$, the closed segment $\left[x,y\right]$ is the set $\left[x,y\right]=\left\{x+t\left(y-x\right);t\in \left[0,1\right]\right\}$. By $\left[x,y\right)$ we denote the set $\left[x,y\right]\setminus \left\{y\right\}$. Given a lower semi-continuous (l.s.c.) function $f:X\to R\cup \left\{+\infty \right\}$ whose domain

is nonempty. The Clarke-Rockafellar generalized directional derivative $f^{\uparrow }\left(x,v\right)$ of $f$ at $x\in \mathrm{\text{dom}}f$ along the direction $v$ is defined by:

where by $y{\to }_{f}x$, we mean $y\to x$ and $f\left(y\right)\to f\left(x\right)$. Here, by $B\left(v,\varepsilon \right)$ we denote the open ball centered at $v$ with radius $\varepsilon$. The Clarke-Rockafellar subdifferential of $f$ at $x\in \mathrm{\text{dom}}f$ is defined by

We adopt the convention $\partial f\left(x\right)=\varnothing$ when $x\notin \mathrm{\text{dom}}f$.

A function $f$ is said to be quasiconvex if for any $x,y\in X$ and $\lambda \in \left[0,1\right]$ we have

$f$ is said to be strictly quasiconvex if the inequality (2) is strict when $x\ne y$. $f$ is said to be pseudoconvex(with respect to the Clarke-Rockafellar subdifferential) if for any $x$ and $y$ in $X$ the following implication holds:

The relation between pseudoconvexity and quasiconvexity has been described in [2, 4, 7, 10] by the following result.

Generally, in generalized convexity, there is a close link between the kind of convexity of a function and a corresponding kind of monotonicity of its subdifferential. Recall that a multifunction $T:X\to X^{\ast }$ is said to be pseudomonotone if for any $x,y\in X$, we have:

We have the following classical result:

In this paper, we want to characterize strictly pseudoconvex functions with respect to the Clarke-Rockafellar subdifferential in two different ways. For this, we introduce the so what we call radially Clarke-Rockafellar subdifferentiable functions (RCRS-functions).

Let $f:X\mapsto R\cup \left\{+\infty \right\}$ be a l.s.c. function. We say that $f$ is radially Clarke-Rockafellar subdifferentiable if for all $x,y\in X$ with $x\ne y$, there is $x_0\in \left(x,y\right)$ such that $\partial f\left(x_0\right)\ne \varnothing$. Recall that an extended-real valued function $f:X\mapsto R\cup \left\{+\infty \right\}$ is said to be radially continuous if for all $x,y\in X$ $f$ is continuous on $\left[x,y\right]$.

2. Characterization of RCRS-strict pseudoconvex functions

In this section, we get analogous results to theorem 1 and theorem 2 for RCRS-strictly pseudoconvex functions.

An extended-real valued function $f:X\mapsto R\cup \left\{+\infty \right\}$ is said to be radially non constant if for all $x,y\in X$ with $x\ne y$, $f\not\equiv$ constant on $\left[x,y\right]$.

We can check immediately that a strict pseudoconvex function is pseudoconvex while the converse is not true in general as we can see for example for the function

We can describe the relation between strict pseudoconvexity and strict quasiconvexity via the following result:

Proof. Let $f$ be a strictly pseudoconvex function, then by theorem 1, $f$ is quasiconvex. Let us prove now that $f$ is strictly quasiconvex. Since $f$ is quasiconvex, then according to Diewert [5], it suffices to prove that $f$ is radially non constant. By the contrary, assume that there exists a closed segment $\left[x,y\right]$ $\left(x\ne y\right)$ on which $f$ is constant. Let $z\in \left(x,y\right)$. Then applying the strict pseudoconvexity property on $x$ and $z$, we deduce

Using the same argument for $z$ and $y$, we obtain

Therefore,

Consequently, for all $z\in \left(x,y\right)$ we have $\partial f\left(z\right)=\varnothing$. But this contradicts the fact that $f$ is a RCRS-function. Thus, $f$ is strictly quasiconvex. On the other hand, $f$ is pseudoconvex. Therefore,

Conversely, assume that $f$ satisfies the condition ii) and $f$ is radially continuous. Then by theorem 1, $f$ is pseudoconvex.

Let us prove now that $f$ is strictly pseudoconvex. Assume by contradiction that there exist $x\ne y$ in $X$ and $x^{\ast }\in \partial f\left(x\right)$ such that

Then, It follows by pseudoconvexity property that

On the other hand, $f$ is quasiconvex. Therefore,

Consequently, $f$ is not radially non constant on $X$. But this contradicts the fact that $f$ is strictly quasiconvex. Thus, we achieve the proof.

Analogously to pseudomonotone multioperators, we define strictly pseudomonotone multioperators as follows:

We have also a relation between strict pseudoconvexity of functions and strict monotonicity of their corresponding Clarke-Rockafellar subdifferentials.

Proof. The first implication can be easily proved, nevertheless we include it here for completeness. Assume that $f$ is strictly pseudoconvex. Let us prove by the contrary that $\partial f$ is strictly pseudomonotone. Suppose that there exist two different points $x,y\in X$, $x^{\ast }\in \partial f\left(x\right)$ and $y^{\ast }\in \partial f\left(y\right)$ such that

Since $f$ is strictly pseudoconvex, then

Contradiction. Thus, $\partial f$ is strictly pseudomonotone.

Conversely, assume that $f$ satisfies the condition ii) and $f$ is radially continuous. Let us prove that $f$ is strictly pseudoconvex. By the contrary, assume that there exist two different points $x$ and $y$ in $X$, and $x^{\ast }\in \partial f\left(x\right)$ such that both inequalities

hold. Then

By theorem 2, it follows that $f$ is pseudoconvex. Therefore,

By theorem 1, $f$ is quasiconvex. Consequently, we can easily see that $f$ must be constant on $\left[x,y\right].$ On the other hand, by (8) and the strict pseudomonotonicity of $\partial f$, we have:

Pick $z_0\in \left(x,y\right)$ such that $\partial f\left(z_0\right)\ne \varnothing$ (such a $z_0$ exists since $f$ is a RCRS-function). Choose any $z_0^{\ast }\in \partial f\left(z_0\right)$. Then, $\langle z_0^{\ast },z_0-x\rangle >0$. Therefore, $\langle z_0^{\ast },y-z_0\rangle >0$. Consequently, there is $\varepsilon >0$ such that

By the pseudoconvexity of $f$, it follows that $y$ is a global minimum of $f$. Hence, $z_0$ is also a global minimum of $f$. Thus, $0\in \partial f\left(z_0\right)$ and this is in contradiction with (9).

3. Maxima of strongly RCRS-strict pseudoconvex functions

In this section, we study a maximization problem over a convex set involving a certain class of RCRS-strictly pseudoconvex functions called class of strongly RCRS-strictly pseudoconvex functions.

Let $f:X\mapsto R\cup \left\{+\infty \right\}$ be a l.s.c. function. We say that $f$ is strongly radially Clarke-Rockafellar subdifferentiable if for all $x,y\in X$ with $x\ne y$ and for all $c:f\left(x\right)<c<f\left(y\right)$, there is $x_0\in \left(x,y\right)$ such that $f\left(x_0\right)=c$ and $\partial f\left(x_0\right)\ne \varnothing$.

Let $C$ be a nonempty convex set of $X$. Consider the following maximization problem:

where the function $f$ is assumed to be strictly pseudoconvex, l.s.c. and strongly radially Clarke-Rockafellar subdifferentiable.

Proof. Assume that $\bar{x}$ is a solution of the problem $\left(\mathcal{P}\right)$. Let $x\in X$ such that $f\left(x\right)=f\left(\bar{x}\right)$ and let $x^{\ast }\in \partial f\left(x\right)$. Then

Since $f$ is strictly pseudoconvex, then

Conversely, suppose that there exists $z\in C$ such that $f\left(z\right)>f\left(\bar{x}\right)$. By the hypotheses, there is $z_0\in C$ such that $f\left(z_0\right)<f\left(\bar{x}\right)$. Since $f$ is strongly radially Clarke-Rockafellar subdifferentiable, then there is some $x_0\in \left(z_0,z\right)$ such that $f\left(x_0\right)=f\left(\bar{x}\right)$ and $\partial f\left(x_0\right)\ne \varnothing$. Pick any ${x_0}^{\ast }\in \partial f\left(x_0\right)$. Then

Which is impossible. To prove that (11) holds when $\bar{x}$ is a maximum, we use only the strict pseudoconvexity of $f$, the other conditions that appear in theorem 7 are needed only to prove that (11) implies that $\bar{x}$ is a maximum. This result is a refinement of both theorem 2.1 of [8] where the function was supposed to be convex continuous and of theorem 4.1 of [7] where the function was assumed to be pseudoconvex and radially continuous.

References

Dept of Mathematics, Faculty of science, Mohamed first university, Oujda, Morocco, (Dynamical system and Optimization Group, GAFO Laboratory)

E-mail address: lahrech@sciences.univ-oujda.ac.ma

National School of Managment, Oujda, Morocco, (Dynamical system and Optimization Group, GAFO Laboratory)

E-mail address: jaddar@sciences.univ-oujda.ac.ma

Dept of Mathematics, Faculty of science, Mohamed first university, Oujda, Morocco, (Dynamical system and Optimization Group, GAFO Laboratory)

E-mail address: ouahab@sciences.univ-oujda.ac.ma

National School of Applied Sciences, Oujda, Morocco, (Dynamical system and Optimization Group, GAFO Laboratory)

E-mail address: ambarki@ensa.univ-oujda.ac.ma

Received October 19, 2006