Lobachevskii Journal of Mathematics

http://ljm.ksu.ru
ISSN 1818-9962

Vol. 24, 2006, 55–62 \begin{matrix}c \\ ◯ \end{matrix} S.Lahrech, A.Jaddar, A.Ouahab, and A.Mbarki

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

SOME REMARKS ABOUT STRICTLY PSEUDOCONVEX

FUNCTIONS WITH RESPECT TO THE CLARKE-ROCKAFELLAR SUBDIFFERENTIAL

(submitted by A. V. Lapin)

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

strongly RCRS-functions.

Key words and phrases. Strictly pseudoconvex functions, RCRS-functions,

2000 Mathematical Subject Classification. 49J52, 49J50.

by X^{∗} its topological dual, while ⟨., .⟩ is the duality pairing between X and X^{∗}. For x and y in X, the closed segment [x, y] is the set [x, y] = { x+t(y −x); t ∈ [0, 1] }. By [x, y) we denote the set [x, y]\{y}. Given a lower semi-continuous (l.s.c.) function f : X → R∪{+∞} whose domain

domf = { x ∈ X; f(x) < +∞ }

is nonempty. The Clarke-Rockafellar generalized directional derivative f^{↑}(x, v) of f at x ∈ domf along the direction v is defined by:

f^{↑}(x, v) = sup_{ε>0} lim sup_{y→_{f} x, t↘0} inf_{u∈B(v,ε)} t^{−1}[f(y + tu) − f(y)],  (1)

where by y →_{f} x, we mean y → x and f(y) → f(x). Here, by B(v, ε) we denote the open ball centered at v with radius ε. The Clarke-Rockafellar subdifferential of f at x ∈ domf is defined by

∂f(x) = { x^{∗} ∈ X^{∗} : ⟨x^{∗}, v⟩ ≤ f^{↑}(x, v) ∀v ∈ X }.

We adopt the convention ∂f(x) = ∅ when x ∉ domf.

A function f is said to be quasiconvex if for any x, y ∈ X and λ ∈ [0, 1] we have

f(λx + (1 − λ)y) ≤ max{ f(x), f(y) }.  (2)

f is said to be strictly quasiconvex if the inequality (2) is strict when x ≠ 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:

(∃x^{∗} ∈ ∂f(x) : ⟨x^{∗}, y − x⟩ ≥ 0) ⟹ f(x) ≤ f(y).  (3)

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

Theorem 1. Let f : X ↦ R ∪ {+∞} be a l.s.c. function. Consider the propositions:

i) f is pseudoconvex.

ii) f is quasiconvex and (0 ∈ ∂f(x) ⟹ x is a global minimum of f).

Then i) implies ii). If moreover, f is radially continuous, then ii) implies i).

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 → X^{∗} is said to be pseudomonotone if for any x, y ∈ X, we have:

[∃x^{∗} ∈ T (x) : ⟨x^{∗}, y − x⟩ ≥ 0] ⟹ ∀y^{∗} ∈ T (y) : ⟨y^{∗}, y − x⟩ ≥ 0. (4)

We have the following classical result:

Theorem 2. [2, 4, 7, 11] Let f : X → R ∪ {+∞} be a l.s.c. function. Consider the propositions:

i) f is pseudoconvex. ii) ∂f is pseudomonotone.

Then i) implies ii). If moreover, f is radially continuous, then ii) implies i).

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 ↦ R ∪ {+∞} be a l.s.c. function. We say that f is radially Clarke-Rockafellar subdifferentiable if for all x, y ∈ X with x ≠ y, there is x_{0} ∈ (x, y) such that ∂f(x_{0}) ≠ ∅. Recall that an extended-real valued function f : X ↦ R ∪ {+∞} is said to be radially continuous if for all x, y ∈ X f is continuous on [x, y].

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 ↦ R ∪ {+∞} is said to be radially non constant if for all x, y ∈ X with x ≠ y, f ≢ constant on [x, y].

Definition 3. A function f : X ↦ R ∪ {+∞} is said to be strictly pseudoconvex(with respect to the Clarke-Rockafellar subdifferential) if for any different points x, y ∈ X, the following implication holds:

(∃x^{∗} ∈ ∂f(x) : ⟨x^{∗}, y − x⟩ ≥ 0) ⟹ f(x) < f(y).  (5)

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

f(x) = \{ \begin{matrix}\sqrt{| x | −1} & if x ∈] − ∞, −1] ∪ [1, +∞[, \\ 0 & if x ∈ [−1, 1]. \end{matrix}  (6)

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

Theorem 4. Let f : X ↦ R ∪ {+∞} be a l.s.c. function such that f is radially Clarke-Rockafellar subdifferentiable. Consider the following assertions:

i) f is strictly pseudoconvex.

ii) f is strictly quasiconvex and (0 ∈ ∂f(x) ⟹ x is a strict global minimum of f).

Then i) implies ii). If moreover, f is radially continuous, then ii) implies i).

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 [x, y] (x ≠ y) on which f is constant. Let z ∈ (x, y). Then applying the strict pseudoconvexity property on x and z, we deduce

∀z^{∗} ∈ ∂f(z) ⟨z^{∗}, x − z⟩ < 0.

Using the same argument for z and y, we obtain

∀z^{∗} ∈ ∂f(z) ⟨z^{∗}, y − z⟩ < 0.

Therefore,

∀z^{∗} ∈ ∂f(z), ⟨z^{∗}, x − y⟩ < 0 and ⟨z^{∗}, x − y⟩ > 0.

Consequently, for all z ∈ (x, y) we have ∂f(z) = ∅. But this contradicts the fact that f is a RCRS-function. Thus, f is strictly quasiconvex. On the other hand, f is pseudoconvex. Therefore,

0 ∈ ∂f(x) ⟹ x is a strict global minimum of f.

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 ≠ y in X and x^{∗} ∈ ∂f(x) such that

⟨x^{∗}, y − x⟩ ≥ 0 and f(x) ≥ f(y).

Then, It follows by pseudoconvexity property that

f(x) = f(y) and ∀z ∈ [x, y], f(z) ≥ f(x) ≥ f(y).

On the other hand, f is quasiconvex. Therefore,

f(z) = f(x) = f(y), ∀z ∈ [x, y].

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:

Definition 5. A multioperator T : X → X^{∗} is said to be strictly pseudomonotone if for any different points x and y in X, the following implication holds:

∃x^{∗} ∈ T (x) : ⟨x^{∗}, y − x⟩ ≥ 0 ⟹ ∀y^{∗} ∈ T (y) : ⟨y^{∗}, y − x⟩ > 0. (7)

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

Theorem 6. Let f : X ↦ R ∪ {+∞} be a l.s.c. function such that f is radially Clarke-Rockafellar subdifferentiable. Consider the following assertions

i) f is strictly pseudoconvex. ii) ∂f is strictly pseudomonotone.

Then i) implies ii). if moreover, f is radially continuous, then ii) implies i).

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 ∂f is strictly pseudomonotone. Suppose that there exist two different points x, y ∈ X, x^{∗} ∈ ∂f(x) and y^{∗} ∈ ∂f(y) such that

⟨x^{∗}, y − x⟩ ≥ 0 and ⟨y^{∗}, x − y⟩ ≥ 0.

Since f is strictly pseudoconvex, then

f(x) < f(y) and f(x) > f(y).

Contradiction. Thus, ∂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^{∗} ∈ ∂f(x) such that both inequalities

⟨x^{∗}, y − x⟩ ≥ 0 and f(x) ≥ f(y)

hold. Then

⟨x^{∗}, z − x⟩ ≥ 0 ∀z ∈ [x, y].  (8)

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

f(x) ≤ f(z) ∀z ∈ [x, y].

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

⟨z^{∗}, z − x⟩ > 0,  ∀z ∈ (x, y) ∀z∗ ∈ ∂f(z).  (9)

Pick z_{0} ∈ (x, y) such that ∂f(z_{0}) ≠ ∅ (such a z_{0} exists since f is a RCRSfunction). Choose any z_{0}^{∗} ∈ ∂f(z_{0}). Then, ⟨z_{0}^{∗}, z_{0} − x⟩ > 0. Therefore, ⟨z_{0}^{∗}, y − z_{0}⟩ > 0. Consequently, there is ε > 0 such that

⟨z_{0}^{∗}, y′ − z_{0}⟩ > 0 ∀y′ ∈ B(y, ε).

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 ∈ ∂f(z_{0}) 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 ↦ R ∪ {+∞} be a l.s.c. function. We say that f is strongly radially Clarke-Rockafellar subdifferentiable if for all x, y ∈ X with x ≠ y and for all c : f(x) < c < f(y), there is x_{0} ∈ (x, y) such that f(x_{0}) = c and ∂f(x_{0}) ≠ ∅.

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

(P)  max_{x∈C} f(x),

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

Theorem 7. Consider x̄ ∈ C such that

−∞ ≤ inf_{C} f < f(x̄).  (10)

Then x̄ is a maximum of f over C if and only if for all x ∈ C such that f(x) = f(x̄) and all x^{∗} ∈ ∂f(x) we have:

⟨x^{∗}, y − x⟩ < 0 ∀y ∈ C \ {x}.  (11)

Proof. Assume that x̄ is a solution of the problem (P). Let x ∈ X such that f(x) = f(x̄) and let x^{∗} ∈ ∂f(x). Then

f(y) ≤ f(x), ∀y ∈ C.

CLARKE-ROCKAFELLAR SUBDIFFERENTIAL 61

Since f is strictly pseudoconvex, then

⟨x^{∗}, y − x⟩ < 0, ∀y ∈ C \ {x}.

Conversely, suppose that there exists z ∈ C such that f(z) > f(x̄). By the hypotheses, there is z_{0} ∈ C such that f(z_{0}) < f(x̄). Since f is strongly radially Clarke-Rockafellar subdifferentiable, then there is some x_{0} ∈ (z_{0}, z) such that f(x_{0}) = f(x̄) and ∂f(x_{0}) ≠ ∅. Pick any x_{0}^{∗} ∈ ∂f(x_{0}). Then

⟨x_{0}^{∗}, z − x_{0}⟩ < 0 and ⟨x_{0}^{∗}, z_{0} − x_{0}⟩ < 0.

Which is impossible. To prove that (11) holds when x̄ 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 x̄ 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

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62 S.LAHRECH, A.JADDAR, A.OUAHAB, AND A.MBARKI

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