Sub-Differential Characterizations of Non-Smooth Lower Semi-Continuous Pseudo-Convex Functions on Real Banach Spaces

Akachukwu Offia^1*, Ugochukwu Osisiogu², Theresa Efor², Friday Oyakhire¹, Monday Ekhator¹, Friday Nkume¹, Sunday Aloke³

¹Department of Mathematics and Statistics, Alex Ekwueme Federal University, Ndufu-Alike, Nigeria.
²Department of Industrial Mathematics and Applied Statistics, Ebonyi State University, Abakaliki, Nigeria.
³Department of Industrial Mathematics, David Umahi Federal University of Health Sciences, Uburu, Nigeria.

DOI: 10.4236/ojop.2023.123007                

Abstract

In this paper, we characterize lower semi-continuous pseudo-convex functions f : X → R ∪ {+ ∞} on convex subset of real Banach spaces K  ⊂ X with respect to the pseudo-monotonicity of its Clarke-Rockafellar Sub-differential. We extend the results on the characterizations of non-smooth convex functions f : X → R ∪ {+ ∞} on convex subset of real Banach spaces K  ⊂ X with respect to the monotonicity of its sub-differentials to the lower semi-continuous pseudo-convex functions on real Banach spaces.

Keywords

Real Banach Spaces, Pseudo-Convex Functions, Pseudo-Monotone Maps, Sub-Differentials, Lower Semi-Continuous Functions and Approximate Mean Value Inequality

1. Introduction

Convex optimization, which studies the problem of minimizing convex functions over convex sets, plays important roles in many branches of applied mathematics. The foremost reason is that; it is very suitable to extremum problems. For instance, some necessary conditions for the existence of a minimum also become sufficient in the in terms of convexity. And convex optimization can be a smooth or a non-smooth convex optimization. Since the concept of convexity does not satisfy some mathematical models, various generalizations of convexity such as quasi-convexity and pseudo-convexity, which retain some important properties of convexity and equally provide a better representation of reality, were introduced in the literature to fill these gaps.

While the quasi-convexity property of a function guarantees the convexity of their sublevel sets, the pseudo-convexity property implies that the critical points are minimizers [1] . One of the features of convexity of functions is the relationship it has with the monotonicity of some maps. For example, a differentiable function is said to be convex if and only if its gradient is a monotone map. In non-smooth analysis, the generalized convexity of functions can be equally characterized in terms of the generalized monotonicity of their related operators [2] .

The concepts of pseudo-convexity, traced to [3] , within his research on analytical functions and independently introduced into the field of optimization by [4] , have many applications in mathematical programming and economic problems [5] [6] [7] . And pseudo-monotonicity, introduced by [8] as a generalization of monotone operators, has been used to describe a property of consumer’s demand correspondence [9] . Although the simplest class of pseudo-monotone operators consists of gradients of pseudo-convex functions, there are some monotone operators that are not sub-differentials [9] . And generalized monotonicity of maps is frequently used in complementarity problems, equilibrium problems and variational inequalities [10] .

2. Preliminaries

Let X be a real Banach space with norm $\Vert .\Vert$ , $X^{*}$ be its topological dual and $\langle x^{*},x\rangle$ be the duality pairing between $x\in X$ and $x^{*}\in X^{*}$ . We denote the closed segment $\left[x,y\right]=\left\{\lambda x+\left(1-\lambda \right)y:\lambda \in \left[0,1\right]\right\}$ for $x,y\in X$ , and define $\left(x,y\right]$ , $\left[x,y\right)$ and $\left(x,y\right)$ similarly.

Definition 2.1 [11] Let $f:X\to ℝ\cup \left\{+\infty \right\}$ be an extended real valued function, the effective domain is defined by

$\text{dom}\left(f\right)=\left\{x\in X:f\left(x\right)<+\infty \right\}$ .

Definition 2.2 [12] A function $f:X\to ℝ\cup \left\{+\infty \right\}$ is said to be lower semi-continuous at $x\in X$ if and only if: $\forall \lambda \in ℝ$ , such that $\lambda <f\left(x\right)$ , $\exists V\subset U\left(x\right)$ : $\lambda <f\left(y\right)$ $\forall y\in V$ .

Definition 2.3 [7] [13] A lower semi-continuous function $f:X\to ℝ\cup \left\{+\infty \right\}$ is said to be quasi-convex, if for any $x,y\in X$ and $z\in \left[x,y\right]$ we have

$f\left(z\right)\le \max \left\{f\left(x\right),f\left(y\right)\right\}$ . (1)

Definition 2.4 [7] [13] [14] A lower semi-continuous function $f:X\to ℝ\cup \left\{+\infty \right\}$ is said to be strictly quasi-convex, if the inequality (1) is strict when $x\ne y$ .

Definition 2.5 [2] Let $T:X\to X^{*}$ be a multivalued operator with domain $D\left(T\right)=\left\{x\in X:T\left(x\right)\ne \varnothing \right\}$ . T is said to be quasi-monotone if for any $x,y\in X$ , $x^{*}\in T$ and $y^{*}\in T\left(y\right)$ , we have

$\langle x^{*},y-x\rangle >0\Rightarrow \langle y^{*},y-x\rangle \ge 0.$

Definition 2.6 [2] Let $T:X\to X^{*}$ be a multivalued operator with domain $D\left(T\right)=\left\{x\in X:T\left(x\right)\ne \varnothing \right\}$ . T is said to be pseudo-monotone if for any $x,y\in X$ , $x^{*}\in T$ and $y^{*}\in T\left(y\right)$ , we have

$\langle x^{*},y-x\rangle \ge 0\Rightarrow \langle y^{*},y-x\rangle \ge 0.$ (2)

Definition 2.7 [7] Let $T:X\to X^{*}$ be a multivalued operator with domain $D\left(T\right)=\left\{x\in X:T\left(x\right)\ne \varnothing \right\}$ . T is said to be strictly pseudo-monotone if for any different two points $x,y\in X$ , $x^{*}\in T$ and $y^{*}\in T\left(y\right)$ , we have

$\langle x^{*},y-x\rangle \ge 0\Rightarrow \langle y^{*},y-x\rangle >0.$ (3)

Definition 2.8 [15] An operator $\partial$ that associates to any lower semi-continuous function $f:X\to ℝ\cup \left\{+\infty \right\}$ and a point $x\in X$ a subset $\partial f\left(x\right)$ of $X^{*}$ is a sub-differential if it satisfies the following properties:

1) $\partial f\left(x\right)=\left\{x^{*}\in X^{*}:\langle x^{*},y-x\rangle +f\left(x\right)\le f\left(y\right),\forall y\in X\right\}$ , whenever f is convex;

2) $0\in \partial f\left(x\right)$ , whenever $x\in \text{dom}f$ is a local minimum of f;

3) $\partial \left(f+g\right)\left(x\right)\subset \partial f\left(x\right)+\partial g\left(x\right)$ , whenever g is a real a real-valued convex continuous function which is $\partial$ -differentiable at x.

Where g-differentiable at x means that both $\partial g\left(x\right)$ and $\partial \left(-g\right)\left(x\right)$ are non-empty. We say that f is $\partial$ -differentiable at x when $\partial f\left(x\right)$ is non-empty while $\partial f\left(x\right)$ are called the sub-gradients of f at x.

Definition 2.9 [15] The Clarke-Rockafellar generalized directional derivative of f at $x_0\in \text{dom}\left(f\right)$ in the direction $d\in X$ is given by

$f^{\uparrow }\left(x_0,d\right)={\sup }_{\epsilon >0}\lim {\sup }_{\begin{array}{c}x\to f^{x_0}\\ \lambda \searrow 0\end{array}}{\inf }_{d^{\prime }\in B_{\epsilon }\left(d\right)}\frac{f\left(x+\lambda d^{\prime }\right)-f\left(x\right)}{\lambda }$ , (4)

where $B_{\epsilon }\left(d\right)=\left\{d^{\prime }\in X:\Vert d^{\prime }-d\Vert <\epsilon \right\}$ , $\lambda \searrow 0$ indicates the fact that $\lambda >0$ and $\lambda \to 0$ , and $x\to f^{x_0}$ means that both $x\to x_0$ and $f\left(x\right)\to f\left(x_0\right)$ ;

While,

Definition 2.10 [15] The Clarke-Rockafellar sub-differential of f at $x_0$ is defined by

$\partial f\left(x_0\right)=\left\{x^{*}\in X^{*}:\left(x^{*},d\right)\le f^{\uparrow }\left(x_0,d\right),\forall d\in X\right\}$ ; (5)

if $x_0\in X\backslash \text{dom}\left(f\right)$ , then

$\partial f\left(x_0\right)=\varnothing$ , [7] .

Definition 2.11 [2] [7] A lower semi-continuous function $f:X\to ℝ\cup \left\{+\infty \right\}$ is said to be quasi-convex (with respect to Clarke-Rockerfeller Sub-differentials) if for any $x,y\in X$ ,

$\exists x^{*}\in \partial f\left(x\right):\langle x^{*},y-x\rangle >0\Rightarrow \forall z\in \left[x,y\right]$ , $f\left(z\right)\le f\left(y\right)$ . (6)

Definition 2.12 [7] [16] A lower semi-continuous function $f:X\to ℝ\cup \left\{+\infty \right\}$ is said to be pseudo-convex (with respect to Clarke-Rockerfeller Subdifferentials) if for any $x,y\in X$ :

$\exists x^{*}\in \partial f\left(x\right):\langle x^{*},y-x\rangle \ge 0\Rightarrow f\left(x\right)\le f\left(y\right)$ . (7)

Definition 2.13 [2] [7] [14] A lower semi-continuous function $f:X\to ℝ\cup \left\{+\infty \right\}$ is said to be strictly pseudo-convex (with respect to Clarke-Rockerfeller Subdifferentials) if for any two different points $x,y\in X$ :

$\exists x^{*}\in \partial f\left(x\right):\langle x^{*},y-x\rangle \ge 0\Rightarrow f\left(x\right)<f\left(y\right)$ , when $x\ne y$ . (8)

Definition 2.14 [7] A lower semi-continuous function $f:X\to ℝ\cup \left\{+\infty \right\}$ is said to be radially continuous if for all $x,y\in X$ , f is continuous on $\left[x,y\right]$ .

Definition 2.15 [7] A function $f:X\to ℝ\cup \left\{+\infty \right\}$ is said to be radially non-constant if for all $x,y\in X$ , with $x\ne y$ , $f\cancel{\equiv }\text{constant}$ on $\left[x,y\right]$ .

Definition 2.16 A sub-differential operator $\partial f:X\to X^{*}$ is said to said to be quasi-monotone if for any $x,y\in X$ , $x^{*}\in \partial f\left(x\right)$ and $y^{*}\in \partial f\left(y\right)$ , we have

$\langle x^{*},y-x\rangle >0\Rightarrow \langle y^{*},y-x\rangle \ge 0.$ (9)

Definition 2.17 A sub-differential operator $\partial f:X\to X^{*}$ is said to said to be quasi-monotone if for any $x,y\in X$ , $x^{*}\in \partial f\left(x\right)$ and $y^{*}\in \partial f\left(y\right)$ , we have

$\langle x^{*},y-x\rangle \ge 0\Rightarrow \langle y^{*},y-x\rangle \ge 0.$ (10)

Theorem 2.1. (Approximate mean value inequality). Let $f:X\to ℝ\cup \left\{+\infty \right\}$ be a Clarke-Rockafellar sub-differentiable lower semi-continuous (l.s.c.) function on a Banach space X. Let $a,b\in X$ with $a\in \text{dom}f$ and $a\ne b$ . Let $\rho \in ℝ$ be such that $\rho \le f\left(b\right)$ . Then, there exist $c\in \left[a,b\right)$ and $x_n\to f^C$ and $x_n^{*}\in \partial f\left(x_n\right)$ such that

1) $\underset{n\to +\infty }{\lim \inf }\langle x_n^{*},c-x_n\rangle \ge 0$ ;

2) $\underset{n\to +\infty }{\lim \inf }\langle x_n^{*},b-a\rangle \ge \rho -f\left(a\right)$ .

Proof. [15] .

Lemma 2.2. Let $f:X\to ℝ\cup \left\{+\infty \right\}$ be a Clarke-Rockafeller sub-differentiable lower semi-continuous (l.s.c.) function on a Banach space X. Let $a,b\in X$ with $f\left(a\right)<f\left(b\right)$ . Then, there exist $c\in \left[a,b\right)$ , and two sequences $c_n\to c$ , and $c_n^{*}\in \partial f\left(c_n\right)$ with

$\langle c_n^{*},x-c_n\rangle >0$ for every $x=c+\lambda \left(b-a\right)$ with $\lambda >0$ .

Proof. By Theorem 2.1, there exists an $x_0\in \left[a,b\right)$ and a sequence $x_n\to f^C$ and $x_n^{*}\in \partial f\left(x_n\right)$ verifying

$\underset{n\to +\infty }{\lim \inf }\langle x_n^{*},c-x_n\rangle \ge 0$ and $\underset{n\to +\infty }{\lim \inf }\langle x_n^{*},b-a\rangle >0$ . (11)

Putting $x=c+\lambda \left(b-a\right)$ with $\lambda >0$ it holds

$\langle x_n^{*},x-x_n\rangle =\langle x_n^{*},c-x_n\rangle +\lambda \langle x_n^{*},b-a\rangle >0$ (12)

for n very large. $■$

We consider the relationship between pseudo-convexity and quasi-convexity.

Theorem 2.3. Let $f:X\to ℝ\cup \left\{+\infty \right\}$ be a lower semi-continuous (l.s.c.) Clarke-Rockafeller subdifferentiable function on a Banach space X. Then, f is quasi-convex if and only if $\partial f$ is quasi-monotone.

Proof. We show that if f is not quasi-convex, then $\partial f$ is not quasi-monotone.

Suppose that there exist some $x,y,z$ in X with $z\in \left[x,y\right]$ and $f\left(z\right)>\max \left\{f\left(x\right),f\left(y\right)\right\}$ . According to Lemma 2.2 applied with $a=x$ and $b=z$ , there exists a sequence $y_n\in \text{dom}\partial f$ and $y_n^{*}\in \partial f\left(y_n\right)$ such that

$y_n\to \bar{y}\in \left[x,z\right]$ , $\bar{y}\ne z$ and $\langle y_n^{*},y-y_n\rangle >0$ . (13)

Let $0<\lambda \le 1$ be such that $z=\bar{y}+\lambda \left(y-\bar{y}\right)$ and set $z_n=y_n+\lambda \left(y-y_n\right)$ , so that $z_n\to z$ . Since f is lower semi-continuous, we may pick $n\in \mathbb{N}$ very large with $f\left(z_n\right)>f\left(y\right)$ . Apply Lemma 2.2 again with $a=y$ and $b=z_n$ to find sequences $x_k\in \text{dom}\partial f$ , $x_k^{*}\in \partial f\left(x_k\right)$ such that

$x_k\to \bar{x}\in \left[y,z_n\right]$ , $\bar{x}\ne z_n$ and $\langle x_k^{*},y_n-x_k\rangle >0$ . (14)

In particular, $\bar{x}\ne y_n$ and

$\langle y_n^{*},\bar{x}-y_n\rangle =\frac{\Vert \bar{x}-y_n\Vert }{\Vert y-y_n\Vert }\langle y_n^{*},y-y_n\rangle >0$ ; (15)

hence, $\langle y_n^{*},x_k-y_n\rangle >0$ for k sufficiently large. But $\langle y_n^{*},y_n-x_k\rangle >0$ , showing that $\partial f$ is not quasi-monotone.

Conversely, we suppose that f is quasi-convex and show that $\partial f$ is quasi-monotone. Let $x^{*}\in \partial f\left(x\right)$ and $y^{*}\in \partial f\left(y\right)$ with $\langle x^{*},y-x\rangle >0$ . We need to verify that $f^{\uparrow }\left(y,x-y\right)\le 0$ . We fix $\epsilon >0$ and $\omega \in \left(0,\epsilon \right)$ such that $\langle x^{*},v-x\rangle >0$ for all $v\in B_{\omega }\left(y\right)$ .

We fix $v\in B_{\omega }\left(y\right)$ . Since $f^{\uparrow }\left(y,x-y\right)>0$ we can find ${\epsilon }^{\prime }\in \left(0,\epsilon -\omega \right)$ , $u\in B_{{\epsilon }^{\prime }}\left(x\right)$ and $t\in \left(0,1\right)$ such that $f\left(u+t\left(v-u\right)\right)>f\left(u\right)$ . From the quasi-convexity of f we deduce that $f\left(u\right)<f\left(v\right)$ , whence,

$f\left(v+\lambda \left(u-v\right)\right)\le f\left(v\right)$ for all $\lambda \in \left(0,1\right)$ ,

so that

${\inf }_{\mu \in B_{\epsilon }\left(x-y\right)}\frac{f\left(v+\lambda \mu \right)-f\left(v\right)}{\lambda }\le \frac{f\left(v+\lambda \left(u-v\right)\right)-f\left(v\right)}{\lambda }\le 0$ for all $\lambda \in \left(0,1\right)$ .

Combining the inequalities and for any $\epsilon >0$ there exists $\omega >0$ such that

$\underset{\underset{\lambda \in \left(0,1\right)}{v\in B_{\omega }\left(y\right)}}{\sup }\left[{\inf }_{\mu \in B_{\epsilon }\left(x-y\right)}\frac{f\left(v+\lambda \mu \right)-f\left(v\right)}{\lambda }\right]\le 0$ ,

which shows that $f^{\uparrow }\left(y,x-y\right)\le 0$ . $■$

3. Sub-Differential Characterization of Pseudo-Convex Functions

Theorem 3.1. Let $f:X\to ℝ\cup \left\{+\infty \right\}$ be a lower semi-continuous (l.s.c.) function on a Banach space X such that f Clarke-Rockafeller suddifferentiable. Consider the following assertions:

(i) $f$ is pseudoconvex.

(ii) $f$ is quasiconvex and ( $0\in \partial f\left(x\right)\Rightarrow x$ is a global minimum of f).

Then, (i) implies (ii). And (ii) implies (i) if $f$ is radially continuous.

Proof. (i) $\Rightarrow$ (ii). We want to prove that $f$ is quasiconvex. Suppose to the contrary that for some $x,y\in X$ , $z\in \left(x,y\right)$ we have $f\left(z\right)>\max \left\{f\left(x\right),f\left(y\right)\right\}$ . Since f is lower semicontinuous, we can find some $\epsilon >0$ such that $f\left(z^{\prime }\right)>\max \left\{f\left(x\right),f\left(y\right)\right\}$ , for all $z^{\prime }\in B_{\epsilon }\left(z\right)$ . Since z cannot be a local nor global minimizers, there exist some $v\in B_{\epsilon }\left(z\right)$ such that $f\left(v\right)<f\left(z\right)$ . From Lemma 2.2, there exist $u_n\to u\in \left[v,z\right)$ and $u_n^{*}\in \partial f\left(u_n^{*}\right)$ such that

$\langle u_n^{*},z-u_n\rangle >0$ .

But since $z\in \left(x,y\right)$ , either of the following must hold

$\langle u_n^{*},x-u_n\rangle >0$ or $\langle u_n^{*},y-u_n\rangle >0$ .

Therefore,

$f\left(u_n\right)\le \max \left\{f\left(x\right),f\left(y\right)\right\}$ .

which is a contradiction.

(ii) $\Rightarrow$ (i). Let $x\in \text{dom}\partial f$ , $y\in X$ , and $x^{*}\in \partial f\left(x\right)$ such that $\langle x^{*},y-x\rangle \ge 0$ . If $0\in \partial f\left(x\right)$ , then x is a global minimum of f and $f\left(x\right)\le f\left(y\right)$ in particular. Otherwise, $\left[0\notin \partial f\left(x\right)\right]$ , there exist $d\in X$ such that $\langle x^{*},d\rangle >0$ . We define a sequence $\left\{y_n\right\}$ by

$y_n=y+\left(\frac{1}{2n\Vert d\Vert }\right)d.$

For every $n\in \mathbb{N}$ , the point $y_n$ satisfies

$y_n\in B_{1/n}\left(y\right)$ ,

$\langle x^{*},y_n-x\rangle =\langle x^{*},y_n-y\rangle +\langle x^{*},y-x\rangle \ge \left(\frac{1}{2n\Vert d\Vert }\right)\langle x^{*},d\rangle >0.$

Using (7), we obtain that, for every n, $f\left(y_n\right)\ge f\left(x\right)$ and by radial continuity of f, $f\left(y\right)\ge f\left(x\right)$ . $■$

Theorem 3.2. Let $f:X\to ℝ\cup \left\{+\infty \right\}$ be a lower semi-continuous (l.s.c.) Clarke-Rockafeller sub-differentiable function. Consider the following assertions:

(i) $f$ is pseudo-convex.

(ii) $\partial f$ is pseudo-monotone

Then, (i) implies (ii). And (ii) implies (i) if $f$ is radially continuous.

Proof. (i) $\Rightarrow$ (ii). Suppose $x^{*}\in \partial f\left(x\right)$ such that $\langle x^{*},y-x\rangle \ge 0$ . By Theorem 3.1, $f$ is quasi-convex. By Theorem 2.3, we conclude that $\partial f$ is quasi-monotone. Hence, $\langle y^{*},y-x\rangle \ge 0$ , for all $y^{*}\in \partial f\left(y\right)$ . Suppose to the contrary that for some $y^{*}\in \partial f\left(y\right)$ , we have $\langle y^{*},y-x\rangle =0$ . From (7), we obtain $f\left(x\right)\ge f\left(y\right)$ .

However, since $f^{\uparrow }\left(x,y-x\right)>0$ , there exist $\epsilon >0$ , such that for some $x_n\to x$ , ${\lambda }_n\searrow 0$ and for all $y^{\prime }\in B_{\epsilon }\left(y\right)$ , we have $f\left(x_n+t_n\left(y^{\prime }-x_n\right)\right)>f\left(x_n\right)$ . By the quasiconvexity of f, it implies that $f\left(y^{\prime }\right)>f\left(x_n\right)$ for every $y^{\prime }\in B_{\epsilon }\left(y\right)$ . In particular, $f\left(y\right)>f\left(x\right)$ because f is lower semicontinuous. Thus, $f\left(y^{\prime }\right)\ge f\left(y\right)$ . This shows that y is a local minimum and also a global minimum, which is a contradiction since we can have that $f\left(y\right)>f\left(x_n\right)$ .

(ii) $\Rightarrow$ (i). Using Theorem, we prove that $f$ is pseudoconvex. Since $\partial f$ is pseudomonotone, $\partial f$ is quasimonotone. By Theorem 3.1, $f$ is quasi-convex. On the other hand, if x is not a minimizer of f, there exists $y\in X$ such that $f\left(y\right)<f\left(x\right)$ . Using Lemma 2.2, we find $u\in \text{dom}\partial f$ and $u^{*}\in \partial f\left(u\right)$ such that $\langle u^{*},x-u\rangle >0$ and by the pseudo-monotonicity of $\partial f$ , $\langle x^{*},x-u\rangle >0$ for every $x^{*}\in \partial f\left(x\right)$ . Hence, 0 does not $\partial f\left(x\right)$ . Consequently, f satisfies condition $0\in \partial f\left(x\right)$ , which implies that x is a global minimum of f, which completes the proof. $■$

Theorem 3.3. Let $f:X\to ℝ\cup \left\{+\infty \right\}$ be a lower semi-continuous (l.s.c.) Clarke-Rockafeller subdifferentiable function on a Banach space X. Consider the following assertions:

(i) $f$ is strictly pseudoconvex.

(ii) $f$ is strictly quasiconvex and ( $0\in \partial f\left(x\right)\Rightarrow x$ is a global minimum of f),

Then, (i) implies (ii). And (ii) implies (i) if $f$ is radially continuous.

Proof. (i) $\Rightarrow$ (ii). We want to prove that $f$ is strictly quasiconvex. Let f be a strictly pseudo-convex function, then by Theorem 3.1, the function $f$ is quasiconvex and satisfies the optimality condition

$0\in \partial f\left(x\right)\Rightarrow$ (x is a global minimum of f).

Since $f$ is quasiconvex, then according to [13] , it suffices to prove that $f$ is radially non-constant. Assume by contradiction that there exists a closed segment $\left[x,y\right]$ with $x\ne y$ where with $f$ is constant. Let $z\in \left(x,y\right)$ and apply the strict pseudo-convexityproperty to x and z, then

$f\left(z\right)=f\left(x\right)\Rightarrow \left(\forall z^{*}\in \partial f\left(z\right):\langle z^{*},x-z\rangle <0\right)$ .

Using the same argument for z and y we obtain

$f\left(z\right)=f\left(y\right)\Rightarrow \left(\forall z^{*}\in \partial f\left(z\right):\langle z^{*},y-z\rangle <0\right)$ .

Since $\partial f\left(z\right)$ is nonempty, it follows that for all $z^{*}\in \partial f\left(z\right)$ , $\langle z^{*},x-y\rangle <0$ and $\langle z^{*},x-y\rangle >0$ ), which is a contradiction.

(ii) $\Rightarrow$ (i). Assume that f satisfies condition ii) and $f$ is radially continuous. Then by Theorem 3.1, $f$ is pseudoconvex. We prove that $f$ is pseudo-convex. Suppose by contradiction that there exist $x\ne y$ in X and $x^{*}\in \partial f\left(x\right)$ such that

$\langle x^{*},y-x\rangle \ge 0$ and $f\left(x\right)\ge f\left(y\right)$ .

Then, it follows by pseudo-convexity property that

$\forall z\in \left[x,y\right]$ , $f\left(z\right)=f\left(x\right)$ .

Since $f$ is quasi-convex, then we have

$\forall z\in \left[x,y\right]$ , $f\left(z\right)\ge f\left(x\right)\ge f\left(y\right)$ .

So $f$ is not radially non-constant on X (since $f$ is constant on $\left[x,y\right]$ ) which contradicts the fact $f$ is strictly quasi-convex.

Theorem 3.4. Let $f:X\to ℝ\cup \left\{+\infty \right\}$ be a lower semi-continuous (l.s.c.) function such that $f$ is radially Clarke-Rockafeller differentiable. Consider the following assertions:

(i) $f$ is strictly pseudo-convex.

(ii) $\partial f$ is strictly pseudomonotone

Then, (i) implies (ii). And (ii) implies (i) if $f$ is radially continuous.

Proof. (i) $\Rightarrow$ (ii). Suppose that $f$ is strictly pseudoconvex. We want to prove that $\partial f$ is strictly pseudomonotone. Suppose to the contrary that there exist two distinct points $x,y\in X$ , $x^{*}\in \partial f\left(x\right)$ and $y^{*}\in \partial f\left(y\right)$ such that

$\langle x^{*},y-x\rangle \ge 0$ and $\langle y^{*},y-x\rangle \le 0$ .

Since $f$ is strictly pseudoconvex, we have that

$f\left(x\right)<f\left(y\right)$ and $f\left(y\right)<f\left(x\right)$ .

Which is a contradiction. Therefore, $\partial f$ is strictly pseudomonotone.

(ii) $\Rightarrow$ (i). Suppose that f satisfies condition (ii) and $f$ is radially continuous. We want to prove that $f$ is strictly pseudoconvex. Suppose to the contrary that there exist two distinct points $x,y\in X$ , and $x^{*}\in \partial f\left(x\right)$ such that

$\langle x^{*},y-x\rangle \ge 0$ and $f\left(x\right)\ge f\left(y\right)$ .

Then,

$\langle x^{*},z-x\rangle \ge 0$ for all $z\in \left[x,y\right]$ . (16)

By theorem 3.2, $f$ is quasiconvex. Consequently, f must be constant on $\left[x,y\right]$ . Contrarily, from (15) and the strict monotonicity of $\partial f\left(x\right)$ , we have

$\langle x^{*},z-x\rangle >0,\forall z\in \left(x,y\right)$ and $\forall z^{*}\in \partial f\left(z\right)$ . (17)

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 radially Clarke-Rockafeller subdifferentiable function). Choose any $z_0^{*}\in \partial f\left(z_0\right)$ . Then, $\langle z_0^{*},z_0-x\rangle >0$ . Therefore, $\langle z_0^{*},y-z_0\rangle >0$ . Consequently, there exist $\epsilon >0$ such that

$\langle z_0^{*},y^{\prime }-z_0\rangle >0$ for all $y^{\prime }\in B_{\epsilon }\left(y\right)$ .

By the pseudo-convexity 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 a contradiction with (17).

4. Conclusion

We extended the relationships between convex functions and corresponding monotone maps to pseudo-convexity and the corresponding pseudo-monotonicity of their sub-differentiable maps. We characterized the lower semi-continuous Clarke-Rockafeller sub-differentiable pseudo-convex functions by the corresponding monotonicity of their Clarke-Rockafeller sub-differentials $\partial f$ , and have shown that if a lower semi-continuous Clarke-Rockafeller sub-differentiable function $f:X\to$ $ℝ\cup \left\{+\infty \right\}$ is radially continuous, then f is pseudo-convex if and only if the sub-differential map $\partial f$ is pseudo-monotone.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

References

[1]	Khanh, P.D. and Phat, V.T. (2020) Second-Order Characterizations of Quasiconvexity and Pseudoconvexity for Differentiable Functions with Lipschitzian Derivatives. Optimization Letters, 14, 2413-2427.
[2]	Daniilidis, A. and Hadjisavvas, N. (1999) On the Subdifferentials of Quasiconvex and Pseudoconvex Functions and Cyclic Monotonicity. Journal of Mathematical Analysis and Applications, 237, 30-42. https://doi.org/10.1006/jmaa.1999.6437
[3]	Levi, E.E. (1910) Studii sui punti singolari essenziali delle funzioni analitiche di due o più variabili complesse. Annali di Matematica Pura ed Applicata (1898-1922), 17, 61-68. https://doi.org/10.1007/BF02419336
[4]	Mangasarian, O.L. (1965) Pseudo-Convex Functions. Journal of the Society for Industrial and Applied Mathematics Series A Control, 3, 281-290. https://doi.org/10.1137/0303020
[5]	Hassouni, A. and Jaddar, A. (2003) On Generalized Monotone Multi-Functions with Applications to Optimality Conditions in Generalized Convex Programming. Journal of Inequalities in Pure Mathematics, 4, Article No. 67.
[6]	Ivanov, V.I. (2019) Characterization of Radially Lower Semi-Continuous Pseudo-Convex Functions. Journal of Optimization Theory and Applications, 184, 368-383.
[7]	Lahrech, S., Jaddar, A., Ouahab, A. and Mbarki, A. (2006) Some Remarks about Strictly Pseudo-Convex Functions with Respect to the Clarke-Rockafellar Sub-Differential. Lobachevskii Journal of Mathematics, 20, 55-62.
[8]	Karamaradian, S. and Schaible, S. (1990) Some Kinds of Monotone Maps. Journal of Optimization Theory and Applications, 66, 37-46. https://doi.org/10.1007/BF00940531
[9]	Hadjisavvas, N., Schaible, S. and Wong, N.C. (2012) Pseudo-Monotone Operators: A Survey of the Theory and Its Applications. Journal of Optimization Theory and Applications, 152, 1-20. https://doi.org/10.1007/s10957-011-9912-5
[10]	Hadjisavvas, N. Komlosi, S. and Schaible, S. (2005) Nonconvex Optimization and Its Applications. Handbook of Generalized Convexity and Generalized Monotonicity. Springer Science + Business Media, Inc., Boston. https://doi.org/10.1007/b101428
[11]	Okelo, N.B. (2019) On Certain Conditions for Convex Optimization in Hilbert Spaces. Khayyam Journal of Mathematics, 5, 108-112.
[12]	Chidume, C.E. (2014) Applicable Functional Analysis. TETFUND Book Project. Ibadan University Press, Ibadan.
[13]	Aussel, D. (1998) Subdifferential Properties of Quasiconvex and Pseudoconvex Functions: Unified Approach. Journal of Optimization Theory and Applications, 97, 29-45. https://doi.org/10.1023/A:1022618915698
[14]	Hassouni, A. and Jaddar, A. (2007) On Pseudo-Convex Functions and Applications to Global Optimization. ESIAM Proceedings, 20, 138-148. https://doi.org/10.1051/proc:072013
[15]	Aussel, D., Corvellec, J. and Lassonde, M. (1995) Mean Value Property and Subdifferential Criteria for Lower Semi-Continuous Functions. Transactions of the American Mathematical Society, 347, 4147-4161. https://doi.org/10.1090/S0002-9947-1995-1307998-0
[16]	Daniilidis, A. and Hadjisawas, N. (1999) Characterization of Non-Smooth Semi-Strictly Quasi-Convex and Strictly Quasi-Convex Functions. Journal of Optimization Theory and Applications, 102, 525-536.