Robust Approximate Optimality Conditions for Uncertain Nonsmooth Optimization with Infinite Number of Constraints

Abstract

This paper deals with robust quasi approximate optimal solutions for a nonsmooth semi-infinite optimization problems with uncertainty data. By virtue of the epigraphs of the conjugates of the constraint functions, we first introduce a robust type closed convex constraint qualification. Then, by using the robust type closed convex constraint qualification and robust optimization technique, we obtain some necessary and sufficient optimality conditions for robust quasi approximate optimal solution and exact optimal solution of this nonsmooth uncertain semi-infinite optimization problem. Moreover, the obtained results in this paper are applied to a nonsmooth uncertain optimization problem with cone constraints.

1. Introduction

Let T be a nonempty infinite index set, $C\subseteq {\mathbb{R}}^n$ be a nonempty convex set, and let f, $g_t:{\mathbb{R}}^n\to \mathbb{R}$, $t\in T$, be continuous functions. We focus on the following semi-infinite optimization problem with infinite number of inequality constraints

$$\mathrm{SIP}\left\{\begin{array}{c}minf\left(x\right)\hfill \\ \mathrm{s}.\mathrm{t}.g_t\left(x\right)\le 0,t\in T,\hfill \\ x\in C.\hfill \end{array}\right.$$

The feasible set of $\mathrm{SIP}$ is defined by

$$\overline{\mathcal{F}}:=\left\{x\in C|g_t\left(x\right)\le 0,\mathrm{for}\mathrm{all}t\in T\right\}.$$

It is well known that this modeling of problems as $\mathrm{SIP}$ is an very interesting research topic in mathematical programming due to the wide range of its applications in various fields such as Chebyshev approximation, engineering design, and optimal control, etc. For the last decades, many papers have been devoted to investigate $\mathrm{SIP}$ from different points of view. We refer the readers to the references [1,2,3,4,5,6,7,8,9,10,11] for more details. However, most practical optimization problems are often contaminated with uncertainty data. Thus, it is meaning to study the theory and application of $\mathrm{SIP}$ with uncertainty data. In this paper, we consider the uncertainty case of $\mathrm{SIP}$

$$\begin{array}{c}\hfill \mathrm{USIP}\left\{\begin{array}{c}minf(x,u)\hfill \\ \mathrm{s}.\mathrm{t}.g_t(x,v_t)\le 0,t\in T,\hfill \\ x\in C,\hfill \end{array}\right.\end{array}$$

where $f:{\mathbb{R}}^n\times {\mathbb{R}}^m\to \mathbb{R}$, and $g_t:{\mathbb{R}}^n\times {\mathbb{R}}^q\to \mathbb{R}$, $t\in T$, are continuous functions. The uncertain parameters u and $v_t$, $t\in T$, belong to the convex compact sets $\mathcal{U}\subseteq {\mathbb{R}}^m$ and ${\mathcal{V}}_t\subseteq {\mathbb{R}}^q$, $t\in T$, respectively.

Nowadays, robust optimization technique has been recognized as one of the powerful deterministic methodology that investigates an optimization problem with data uncertainty in the objective or constraint functions. Following this methodology, many interesting results have been obtained for different kinds of uncertain optimization problems, see, for example, [12,13,14,15,16,17,18,19,20,21,22,23,24,25]. Here, we only mention the works on optimality due to [17,19,21,23,25]. More precisely, by virtue of a so-called Extended Nonsmooth Mangasarian-Fromovitz constraint qualification, Lee and Pham [17] obtained a necessary optimality condition for a class of nonsmooth optimization problems in the face of uncertainty data. Chuong [19] obtained some necessary and sufficient optimality conditions for robust Pareto efficient solutions of an uncertain multiobjective optimization problem in terms of multipliers and limiting subdifferential properties. By using a robust type constraint qualifications, Sun et al. [21] investigated some characterizations of robust optimal solutions of an fractional optimization problem under uncertainty data both in the objective and constraint functions. Fakhar et al. [23] deduced some optimality conditions for robust (weakly) efficient solutions of a class of uncertain nonsmooth multiobjective optimization problems in terms of a new concept of generalized convexity. They also presented the viability of their methodology for portfolio optimization. Wei et al. [25] established some weak alternative theorems and optimality conditions for a general scalar robust optimization problem by using the image space analysis approach. However, there is a few papers to deal with robust approximate optimal solutions of uncertain optimization problems, see, for example, [26,27,28,29]. Lee and Lee [26,29] established optimality theorems and duality results of robust approximate optimal solutions for uncertain convex optimization problems under closed convex cone constraint qualifications (see also [27]). Sun et al. [28] introduced a robust type closed convex constraint qualification condition and then established some optimality conditions for robust approximate optimal solutions of a convex optimization problem with uncertainty data.

Motivated by the work reported in [27,28,29], this paper is devoted to deal with robust quasi approximate optimal solution of $\mathrm{USIP}$ by virtue of its robust counterpart

$$\begin{array}{c}\hfill \mathrm{RUSIP}\left\{\begin{array}{c}min\left\{\underset{u\in \mathcal{U}}{max}f(x,u)\right\}\hfill \\ \mathrm{s}.\mathrm{t}.g_t(x,v_t)\le 0,\forall (t,v_t)\in \mathrm{gph}\mathcal{V},\hfill \\ x\in C,\hfill \end{array}\right.\end{array}$$

where $\mathcal{V}:T\to 2^{{\mathbb{R}}^q}$ is an uncertainty set-valued mapping with $\mathcal{V}\left(t\right):={\mathcal{V}}_t$, for all $t\in T.$ We make two major contributions to invistigate robust quasi approximate optimal solutions for $\mathrm{USIP}$. By using the epigraphs of the conjugate functions, we first introduce some new robust type constraint qualification conditions. Then, we obtain some robust forms of necessary and sufficient optimality conditions for quasi approximate optimal solutions of $\mathrm{USIP}$, which gives a new generalization of the approximate optimality conditions for semi-infinite optimization problems to uncertain semi-infinite optimization problems. Moreover, we apply the approach used in this paper to investigate uncertain optimization problems with cone constraints. We also show that several results on characterizations of (robust) approximate optimal solution of (uncertain) optimization problems reported in recent literature can be obtained by the use of our approach.

We organize this paper as follows. Section 2 give some basic definitions and preliminary results used in this paper. Section 3 and Section 4 obtain necessary and sufficient optimality conditions for robust quasi approximate optimal solutions of $\mathrm{USIP}$. Section 5 is devoted to apply the proposed methods to an uncertain optimization problem with cone constraints.

2. Preliminaries

In this paper, we recall some notations and preliminary results, see, e.g., [3,30]. Let ${\mathbb{R}}^n$ be the Euclidean space of dimension n and let ${\mathbb{R}}_{+}^n$ be the nonnegative orthant of ${\mathbb{R}}^n$. The symbol ${\mathbb{B}}^{*}$ stands for the closed unit ball of ${\mathbb{R}}^n$. The inner product in ${\mathbb{R}}^n$ is defined by $\langle ·,·\rangle$. The norm of an element $\xi$ of ${\mathbb{R}}^n$ is given by

$$\parallel \xi \parallel :=\sup \left\{\langle \xi ,d\rangle \mid d\in {\mathbb{R}}^n,\parallel d\parallel \le 1\right\}.$$

Let $D\subseteq {\mathbb{R}}^n$. The closure (resp. convex hull, convex cone hull) of D is denoted by $\mathrm{cl}D$ (resp. $\mathrm{co}D$, $\mathrm{cone}D$). The dual cone of D is defined by

$$D^{*}=\{x^{*}\in {\mathbb{R}}^n|\langle x^{*},x\rangle \ge 0,\forall x\in D\}.$$

The indicator function ${\delta }_D:{\mathbb{R}}^n\to \mathbb{R}\cup \{+\infty \}$ of D is defined by

$${\delta }_D\left(x\right)=\left\{\begin{array}{c}0,\mathrm{if}x\in D,\hfill \\ +\infty ,\mathrm{if}x\notin D.\hfill \end{array}\right.$$

For the nonempty infinite index set T. Let ${\mathbb{R}}^{\left(T\right)}$ be the following linear space [1],

$${\mathbb{R}}^{\left(T\right)}:=\{\lambda ={\left({\lambda }_t\right)}_{t\in T}|{\lambda }_t=0\mathrm{for} \mathrm{all}t\in T\mathrm{except}\mathrm{for}\mathrm{finitely}\mathrm{many}{\lambda }_t\ne 0\}.$$

The positive cone of ${\mathbb{R}}^{\left(T\right)}$ is defined by

$${\mathbb{R}}_{+}^{\left(T\right)}:=\left\{\lambda \in {\mathbb{R}}^{\left(T\right)}|{\lambda }_t\ge 0\mathrm{for} \mathrm{all}t\in T\right\}.$$

The supporting set of $\lambda \in {\mathbb{R}}^{\left(T\right)}$ is defined by

$$T\left(\lambda \right):=\{t\in T|{\lambda }_t\ne 0\},$$

which is a finite subset of $T.$

For an extended real-valued function $f:{\mathbb{R}}^n\to \mathbb{R}\cup \{+\infty \}$. f is said to be proper, if dom $f=\{x\in {\mathbb{R}}^n|f\left(x\right)<+\infty \}\ne \varnothing$. The epigraph of f is defined by epi $f=\{(x,r)\in {\mathbb{R}}^n\times \mathbb{R}|f\left(x\right)\le r\}.$ f is said to be a convex function if $\mathrm{epi}f$ is a convex set. The function f is said to be concave whenever $-f$ is convex. More generally, f is said to be lower semicontinuous if $\mathrm{epi}f$ is closed. The Legendre-Fenchel conjugate function of f is $f^{*}:{\mathbb{R}}^n\to \mathbb{R}\cup \{\pm \infty \}$ defined by

$$f^{*}\left(x^{*}\right)=\underset{x\in {\mathbb{R}}^n}{\sup }\left\{\langle x^{*},x\rangle -f\left(x\right)\right\},\forall x^{*}\in {\mathbb{R}}^n.$$

For any $\epsilon \ge 0,$ the $\epsilon$-subdifferential of f at $\overline{x}\in \mathrm{dom} f$ is given by

$${\partial }^{\epsilon }f\left(\overline{x}\right)=\left\{x^{*}\in {\mathbb{R}}^n|f\left(x\right)-f\left(\overline{x}\right)\ge \langle x^{*},x-\overline{x}\rangle -\epsilon ,\forall x\in {\mathbb{R}}^n\right\}.$$

If $\epsilon =0$, the set $\partial f\left(\overline{x}\right):={\partial }^{0}f\left(\overline{x}\right)$ is the subdifferential of f at $\overline{x}\in \mathrm{dom} f$, that is,

$$\partial f\left(\overline{x}\right)=\left\{x^{*}\in {\mathbb{R}}^n|f\left(x\right)-f\left(\overline{x}\right)\ge \langle x^{*},x-\overline{x}\rangle ,\forall x\in {\mathbb{R}}^n\right\}.$$

Lemma 1

([31]). Let$f:{\mathbb{R}}^n\to \mathbb{R}\cup \{+\infty \}$ be a proper, lower semicontinuous, and convex function, and let $\overline{x}\in dom f$. Then,

$$\begin{array}{c}\hfill \mathrm{epi}{f}^{*}=\bigcup _{\epsilon \ge 0}\left\{\left(\xi ,\langle \xi ,\overline{x}\rangle +\epsilon -f\left(\overline{x}\right)\right)|\xi \in {\partial }^{\epsilon }f\left(\overline{x}\right)\right\}.\end{array}$$

Lemma 2

([3]). Let I be an arbitrary index set and let $f_i:{\mathbb{R}}^n\to \mathbb{R}\cup \{+\infty \}$, $i\in I$, be proper, lower semicontinuous and convex functions on ${\mathbb{R}}^n$. Assume that there exists $x_0\in {\mathbb{R}}^n$, such that ${\sup }_{i\in I}{f}_i\left(x_0\right)<\infty$. Then,

$$\mathrm{epi}{(\underset{i\in I}{\sup }{f}_i)}^{*}=\mathrm{cl}\left(\mathrm{co}\bigcup _{i\in I}\mathrm{epi}{f}_i^{*}\right),$$

where ${\sup }_{i\in I}{f}_i:{\mathbb{R}}^n\to \mathbb{R}\cup \{+\infty \}$ is defined by $\left({\sup }_{i\in I}{f}_i\right)\left(x\right)={\sup }_{i\in I}{f}_i\left(x\right)$ for all $x\in {\mathbb{R}}^n$.

Lemma 3

([3]). Let $f_1,f_2:{\mathbb{R}}^n\to \mathbb{R}\cup \{+\infty \}$ be proper, and convex functions such that $dom{f}_1\cap dom{f}_2\ne \varnothing .$

(i) 

If $f_1$ and $f_2$ are lower semicontinuous, then,

$$\mathrm{epi}{(f_1+f_2)}^{*}=\mathrm{cl}(\mathrm{epi}{f}_1^{*}+\mathrm{epi}{f}_2^{*}).$$

(ii) 

If one of $f_1$ and $f_2$ is continuous at some $\overline{x}\in \mathrm{dom}{f}_1\cap \mathrm{dom}{f}_2$, then,

$$\mathrm{epi}{(f_1+f_2)}^{*}=\mathrm{epi}{f}_1^{*}+\mathrm{epi}{f}_2^{*}.$$

3. Necessary Approximate Optimality Conditions

This section is devoted to establish some necessary optimality conditions for a robust quasi approximate optimal solution of $\mathrm{USIP}$. The following conceptions and results will be used in this sequel.

Definition 1.

The robust feasible set of $\mathrm{USIP}$ is given by

$$\mathcal{F}:=\left\{x\in C|g_t(x,v_t)\le 0,\forall v_t\in {\mathcal{V}}_t,t\in T\right\}.$$

Definition 2.

(i) Let $\epsilon \ge 0$. A point $\overline{x}\in \mathcal{F}$ is called a robust quasi ε-optimal solution of $\mathrm{USIP}$, iff $\overline{x}\in \mathcal{F}$ is a quasi ε-optimal solution of $\mathrm{RUSIP}$, i.e.,

$$\underset{u\in \mathcal{U}}{max}f(\overline{x},u)\le \underset{u\in \mathcal{U}}{max}f(x,u)+\sqrt{\epsilon }\parallel x-\overline{x}\parallel ,\forall x\in \mathcal{F}.$$

(ii) A point $\overline{x}\in \mathcal{F}$ is called a robust optimal solution of $\mathrm{USIP}$, iff $\overline{x}\in \mathcal{F}$ is an optimal solution of $\mathrm{RUSIP}$, i.e.,

$$\underset{u\in \mathcal{U}}{max}f(\overline{x},u)\le \underset{u\in \mathcal{U}}{max}f(x,u),\forall x\in \mathcal{F}.$$

Remark 1.

If $\mathcal{U}$ is a singleton set, the robust quasi ε-optimal solution coincides the concept defined by Lee and Lee [29]. Moreover, if $\mathcal{U}$ and ${\mathcal{V}}_t$, $t\in T$ are singletons, the robust quasi ε-optimal solution of $\mathrm{USIP}$ deduces to be the usual one of quasi ε-optimal solution of $SIP$, that is

$$f\left(\overline{x}\right)\le f\left(x\right)+\sqrt{\epsilon }\parallel x-\overline{x}\parallel ,\forall x\in \overline{\mathcal{F}}.$$

For more details, please see [5,32].

The following constraint qualification will be used in the study of robust quasi approximate optimal solution of $\mathrm{USIP}$.

Definition 3.

We say that robust type constraint qualification condition $\mathrm{RCQC}$ holds, iff

$$\bigcup _{v\in \mathcal{V},\lambda \in {\mathbb{R}}_{+}^{\left(T\right)}}\mathrm{epi}{\left(\sum _{t\in T}{\lambda }_t{g}_t(·,v_t)\right)}^{*}+\mathrm{epi}{\delta }_C^{*}\mathit{is}\mathit{a}\mathit{closed}\mathit{convex}\mathit{set},$$

where $v\in \mathcal{V}$ means that v is a selection of $\mathcal{V}$, i.e., $v:T\to {\mathbb{R}}^q$ and $v_t\in {\mathcal{V}}_t$ for all $t\in T$.

Remark 2.

In the special case when $C={\mathbb{R}}^n$, $\mathrm{RCQC}$ coincides the so-called closed convex cone constraint qualification defined by Lee and Lee [29].

Now, following [29], we give some characterizations of $\mathrm{RCQC}$.

Proposition 1.

Let $g_t:{\mathbb{R}}^n\times {\mathbb{R}}^q\to \mathbb{R}$, $t\in T$, be continuous functions. Assume that ${\mathcal{V}}_t\subseteq {\mathbb{R}}^n$, $t\in T$, is convex, and for any $v_t\in {\mathcal{V}}_t$, $g_t(·,v_t)$ is a convex function, and for any $x\in {\mathbb{R}}^n$, $g_t(x,·)$ is concave on ${\mathcal{V}}_t$. Then,

$$\bigcup _{v\in \mathcal{V},\lambda \in {\mathbb{R}}_{+}^{\left(T\right)}}\mathrm{epi}{\left(\sum _{t\in T}{\lambda }_t{g}_t(·,v_t)\right)}^{*}+\mathrm{epi}{\delta }_C^{*}$$

is a convex set.

Proof. 

By ([29] Proposition 2.5) and ([28] Proposition 3.6), we can easily get the desired result. □

Naturally, we can obtain the following result by virtue of ([29] Proposition 2.6).

Proposition 2.

Let T be a compact metric space, and let $\mathcal{V}$ be compact-valued and uniformly upper semi-continuous on T. Let $g_t:{\mathbb{R}}^n\times {\mathbb{R}}^q\to \mathbb{R}$, $t\in T$, be continuous functions such that for each $v_t\in {\mathcal{V}}_t$, $g_t(·,v_t)$ is a convex function. Suppose that there exists $\widehat{x}\in {\mathbb{R}}^n$ such that $g_t(\widehat{x},v_t)<0,forany{v}_t\in {\mathcal{V}}_t,t\in T.$ Then,

$$\bigcup _{v\in \mathcal{V},\lambda \in {\mathbb{R}}_{+}^{\left(T\right)}}\mathrm{epi}{\left(\sum _{t\in T}{\lambda }_t{g}_t(·,v_t)\right)}^{*}+\mathrm{epi}{\delta }_C^{*}$$

is a closed set.

Now, we present a robust Farkas Lemma for convex functions. Since its proof is similar to ([29] Lemma 3.1), we omit it.

Lemma 4.

Let $h:{\mathbb{R}}^n\to \mathbb{R}$ be a convex function, and let $g_t:{\mathbb{R}}^n\times {\mathbb{R}}^q\to \mathbb{R}$, $t\in T$, be continuous functions such that for any $v_t\in {\mathcal{V}}_t$, $g_t(·,v_t)$ is a convex function. Then, the following statements are equivalent:

(i)

$\left\{x\in C|g_t(x,v_t)\le 0,\forall v_t\in {\mathcal{V}}_t,t\in T\right\}\subseteq \left\{x\in {\mathbb{R}}^n|h\left(x\right)\ge 0\right\}.$

(ii)

$(0,0)\in \mathrm{epi}{h}^{*}+\mathrm{cl}\mathrm{co}\left({\bigcup }_{v\in \mathcal{V},\lambda \in {\mathbb{R}}_{+}^{\left(T\right)}}\mathrm{epi}{\left({\sum }_{t\in T}{\lambda }_t{g}_t(·,v_t)\right)}^{*}+\mathrm{epi}{\delta }_C^{*}\right).$

Now, we gives a necessary approximate optimality condition for a robust quasi $\epsilon$-optimal solution of $\mathrm{USIP}$ under the condition $\mathrm{RCQC}$.

Theorem 1.

Let $\epsilon \ge 0$. Let $f:{\mathbb{R}}^n\times {\mathbb{R}}^m\to \mathbb{R}$ be a continuous convex-concave function, and let $g_t:{\mathbb{R}}^n\times {\mathbb{R}}^q\to \mathbb{R}$, $t\in T$, be continuous functions such that $g_t(·,v_t)$ is convex on ${\mathbb{R}}^n$, for any $v_t\in {\mathcal{V}}_t$. Assume that $\mathrm{RCQC}$ holds. If $\overline{x}\in \mathcal{F}$ is a robust quasi ε-optimal solution of $\mathrm{USIP}$, then, there exist $\overline{u}\in \mathcal{U}$, ${\left({\overline{\lambda }}_t\right)}_{t\in T}\in {\mathbb{R}}_{+}^{\left(T\right)}$, and ${\overline{v}}_t\in {\mathcal{V}}_t$, $t\in T$ such that

$$\begin{array}{c}\hfill 0\in {\partial }_{x}f(\overline{x},\overline{u})+\sum _{t\in T}{\partial }_x\left({\overline{\lambda }}_t{g}_t\right)(\overline{x},{\overline{v}}_t)+\partial {\delta }_C\left(\overline{x}\right)+\sqrt{\epsilon }{\mathbb{B}}^{*},\end{array}$$

and

$$\begin{array}{c}\hfill g_t(\overline{x},{\overline{v}}_t)=0,\forall t\in T\left(\overline{\lambda }\right).\end{array}$$

Proof. 

Assume that $\overline{x}\in \mathcal{F}$ is a robust quasi $\epsilon$-optimal solution of $\mathrm{USIP}$. Then,

$$\underset{u\in \mathcal{U}}{max}f(\overline{x},u)\le \underset{u\in \mathcal{U}}{max}f(x,u)+\sqrt{\epsilon }\parallel x-\overline{x}\parallel ,\forall x\in \mathcal{F}.$$

This means that

$$\mathcal{F}\subseteq \left\{x\in C|\underset{u\in \mathcal{U}}{max}f(x,u)+\sqrt{\epsilon }\parallel x-\overline{x}\parallel -\underset{u\in \mathcal{U}}{max}f(\overline{x},u)\ge 0\right\}.$$

Let

$$\phi \left(x\right):=\underset{u\in \mathcal{U}}{max}f(x,u)+\sqrt{\epsilon }\parallel x-\overline{x}\parallel -\underset{u\in \mathcal{U}}{max}f(\overline{x},u).$$

By Lemma 4, we get

$$\begin{array}{c}\hfill (0,0)\in \mathrm{epi}{\phi }^{*}+\mathrm{cl}\mathrm{co}\left(\bigcup _{v\in \mathcal{V},\lambda \in {\mathbb{R}}_{+}^{\left(T\right)}}\mathrm{epi}{\left(\sum _{t\in T}{\lambda }_t{g}_t(·,v_t)\right)}^{*}+\mathrm{epi}{\delta }_C^{*}\right).\end{array}$$

(1)

Since $\mathrm{RCQC}$ holds, it follows from (1) and Lemma 3 that

$$\begin{array}{c}\hfill (0,0)\in \mathrm{epi}{\phi }^{*}+\bigcup _{v\in \mathcal{V},\lambda \in {\mathbb{R}}_{+}^{\left(T\right)}}\sum _{t\in T}\mathrm{epi}{\left({\lambda }_t{g}_t(·,v_t)\right)}^{*}+\mathrm{epi}{\delta }_C^{*}.\end{array}$$

(2)

On the other hand, by using Lemma 2 and the similar method of ([27] Theorem 1),

$$\begin{array}{c}\hfill \mathrm{epi}{\phi }^{*}=\mathrm{cl}\left(\mathrm{co}\bigcup _{u\in \mathcal{U}}\mathrm{epi}{\left(f(·,u)\right)}^{*}\right)+\sqrt{\epsilon }{\mathbb{B}}^{*}\times \left[\sqrt{\epsilon }\parallel \overline{x}\parallel +\underset{u\in \mathcal{U}}{max}f(\overline{x},u),+\infty \right)\end{array}$$

Moreover, since f is a continuous convex-concave function, it is easy to see ${\bigcup }_{u\in \mathcal{U}}\mathrm{epi}{\left(f(·,u)\right)}^{*}$ is a closed convex set. Then,

$$\begin{array}{c}\hfill \mathrm{epi}{\phi }^{*}=\bigcup _{u\in \mathcal{U}}\mathrm{epi}{\left(f(·,u)\right)}^{*}+\sqrt{\epsilon }{\mathbb{B}}^{*}\times \left[\sqrt{\epsilon }\parallel \overline{x}\parallel +\underset{u\in \mathcal{U}}{max}f(\overline{x},u),+\infty \right)\end{array}$$

(3)

Together with (2) and (3), we have

$$\begin{array}{c}\left(0,-\sqrt{\epsilon }\parallel \overline{x}\parallel -\underset{u\in \mathcal{U}}{max}f(\overline{x},u)\right)\in \bigcup _{u\in \mathcal{U}}\mathrm{epi}{\left(f(·,u)\right)}^{*}+\hfill \\ \bigcup _{v\in \mathcal{V},\lambda \in {\mathbb{R}}_{+}^{\left(T\right)}}\sum _{t\in T}\mathrm{epi}{\left({\lambda }_t{g}_t(·,v_t)\right)}^{*}+\mathrm{epi}{\delta }_C^{*}+\sqrt{\epsilon }{\mathbb{B}}^{*}\times {\mathbb{R}}_{+}.\hfill \end{array}$$

Therefore, there exist $\overline{u}\in \mathcal{U}$, ${\left({\overline{\lambda }}_t\right)}_{t\in T}\in {\mathbb{R}}_{+}^{\left(T\right)}$, and ${\overline{v}}_t\in {\mathcal{V}}_t$, $t\in T$ such that

$$\begin{array}{c}\hfill \left(0,-\sqrt{\epsilon }\parallel \overline{x}\parallel -\underset{u\in \mathcal{U}}{max}f(\overline{x},u)\right)\in \mathrm{epi}{\left(f(·,\overline{u})\right)}^{*}+\sum _{t\in T}\mathrm{epi}{\left({\overline{\lambda }}_t{g}_t(·,{\overline{v}}_t)\right)}^{*}+\mathrm{epi}{\delta }_C^{*}+\sqrt{\epsilon }{\mathbb{B}}^{*}\times {\mathbb{R}}_{+}.\end{array}$$

This follows that there exist $({\xi }_0^{*},r_0)\in \mathrm{epi}{\left(f(·,\overline{u})\right)}^{*}$, $({\xi }_t^{*},r_t)\in \mathrm{epi}{\left({\overline{\lambda }}_t{g}_t(·,{\overline{v}}_t)\right)}^{*}$, $({\gamma }^{*},s)\in \mathrm{epi}{\delta }_C^{*}$, and $(b^{*},b)\in {\mathbb{B}}^{*}\times {\mathbb{R}}_{+}$, such that

$$\begin{array}{c}\hfill \left(0,-\sqrt{\epsilon }\parallel \overline{x}\parallel -\underset{u\in \mathcal{U}}{max}f(\overline{x},u)\right)=\left({\xi }_0^{*}+\sum _{t\in T}{\xi }_t^{*}+{\gamma }^{*}+\sqrt{\epsilon }{b}^{*},r_0+\sum _{t\in T}{r}_t+s+b\right).\end{array}$$

(4)

Moreover, by Lemma 1, there exist ${\epsilon }_0\ge 0$, ${\epsilon }_t\ge 0$, $t\in T$, and ${\epsilon }_s\ge 0$ such that

$${\xi }_0^{*}\in {\partial }_x^{{\epsilon }_0}f(\overline{x},\overline{u}), \mathrm{and} r_0=\langle {\xi }_0^{*},\overline{x}\rangle +{\epsilon }_0-f(\overline{x},\overline{u}),$$

$${\xi }_t^{*}\in {\partial }_x^{{\epsilon }_t}\left({\overline{\lambda }}_t{g}_t\right)(\overline{x},{\overline{v}}_t), \mathrm{and} r_t=\langle {\xi }_t^{*},\overline{x}\rangle +{\epsilon }_t-\left({\overline{\lambda }}_t{g}_t\right)(\overline{x},{\overline{v}}_t),$$

$${\gamma }^{*}\in {\partial }^{{\epsilon }_s}{\delta }_C\left(\overline{x}\right), \mathrm{and} s=\langle {\gamma }^{*},\overline{x}\rangle +{\epsilon }_s.$$

From (4), we deduce that

$$\begin{array}{c}\hfill 0\in {\partial }_x^{{\epsilon }_0}f(\overline{x},\overline{u})+\sum _{t\in T}{\partial }_x^{{\epsilon }_t}\left({\overline{\lambda }}_t{g}_t\right)(\overline{x},{\overline{v}}_t)+{\partial }^{{\epsilon }_s}{\delta }_C\left(\overline{x}\right)+\sqrt{\epsilon }{\mathbb{B}}^{*},\end{array}$$

(5)

and

$$\begin{array}{cc}& -\sqrt{\epsilon }\parallel \overline{x}\parallel -\underset{u\in \mathcal{U}}{max}f(\overline{x},u)\hfill \\ =& r_0+\sum _{t\in T}{r}_t+s+b\hfill \\ =& 〈{\xi }_0^{*}+\sum _{t\in T}{\xi }_t^{*}+{\gamma }^{*},\overline{x}〉+{\epsilon }_0+\sum _{t\in T}{\epsilon }_t+{\epsilon }_s-f(\overline{x},\overline{u})-\sum _{t\in T}{\overline{\lambda }}_t{g}_t(\overline{x},{\overline{v}}_t)+b\hfill \\ =& -\sqrt{\epsilon }\langle b^{*},\overline{x}\rangle +{\epsilon }_0+\sum _{t\in T}{\epsilon }_t+{\epsilon }_s-f(\overline{x},\overline{u})-\sum _{t\in T}{\overline{\lambda }}_t{g}_t(\overline{x},{\overline{v}}_t)+b\hfill \\ \ge & -\sqrt{\epsilon }\langle b^{*},\overline{x}\rangle +{\epsilon }_0+\sum _{t\in T}{\epsilon }_t+{\epsilon }_s-\underset{u\in \mathcal{U}}{max}f(\overline{x},u)-\sum _{t\in T}{\overline{\lambda }}_t{g}_t(\overline{x},{\overline{v}}_t)+b.\hfill \end{array}$$

Together with ${\overline{\lambda }}_t{g}_t(\overline{x},{\overline{v}}_t)\le 0,$ we get

$$\begin{array}{c}\hfill 0\le {\epsilon }_0+\sum _{t\in T}{\epsilon }_t+{\epsilon }_s-\sum _{t\in T}{\overline{\lambda }}_t{g}_t(\overline{x},{\overline{v}}_t)\le \sqrt{\epsilon }\langle b^{*},\overline{x}\rangle -\sqrt{\epsilon }\parallel \overline{x}\parallel -b\le 0.\end{array}$$

Then, ${\epsilon }_0={\epsilon }_t={\epsilon }_s=0$ and ${\overline{\lambda }}_t{g}_t(\overline{x},{\overline{v}}_t)=0$. Thus, it follows from (5) that

$$\begin{array}{c}\hfill 0\in {\partial }_{x}f(\overline{x},\overline{u})+\sum _{t\in T}{\partial }_x\left({\overline{\lambda }}_t{g}_t\right)(\overline{x},{\overline{v}}_t)+\partial {\delta }_C\left(\overline{x}\right)+\sqrt{\epsilon }{\mathbb{B}}^{*}.\end{array}$$

This completes the proof. □

Remark 3.

It is worth observing that robust approximate optimality conditions for $\mathrm{USIP}$ obtained in Theorem 1 seems to be new (the same as for Theorems 2, 3 and 4). The robust approximate optimality result similar to the one in Theorems 1 and 2 appeared in ([27] Theorem 3.2) when $C={\mathbb{R}}^n$, $\mathcal{U}$ is singleton and T is a nonempty finite index set.

Now, we present an numerical example to illustrate our necessary robust approximate optimality condition for $\mathrm{USIP}$.

Example 1

([29]). Let ${\mathbb{R}}^n={\mathbb{R}}^m={\mathbb{R}}^q=\mathbb{R}$, $C={\mathbb{R}}_{+}$, $\mathcal{U}=[0,1]$, and ${\mathcal{V}}_t=[1-\frac{t}{2},1+\frac{t}{2}]$ for any $t\in T:=[0,1]$. Moreover, for any $x\in \mathbb{R}$, $u\in \mathcal{U}$ and $v_t\in {\mathcal{V}}_t$, let

$$f(x,u)=x^2-ux,$$

and

$$g_t(x,v_t)=t{x}^2-v_{t}x.$$

Then, for $\mathrm{USIP}$, it is easy to verify that $\mathcal{F}=[0,\frac{1}{2}]$, and $\mathrm{RCQC}$ holds. So, the conditions of Theorem 1 are satisfied.

On the other hand, let $\epsilon =2$ and $\overline{x}=0\in \mathcal{F}$. Obviously, $\overline{x}$ is a robust quasi ε-optimal solution of $\mathrm{USIP}$. Moreover, for instance, there exist $\overline{u}=0$, ${\overline{\lambda }}_t=1$, $t=0$, ${\overline{\lambda }}_t=0$, $t\in (0,1]$, and ${\overline{v}}_t=1+\frac{t}{4}$, $t\in [0,1]$ such that

$$\begin{array}{c}\hfill 0\in {\partial }_{x}f(\overline{x},\overline{u})+\sum _{t\in T}{\partial }_x\left({\overline{\lambda }}_t{g}_t\right)(\overline{x},{\overline{v}}_t)+\partial {\delta }_C\left(\overline{x}\right)+\sqrt{\epsilon }{\mathbb{B}}^{*},\end{array}$$

and

$$\begin{array}{c}\hfill g_t(\overline{x},{\overline{v}}_t)=0,\forall t\in T\left(\overline{\lambda }\right).\end{array}$$

Corollary 1.

Let T be a compact metric space, and let $\mathcal{V}$ be compact-valued and uniformly upper semi-continuous on T. Let $f:{\mathbb{R}}^n\times {\mathbb{R}}^m\to \mathbb{R}$ be a continuous convex-concave function, and let $g_t:{\mathbb{R}}^n\times {\mathbb{R}}^q\to \mathbb{R}$, $t\in T$, be continuous functions such that for each $v_t\in {\mathcal{V}}_t$, $g_t(·,v_t)$ is a convex function, and for each $x\in {\mathbb{R}}^n$, $g_t(x,·)$ is concave on ${\mathcal{V}}_t$. Suppose that there exists $\widehat{x}\in {\mathbb{R}}^n$ such that $g_t(\widehat{x},v_t)<0, for any v_t\in {\mathcal{V}}_t,t\in T.$ If $\overline{x}\in \mathcal{F}$ is a robust quasi ε-optimal solution of $\mathrm{USIP}$, then, there exist $\overline{u}\in \mathcal{U}$, ${\left({\overline{\lambda }}_t\right)}_{t\in T}\in {\mathbb{R}}_{+}^{\left(T\right)}$, and ${\overline{v}}_t\in {\mathcal{V}}_t$, $t\in T$ such that

$$\begin{array}{c}\hfill 0\in {\partial }_{x}f(\overline{x},\overline{u})+\sum _{t\in T}{\partial }_x\left({\overline{\lambda }}_t{g}_t\right)(\overline{x},{\overline{v}}_t)+\partial {\delta }_C\left(\overline{x}\right)+\sqrt{\epsilon }{\mathbb{B}}^{*},\end{array}$$

and

$$\begin{array}{c}\hfill g_t(\overline{x},{\overline{v}}_t)=0,\forall t\in T\left(\overline{\lambda }\right).\end{array}$$

Proof. 

Combing Proposition 1, Proposition 2 and Theorem 1, we can easily obtain the desired result. □

When $\mathcal{U}$ and $\mathcal{V}$ are singletons, the following result holds naturally.

Corollary 2.

Let $\epsilon \ge 0$. Let $f:{\mathbb{R}}^n\to \mathbb{R}$ be a convex function, and let $g_t:{\mathbb{R}}^n\to \mathbb{R}$, $t\in T$, be continuous convex functions. Assume that $\mathrm{cone}\left({\bigcup }_{t\in T}\mathrm{epi}{g}_t^{*}\right)+\mathrm{epi}{\delta }_C^{*}$ is a closed set. If $\overline{x}\in \overline{\mathcal{F}}$ is a quasi ε-optimal solution of $\mathrm{SIP}$, then, there exist ${\left({\overline{\lambda }}_t\right)}_{t\in T}\in {\mathbb{R}}_{+}^{\left(T\right)}$, such that

$$\begin{array}{c}\hfill 0\in {\partial }_{x}f\left(\overline{x}\right)+\sum _{t\in T}{\overline{\lambda }}_t{\partial }_x{g}_t\left(\overline{x}\right)+\partial {\delta }_C\left(\overline{x}\right)+\sqrt{\epsilon }{\mathbb{B}}^{*}, \mathrm{and} g_t\left(\overline{x}\right)=0,\forall t\in T\left(\overline{\lambda }\right).\end{array}$$

The following theorem establish necessary condition for robust optimal solution of $\mathrm{USIP}$. This result can be considered as a version of robust optimality condition for nonsmooth and nonlinear semi-infinite optimization problems which have not yet been considered in the literature.

Theorem 2.

Let $f:{\mathbb{R}}^n\times {\mathbb{R}}^m\to \mathbb{R}$ be a continuous convex-concave function, and let $g_t:{\mathbb{R}}^n\times {\mathbb{R}}^q\to \mathbb{R}$, $t\in T$, be continuous functions such that $g_t(·,v_t)$ is convex on ${\mathbb{R}}^n$, for any $v_t\in {\mathcal{V}}_t$. Assume that $\mathrm{RCQC}$ holds. If $\overline{x}\in \mathcal{F}$ is a robust optimal solution of $\mathrm{USIP}$, then, there exist $\overline{u}\in \mathcal{U}$, ${\left({\overline{\lambda }}_t\right)}_{t\in T}\in {\mathbb{R}}_{+}^{\left(T\right)}$, and ${\overline{v}}_t\in {\mathcal{V}}_t$, $t\in T$ such that

$$\begin{array}{c}\hfill 0\in {\partial }_{x}f(\overline{x},\overline{u})+\sum _{t\in T}{\partial }_x\left({\overline{\lambda }}_t{g}_t\right)(\overline{x},{\overline{v}}_t)+\partial {\delta }_C\left(\overline{x}\right),\end{array}$$

and

$$\begin{array}{c}\hfill g_t(\overline{x},{\overline{v}}_t)=0,\forall t\in T\left(\overline{\lambda }\right).\end{array}$$

The particular case of Theorem 1 corresponding to the cases where $\epsilon =0$, $\mathcal{U}$ and $\mathcal{V}$ are singletons are of special interest. The interested reader is referred to [4,6,7,8,10] for necessary optimality conditions of $\mathrm{SIP}$ in terms of different conditions.

Corollary 3.

Let $f:{\mathbb{R}}^n\to \mathbb{R}$ be a convex function, and let $g_t:{\mathbb{R}}^n\to \mathbb{R}$, $t\in T$, be continuous convex functions. Assume that $\mathrm{cone}\left({\bigcup }_{t\in T}\mathrm{epi}{g}_t^{*}\right)+\mathrm{epi}{\delta }_C^{*}$ is a closed set. If $\overline{x}\in \overline{\mathcal{F}}$ is a optimal solution of $\mathrm{SIP}$, then, there exist ${\left({\overline{\lambda }}_t\right)}_{t\in T}\in {\mathbb{R}}_{+}^{\left(T\right)}$, such that

$$\begin{array}{c}\hfill 0\in {\partial }_{x}f\left(\overline{x}\right)+\sum _{t\in T}{\overline{\lambda }}_t{\partial }_x{g}_t\left(\overline{x}\right)+\partial {\delta }_C\left(\overline{x}\right), and{g}_t\left(\overline{x}\right)=0,\forall t\in T\left(\overline{\lambda }\right).\end{array}$$

4. Sufficient Approximate Optimality Conditions

This section is devoted to give some sufficient approximate optimality condition for a robust quasi approximate optimal solution of $\mathrm{USIP}.$ Now, we first introduce a new concept of generalized robust approximate $\mathrm{KKT}$ conditions for $\mathrm{USIP}$.

Definition 4.

Let $\epsilon \ge 0$. A point $(\overline{x},\overline{\lambda },\overline{u},\overline{v})\in \mathcal{F}\times {\mathbb{R}}_{+}^{\left(T\right)}\times \mathcal{U}\times \mathcal{V}$ is said to satisfy the generalized robust approximate $KKT$ conditions for $\mathrm{USIP}$, iff

$$\begin{array}{c}\hfill 0\in {\partial }_{x}f(\overline{x},\overline{u})+\sum _{t\in T}{\partial }_x\left({\overline{\lambda }}_t{g}_t\right)(\overline{x},{\overline{v}}_t)+\partial {\delta }_C\left(\overline{x}\right)+\sqrt{\epsilon }{\mathbb{B}}^{*},\end{array}$$

(6)

and

$$\begin{array}{c}\hfill g_t(\overline{x},{\overline{v}}_t)=0,t\in T\left(\overline{\lambda }\right).\end{array}$$

(7)

Remark 4.

Let $\epsilon =0$ in Definition 4. Then, the generalized robust approximate $KKT$ conditions for $\mathrm{USIP}$ coincides with the generalized robust $KKT$ conditions for $\mathrm{USIP}$.

Theorem 3.

Let $\epsilon \ge 0$. Suppose that $(\overline{x},\overline{\lambda },\overline{u},\overline{v})\in \mathcal{F}\times {\mathbb{R}}_{+}^{\left(T\right)}\times \mathcal{U}\times \mathcal{V}$ satisfies the generalized robust approximate $KKT$ conditions and $f(\overline{x},\overline{u})={max}_{u\in \mathcal{U}}{f}_i(\overline{x},u)$. Then, $\overline{x}\in \mathcal{F}$ is a robust quasi ε-optimal solution of $\mathrm{USIP}$.

Proof. 

Since $(\overline{x},\overline{\lambda },\overline{u},\overline{v})\in \mathcal{F}\times {\mathbb{R}}_{+}^{\left(T\right)}\times \mathcal{U}\times \mathcal{V}$ satisfies the generalized robust approximate $\mathrm{KKT}$ conditions, there exist ${\xi }_0^{*}\in {\partial }_{x}f(\overline{x},\overline{u})$, ${\xi }_t^{*}\in {\partial }_x\left({\overline{\lambda }}_t{g}_t\right)(\overline{x},{\overline{v}}_t)$, and ${\gamma }^{*}\in \partial {\delta }_C\left(\overline{x}\right)$ and ${\zeta }^{*}\in {\mathbb{B}}^{*}$, such that

$$\begin{array}{c}\hfill {\xi }_0^{*}+\sum _{t\in T}{\xi }_t^{*}+{\gamma }^{*}+\sqrt{\epsilon }{\zeta }^{*}=0.\end{array}$$

(8)

Since ${\xi }_0^{*}\in {\partial }_{x}f(\overline{x},\overline{u})$, ${\xi }_t^{*}\in {\partial }_x\left({\overline{\lambda }}_t{g}_t\right)(\overline{x},{\overline{v}}_t)$, and ${\gamma }^{*}\in \partial {\delta }_C\left(\overline{x}\right)$, we obtain that, for any $x\in \mathcal{F}$,

$$f(x,\overline{u})-f(\overline{x},\overline{u})\ge 〈{\xi }_0^{*},x-\overline{x}〉,$$

$$\left({\overline{\lambda }}_t{g}_t\right)(x,{\overline{v}}_t)-\left({\overline{\lambda }}_t{g}_t\right)(\overline{x},{\overline{v}}_t)\ge 〈{\xi }_t^{*},x-\overline{x}〉,$$

and

$${\delta }_C\left(x\right)-{\delta }_C\left(\overline{x}\right)\ge \langle {\gamma }^{*},x-\overline{x}\rangle .$$

Then,

$$\begin{array}{cc}& f(x,\overline{u})-f(\overline{x},\overline{u})+\sum _{t\in T}{\overline{\lambda }}_t{g}_t(x,{\overline{v}}_t)-\sum _{t\in T}{\overline{\lambda }}_t{g}_t(\overline{x},{\overline{v}}_t)\hfill \\ \ge & 〈{\xi }_0^{*}+\sum _{t\in T}{\xi }_t^{*}+{\gamma }^{*},x-\overline{x}〉,\forall x\in \mathcal{F}.\hfill \end{array}$$

Moreover, together with ${\overline{\lambda }}_t{g}_t(x,{\overline{v}}_t)\le 0$ and (8), we get,

$$\begin{array}{c}\hfill f(x,\overline{u})-f(\overline{x},\overline{u})-\sum _{t\in T}{\overline{\lambda }}_t{g}_t(\overline{x},{\overline{v}}_t)\ge -\sqrt{\epsilon }〈{\zeta }^{*},x-\overline{x}〉,\forall x\in \mathcal{F}.\end{array}$$

It follows from (7), $f(\overline{x},\overline{u})={max}_{u\in \mathcal{U}}{f}_i(\overline{x},u)$ and $f(x,\overline{u})\le {max}_{u\in \mathcal{U}}{f}_i(x,u)$ that

$$\begin{array}{c}\hfill \underset{u\in \mathcal{U}}{max}{f}_i(x,u)-\underset{u\in \mathcal{U}}{max}{f}_i(\overline{x},u)\ge -\sqrt{\epsilon }〈{\zeta }^{*},x-\overline{x}〉,\forall x\in \mathcal{F}.\end{array}$$

Note that $〈{\zeta }^{*},x-\overline{x}〉\le \parallel x-\overline{x}\parallel$, $\forall x\in \mathcal{F}$. Thus,

$$\begin{array}{c}\hfill \underset{u\in \mathcal{U}}{max}{f}_i(x,u)-\underset{u\in \mathcal{U}}{max}{f}_i(\overline{x},u)\ge -\sqrt{\epsilon }\parallel x-\overline{x}\parallel ,\forall x\in \mathcal{F},\end{array}$$

Thus, $\overline{x}$ is a robust $\epsilon$-optimal solution of $\mathrm{USIP}$ and the proof is complete. □

Remark 5.

It is worth observing that if, in addition, $\mathcal{U}$ is singleton and T is a nonempty finite index set, the approximate optimality conditions given in Theorem 3 was established in [27]. So, our results can be regarded as a generalization of the results obtained in [27].

In the case that $\mathcal{U}$ and $\mathcal{V}$ are singletons, we get the following optimality conditions which have been studied in [5] under different kinds of constraint qualifications.

Corollary 4.

Consider the problem $\mathrm{SIP}$, and let $\epsilon \ge 0$ and $\overline{x}\in \overline{\mathcal{F}}$. Let $f:{\mathbb{R}}^n\to \mathbb{R}$ be convex at $\overline{x}$, and let $g_t:{\mathbb{R}}^n\to \mathbb{R}$, $t\in T$, be continuous convex functions. If there exists ${\left({\overline{\lambda }}_t\right)}_{t\in T}\in {\mathbb{R}}_{+}^{\left(T\right)}$ such that

$$\begin{array}{c}\hfill 0\in \partial f\left(\overline{x}\right)+\sum _{t\in T}{\overline{\lambda }}_t\partial g_t\left(\overline{x}\right)+\partial {\delta }_C\left(\overline{x}\right)+\sqrt{\epsilon }{\mathbb{B}}^{*}, and g_t\left(\overline{x}\right)=0,t\in T\left(\overline{\lambda }\right).\end{array}$$

Then, $\overline{x}\in \overline{\mathcal{F}}$ is a quasi ε-optimal solution of $\mathrm{SIP}$.

The following theorem establish a sufficient condition for robust optimal solution of $\mathrm{USIP}$.

Theorem 4.

Let $\epsilon \ge 0$. Suppose that $(\overline{x},\overline{\lambda },\overline{u},\overline{v})\in \mathcal{F}\times {\mathbb{R}}_{+}^{\left(T\right)}\times \mathcal{U}\times \mathcal{V}$ satisfies the generalized robust $KKT$ conditions and $f(\overline{x},\overline{u})={max}_{u\in \mathcal{U}}{f}_i(\overline{x},u)$. Then, $\overline{x}\in \mathcal{F}$ is a robust optimal solution of $\mathrm{USIP}$.

Similarly, when $\epsilon =0$, $\mathcal{U}$ and $\mathcal{V}$ are singletons, the following sufficient optimality conditions for $\mathrm{SIP}$ are obtained. Please see [4,6,7,8,10] for more details.

Corollary 5.

Consider the problem $\mathrm{SIP}$, and $\overline{x}\in \overline{\mathcal{F}}$. Let $f:{\mathbb{R}}^n\to \mathbb{R}$ be convex at $\overline{x}$, and let $g_t:{\mathbb{R}}^n\to \mathbb{R}$, $t\in T$, be continuous convex functions. If there exists ${\left({\overline{\lambda }}_t\right)}_{t\in T}\in {\mathbb{R}}_{+}^{\left(T\right)}$ such that

$$\begin{array}{c}\hfill 0\in \partial f\left(\overline{x}\right)+\sum _{t\in T}{\overline{\lambda }}_t\partial g_t\left(\overline{x}\right)+\partial {\delta }_C\left(\overline{x}\right), and g_t\left(\overline{x}\right)=0,t\in T\left(\overline{\lambda }\right).\end{array}$$

Then, $\overline{x}\in \overline{\mathcal{F}}$ is a optimal solution of $\mathrm{SIP}$.

5. Applications

In this section, we apply the obtained results to an uncertain optimization problem with cone constraints. Let $C\subseteq {\mathbb{R}}^n$ be a nonempty convex set, and let $K\subseteq {\mathbb{R}}^p$ be a nonempty closed and convex cone. Consider an uncertain conic optimization problem

$$\begin{array}{c}\hfill \mathrm{UCOP}\left\{\begin{array}{c}minf(x,u)\hfill \\ \mathrm{s}.\mathrm{t}.g(x,v)\in -K,\hfill \\ x\in C,\hfill \end{array}\right.\end{array}$$

where $f:{\mathbb{R}}^n\times {\mathbb{R}}^m\to \mathbb{R}$ and $g:{\mathbb{R}}^n\times {\mathbb{R}}^q\to {\mathbb{R}}^p$ are continuous functions. The uncertain parameters u and v belong to the convex and compact uncertain sets $\mathcal{U}\subseteq {\mathbb{R}}^m$ and $\mathcal{V}\subseteq {\mathbb{R}}^q$, respectively.

As in [3], for each $\lambda \in K^{*}$, $\langle \lambda ,g\rangle$ will be denoted by $\left(\lambda g\right)$. Note that for any $x\in C$ and $v\in \mathcal{V}$, $g(x,v)\in -K$ if and only if $\left(\lambda g\right)(x,v)\le 0,$$\forall \lambda \in K^{*}$. Then, $\mathrm{UCOP}$ can be reformulated as an example of $\mathrm{USIP}$ by setting

$$T:=K^{*},g_{\lambda }(x,v_{\lambda }):=\left(\lambda g\right)(x,v),\mathrm{for}\mathrm{any}\lambda \in T=K^{*}.$$

In this section, we also use $\mathcal{F}$ to denote the feasible set of $\mathrm{UCOP}$:

$$\mathcal{F}:=\left\{x\in C|\left(\lambda g\right)(x,v)\le 0,\forall v\in \mathcal{V},\lambda \in K^{*}\right\}=\left\{x\in C\left|g\right(x,v)\in -K,\forall v\in \mathcal{V}\right\}.$$

Moreover, let $\beta :={\left({\beta }_{\lambda }\right)}_{\lambda \in K^{*}}\in {\mathbb{R}}_{+}^{\left(K^{*}\right)}$, for any $\lambda \in K^{*}$. Then,

$$\tilde{\lambda }:=\sum _{\lambda \in K^{*}}{\beta }_{\lambda }\lambda \in K^{*},$$

and so

$$\bigcup _{v\in \mathcal{V},\beta \in {\mathbb{R}}_{+}^{\left(K^{*}\right)}}\mathrm{epi}{\left(\sum _{\lambda \in K^{*}}{\beta }_{\lambda }\left(\lambda g\right)(·,v)\right)}^{*}=\bigcup _{v\in \mathcal{V},\tilde{\lambda }\in K^{*}}\mathrm{epi}{\left(\left(\tilde{\lambda }g\right)(·,v)\right)}^{*}.$$

Thus, $\mathrm{RCQC}$ deduces to the constraint qualification $\mathrm{RCCCQ}$ introduced in [28], that is

$$\bigcup _{v\in \mathcal{V},\lambda \in K^{*}}\mathrm{epi}{\left(\left(\lambda g\right)(·,v)\right)}^{*}+\mathrm{epi}{\delta }_C^{*}\mathrm{is}\mathrm{closed}\mathrm{and}\mathrm{convex}.$$

Similarly, we can get the corresponding results for robust quasi $\epsilon$-optimal solutions of $\mathrm{UCOP}$.

Theorem 5.

Let $\epsilon \ge 0$. Let $f:{\mathbb{R}}^n\times {\mathbb{R}}^m\to \mathbb{R}$ be a continuous convex-concave function, and let $g:{\mathbb{R}}^n\times {\mathbb{R}}^q\to {\mathbb{R}}^p$, be a continuous function such that $g(·,v)$ is K-convex on ${\mathbb{R}}^n$, for any $v\in \mathcal{V}$. Assume that $\mathrm{RCCCQ}$ holds. If $\overline{x}\in \mathcal{F}$ is a robust quasi ε-optimal solution of $\mathrm{UCOP}$, then, there exist $\overline{u}\in \mathcal{U}$, $\overline{v}\in \mathcal{V}$, and $\overline{\lambda }\in K^{*}$, such that

$$\begin{array}{c}\hfill 0\in {\partial }_{x}f(\overline{x},\overline{u})+{\partial }_x\left(\overline{\lambda }g\right)(\overline{x},\overline{v})+\partial {\delta }_C\left(\overline{x}\right)+\sqrt{\epsilon }{\mathbb{B}}^{*},\end{array}$$

and

$$\begin{array}{c}\hfill \overline{\lambda }g(\overline{x},\overline{v})=0.\end{array}$$

The following theorem establish necessary condition for robust optimal solution of $UCOP$.

Theorem 6.

Let $f:{\mathbb{R}}^n\times {\mathbb{R}}^m\to \mathbb{R}$ be a continuous convex-concave function, and let $g:{\mathbb{R}}^n\times {\mathbb{R}}^q\to {\mathbb{R}}^p$ be a continuous function such that $g(·,v)$ is K-convex on ${\mathbb{R}}^n$, for any $v\in \mathcal{V}$. Assume that $\mathrm{RCCCQ}$ holds. If $\overline{x}\in \mathcal{F}$ is a robust optimal solution of $\mathrm{UCOP}$, then, there exist $\overline{u}\in \mathcal{U}$, $\overline{v}\in \mathcal{V}$, and $\overline{\lambda }\in K^{*}$, such that

$$\begin{array}{c}\hfill 0\in {\partial }_{x}f(\overline{x},\overline{u})+{\partial }_x\left(\overline{\lambda }g\right)(\overline{x},\overline{v})+\partial {\delta }_C\left(\overline{x}\right),\end{array}$$

and

$$\begin{array}{c}\hfill \overline{\lambda }g(\overline{x},\overline{v})=0.\end{array}$$

Definition 5.

Let $\epsilon \ge 0$. A point $(\overline{x},\overline{\lambda },\overline{u},\overline{v})\in \mathcal{F}\times K^{*}\times \mathcal{U}\times \mathcal{V}$ satisfies the generalized robust approximate $KKT$ conditions for $\mathrm{UCOP}$, iff

$$\begin{array}{c}\hfill 0\in {\partial }_{x}f(\overline{x},\overline{u})+{\partial }_x\left(\overline{\lambda }g\right)(\overline{x},\overline{v})+\partial {\delta }_C\left(\overline{x}\right)+\sqrt{\epsilon }{\mathbb{B}}^{*},\end{array}$$

and

$$\begin{array}{c}\hfill \overline{\lambda }g(\overline{x},\overline{v})=0.\end{array}$$

Remark 6.

Note that if $\epsilon =0$ in Definition 5, we obtain the concept of generalized robust $KKT$ conditions for $\mathrm{UCOP}$.

Similarly, we obtain the following sufficient optimality conditions for $\mathrm{UCOP}$.

Theorem 7.

Let $\epsilon \ge 0$. Suppose that $(\overline{x},\overline{\lambda },\overline{u},\overline{v})\in \mathcal{F}\times K^{*}\times \mathcal{U}\times \mathcal{V}$ satisfies the generalized robust approximate $KKT$ conditions and $f(\overline{x},\overline{u})={max}_{u\in \mathcal{U}}{f}_i(\overline{x},u)$. Then, $\overline{x}\in \mathcal{F}$ is a robust quasi ε-optimal solution of $\mathrm{UCOP}$.

Theorem 8.

Suppose that $(\overline{x},\overline{\lambda },\overline{u},\overline{v})\in \mathcal{F}\times K^{*}\times \mathcal{U}\times \mathcal{V}$ satisfies the generalized robust $KKT$ conditions and $f(\overline{x},\overline{u})={max}_{u\in \mathcal{U}}{f}_i(\overline{x},u)$. Then, $\overline{x}\in \mathcal{F}$ is a robust optimal solution of $\mathrm{UCOP}$.

Remark 7.

In the case that $\mathcal{U}$ is a singleton, optimality conditions of other kinds of robust approximate optimal solutions for $\mathrm{UCOP}$ has been considered in [28].

6. Conclusions

In this paper, a nonsmooth semi-infinite optimization problem under data uncertainty $\mathrm{USIP}$ is considered. By using a new robust type constraint qualification condition $\mathrm{RCQC}$ and the notions of the subdifferential of convex functions, some necessary and sufficient approximate optimality conditions for robust quasi $\epsilon$-optimal solutions of $\mathrm{USIP}$ are established. The results obtained in this paper improve the corresponding results reported in recent literature.

Our research paves the way for further study. It would be interesting to consider other concepts of approximate optimal solutions, such as almost robust (quasi) approximate optimal solution, or almost robust regular approximate optimal solution, for $\mathrm{USIP}$ in the future. On the other hand, since fractional semi-infinite optimization is one of an important model for semi-infinite optimization problems, so it could be possible to investigate fractional semi-infinite optimization problems with data uncertainty in the future.