Karush-Kuhn-Tucker optimality conditions for a class of robust optimization problems with an interval-valued objective function

Abstract In this article, we study the nonlinear and nonsmooth interval-valued optimization problems in the face of data uncertainty, which are called interval-valued robust optimization problems (IVROPs). We introduce the concept of nondominated solutions for the IVROP. If the interval-valued objective function f and constraint functions g i {g}_{i} are nonsmooth on Banach space E, we establish a nonsmooth and robust Karush-Kuhn-Tucker optimality theorem.

However, more and more researchers have recently started to study the interval-valued optimization problems because of their important applications (see [36][37][38][39][40][41][42]). In this article, we shall investigate the following interval-valued robust optimization problem (IVROP): , 0, , 1, 2, , , where objective function f is a locally Lipschitz interval-value function. If the constraint functions g i are nonlinear and nonsmooth on the Banach space E for each ∈ { … } i m 1, 2, , , we shall establish the robust KKT necessary optimality conditions.
In order to highlight the generality of our IVROPs, we present several particular cases. for all ∈ x E, then problem (1.1) reduces to the following problem: subject to , 0, , 1, 2, , , which is called robust optimization problem (ROP). The KKT necessary optimality conditions of this kind of ROP have been studied by many scholars, for more details, we refer to [43][44][45][46][47].
Case 2. For each = … i m 1, 2, , , if constraint functions are independent of v i , then problem (1.1) reduces to the following interval-valued optimization problem: Under the assumptions that each ( ) g x i is convex and continuously differentiable, Ishibuchi and Tanaka in [36], and Inuiguchi and Kume in [37], and Wu in [38,39] studied the KKT necessary optimality conditions of the interval-valued optimization.

Preliminaries
In this section, we first recall some useful notions and well-known results in nonsmooth analysis and nonlinear analysis. Let E be a given real reflexive Banach space. 〈⋅ ⋅〉 , is the duality pairing between * E and , .
: is a locally Lipschitz functional. We denote by°( We shall also denote the one-side directional derivative of ϕ at u by ′( ) ϕ u v ; , i.e., whenever this limit exists.
We point out that for each ∈ u E, we have ∂ ( ) ≠ ∅ ϕ u (see [48]). The next theorem provides some basic properties for Clarke's generalized directional derivatives and Clarke's generalized gradients (for details, see Clarke [48], Proposition 2.1.2).
: be locally Lipchitz of rank L u near the point ∈ u E. Then, is finite convex, positively homogeneous, subadditive and satisfies Now, we introduce the invex set and invex function on Banach space E.
. We say that , then an invex set with respect to η will be reduced to a convex set, while a preconcave function will be reduced to a concave function. Now, we give the well-known Gorden's alternative theorem.
Theorem 2.7. (Gorden's alternative theorem [46]) Let → f X : i for = … i m 1, , be convex functions, X be a convex subset of Banach space E. Define ; Especially, each ∈ x can be regarded as a closed interval xa a A . In fact, interval functions have other properties, and for more details one can refer to [37][38][39].
A mapping . First, we consider the following formulation of interval-valued optimization problems on Banach space E: where ( ) f x is an interval-valued function. We need to make clear the meaning of minimization problems (IVOP). For the purpose, we introduce a partial order relationship in ( ) and ≠ A B, i.e., one of the following is satisfied: or a1 and ; or a2 and ; or a3 and .
Now, we define the concept of solutions for interval-valued optimization problems, which was introduced by Wu in [38,39].
. We say that * x is a nondominated solution of the IVOP if and only if there exists no ∈ x U such that the set of all nondominated objective values of f. Example 2.1. Let us consider the following interval-valued optimization problem: as: It is not hard to see that , and 2, 2 , , and for all feasible solutions ( ) is a nondominated solution of problem (P1).
Example 2.2. Let us consider another interval-valued optimization problem: , , , : as: It is not hard to see that problem (P2) merely has nondominated solutions ( Figure 2

KKT nonsmooth robust optimality conditions with interval-valued functions
In this section, we will give a nonsmooth KKT optimality theorem for IVROPs (1.1).   Assumptions: ix are Clarke's generalized directional derivative and one-side directional derivative of g i with respect to x, respectively; and The compactness of V i implies that ϕ i is well defined on E. By ( ) A 2 , we readily get that ϕ i is Lipschitz on E (with constant L i ). On the other hand, we define the constraint set of the (IVROP) by From the definition of function ϕ i , we have For any ∈ * We also define an Extended Nonsmooth Mangasarian-Fromovitz constraint qualification (ENMFCQ) at ∈ * x S as follows: Now, we are going to prove the following KKT necessary optimality theorem of IVROPs. The proof of the above theorem relies on the following two lemmas.

Lemma 3.2. Let U be a subset of E and let V be a sequentially compact subset of topological space T and invex set with respect to
. And let × → g U V : satisfy conditions ( ) ( ) A A - 1 4 . So, we can also define maximum function → ϕ U : by In addition, assume that the function ( ⋅) g x, is preconcave on V for each ∈ x E. Then, the following statements hold: is invex with respect to η and sequentially compact in T.
is convex and weak* compact.
is invex with respect to η and sequentially compact in T.
Let ∈ x U be fixed and V an invex set with respect to η. Therefore, for any . Since the function ( ⋅) g x, is preconcave, we obtain that Therefore, ( , which implies + ( x is convex and weak* compact. According to the definition of ∂ ( x 2 2 . By (i), we get that the set ( ) V x is invex with respect to η. Therefore, and   which implies , , i.e., x is convex. In the following, we shall use Alaoglu's theorem to prove that ∂ ( ( )) g x V x , x is weak* compact. We first prove that the set ∂ ( x is weak * closed. Next, we will prove that the set ∂ ( , ; l i ms u p , , , .
Hereafter, ∥⋅∥ * stands for the norm in the Banach space * E . Consequently, we obtain , a n d .
x Therefore, we have shown that the set ∂ ( ( )) g x V x , x is weak * closed and bounded in * E . By Alaoglu's theorem, we obtain that ∂ ( We first prove the inclusion , ; , ; l i m , , lim sup ; . Therefore, for any ∈ ( ) v V x , one has ; f o ra l l , , x Conversely, for any ∈ ∂ ( ) ξ ϕ x°( , , . By the assumptions ( ) A 3 and Theorem 2.1 of [48] we have , .
x Consequently, x Particularly, we get x is convex and weak* compact in * E . Applying Sion's minimax theorem (see [49]), we deduce the existence of an element ∈ ∂ ( 1 , x Therefore, , .
x This completes the proof. □ Next, we shall consider the following nonsmooth interval-valued optimization problem: For this problem, we have is a nondominated solution of the IVOP, then the following system ( ) S2 : has no solution ∈ d E.
Proof. Arguing by contradiction. Let us assume that system ( ) S2 has at least one solution, i.e., there exists Consequently, there exists an integer number Similarly, we get that such that , . , such that ; ; 0 , .
Hence,  If we assume that the ENMFCQ at * x holds, we are going to show that (3.14) holds. We argue by contradiction, if it is not true, i.e.,