On strong KKT type sufficient optimality conditions for multiobjective semi-infinite programming problems with vanishing constraints

In this paper, we consider a nonsmooth multiobjective semi-infinite programming problem with vanishing constraints (MOSIPVC). We introduce stationary conditions for the MOSIPVCs and establish the strong Karush-Kuhn-Tucker type sufficient optimality conditions for the MOSIPVC under generalized convexity assumptions.


Introduction
Multiobjective semi-infinite programming problems (MOSIPs) arise when more than one objective function is to be optimized over the feasible region described by an infinite number of constraints. If there is only one objective function in a MOSIP, then it is known as semi-infinite programming problem (SIP). SIPs have played an important role in several areas of modern research, such as transportation theory [], engineering design [], robot trajectory planning [] and control of air pollution []. We refer to the books [, ] for more details as regards SIPs and their applications and to some recent papers [-] for details as regards MOSIPs.
Achtziger and Kanzow [] introduced the mathematical programs with vanishing constraints (MPVCs) and showed that many problems from structural topology optimization can be reformulated as MPVCs. Hoheisel  , we extend the concept of the strong KKT optimality conditions for the MOSIPs with vanishing constraints (MOSIPVCs) that do not involve any constraint qualification. The paper is organized as follows. In Section , we present some known definitions and results which will be used in the sequel. In Section , we define stationary points and establish strong KKT type optimality for MOSIPVC. In Section , we conclude the results of the paper.

Definitions and preliminaries
In this paper, we consider the following MOSIPVC: Letx ∈ M. The following index sets will be used in the sequel.
T(x) := t ∈ T : g t (x) =  , Furthermore, the index set I + (x) can be divided as follows: Similarly, the index set I  (x) can be partitioned as follows: The Clarke directional derivative of a locally Lipschitz function f : R n → R aroundx in the direction v ∈ R n and the Clarke subdifferential of f atx are, respectively, given by We recall the following results from [].
Theorem . Let f and g be locally Lipschitz from R n to R aroundx. Then the following properties hold: The following definitions and lemma from Kanzi and Nobakhtian [] will be used in the sequel.
. f is said to be strictly generalized convex atx if, for each x ∈ R n , x =x and any . f is said to be generalized quasiconvex atx if, for each x ∈ R n and any ξ ∈ ∂ c f (x), . f is said to be strictly generalized quasiconvex atx if, for each x ∈ R n and any Lemma . Let f  be strictly generalized convex and f  , f  , . . . , f s be generalized convex function at x. If λ  >  and λ l ≥  for l = , . . . , s, then s l= λ l f l is strictly generalized convex at x.

Strong KKT type sufficient optimality conditions
We . . , l such that the following conditions hold:  In the following theorem, we establish the strong KKT type sufficient optimality result for the MOSIPVC under generalized convexity assumptions.

. , l, are generalized convex atx on M and at least one of them is strictly generalized convex atx on M. Thenx is a weakly efficient solution for the MOSIPVC.
Proof Sincex is a MOSIPVC M-stationary point, there existξ Suppose on the contrary thatx is not a weakly efficient solution for the MOSIPVC, that is, there existsx ∈ M, such that Sincex is a MOSIPVC M-stationary point andx is a feasible point of the MOSIPVC, we have Therefore, from (.), (.) and (.), we obtain Thus, we arrive at a contradiction and hence the result.
The following result is a direct consequence of Theorem ., where the MOSIPVC Mstationary point is replaced by a MOSIPVC S-stationary point. The following example satisfies the assumptions of Theorem ..

Corollary . Letx be a MOSIPVC S-stationary point. Suppose that f i
Example . Consider the following problem in R  : Note that f  (x) = |x  |, f  (x) = |x  | + |x  | and the feasible region of the MOSIPVC (.) is given by which is represented by the shaded region in Figure    The following example satisfies the assumptions of Theorem ..
Example . Consider the following problem in R  : Note that f  (x) = |x  |, f  (x) = |x  | and the feasible region of the MOSIPVC (.) is given by which is represented by the shaded region in Figure

Results and discussion
In this paper, we consider a MOSIPVC. We introduce stationary conditions for the MOSIPVC and establish the strong KKT type sufficient optimality conditions for the MOSIPVC under generalized convexity assumptions. We extend the concept of the strong KKT optimality conditions for the MOSIPVC that do not involve any constraint qualification. Furthermore, the results of this paper may be extended to strong KKT type necessary optimality conditions for the MOSIPVC involving constraint qualification.