constraint qualification

In this paper, the class of nonconvex vector-valued optimization problems with inequality constraints is considered. We introduce two constraint qualifications and derive the weak and strong Karush-Kuhn-Tucker type of necessary conditions for a (weakly) efficient solution to the considered problem. All results are given in terms of Dini directional derivative and Clarke subdifferential.

This paper introduces several Abadie-type constraint qualifications and derives necessary optimality conditions in the Karush-Kuhn-Tucker‎ ‎for both weakly efficient solutions and efficient solutions of a nonsmooth multi-objective semi-infinite programming problem characterized by locally Lipschitz data‎. ‎The findings are expressed in terms of the Micheal-Penot subdifferential‎.

The primary objective of this paper is to enhance several well-known geometric constraint qualifications and necessary optimality conditions for nonsmooth semi-infinite optimization problems (SIPs). We focus on defining novel algebraic Mangasarian-Fromovitz type constraint qualifications, and on presenting two Karush-Kuhn-Tucker type necessary optimality conditions for nonsmooth SIPs defined by locally Lipschitz functions. Then, by employing a new type of generalized invex functions, we present sufficient conditions for the optimality of a feasible point of the considered problems. It is noteworthy that the new class of invex functions we considered encompasses several classes of invex functions introduced previously. Our results are based on the Michel-Penot subdifferential.

This paper examines normal cones of the feasible set for mathematical programming problems with switching constraints (MPSC)‎. ‎Functions involved are assumed to be continuously differentiable‎. ‎The primary focus is on providing the upper estimate of the Mordukhovich normal cone for the feasible set of MPSCs‎. ‎First‎, ‎a constraint qualification‎, ‎called the ``MPSC-No Nonzero Abnormal Multiplier Constraint Qualification''‎, ‎is considered for the problem‎. ‎Based on this qualification‎, ‎the main result of the paper is presented‎. ‎Finally‎, ‎an optimality condition‎, ‎called the ``necessary M-stationarity condition'' is proposed for optimal solutions of the considered problems‎. ‎Since other optimization problems with multiplicative constraints can be rewritten in the form of MPSCs‎, ‎results obtained in this paper can be extended to a wider class of problems involving multiplicative constraints‎.

This paper aims to study a broad class of multiobjective mathematical problems with switching constraints in which all emerging functions are assumed to be locally Lipschitz. First, we are interested in some Abadie, Guignard, and Cottle types qualification conditions for the problem. Then, these constraint qualifications are applied to obtain several stationarity conditions. The results are based on Clarke's subdifferential.

This paper aims to establish first-order necessary optimality conditions for non-smooth multi-objective generalized semi-infinite programming problems‎. ‎These problems involve inequality constraints whose index set depends on the decision vector‎, ‎and all emerging functions are assumed to be locally Lipschitz‎. ‎We introduce a new constraint qualification for these problems‎. ‎Building upon this qualification‎, ‎we derive an upper estimate for the Clarke sub-differential of the value function of the problem‎. ‎Furthermore‎, ‎we demonstrate the necessary optimality conditions for properly efficient solutions to the problem‎.

In this paper we consider the multi-objective generalizedsemi-infinite optimization problems with nondifferentiable convex data,whereas Soroush (Journal of Mathematical Extension 16(9): 1-14, 2022)investigated them in single-objective case. We introduce some upperlevelqualification conditions for the problems, and based on these qualifications,we demonstrate some first-order necessary optimality conditionsat weakly efficient and efficient solutions of the considered problem.

This paper addresses a non-smooth multi-objective semi-infinite programming problem that involves a feasible set defined by inequality constraints‎. ‎Our focus is on introducing a new weak Slater constraint qualification and deriving the necessary and sufficient conditions for (weakly‎, ‎properly) efficient solutions to the problem using (weak and strong) Karush-Kuhn-Tucker types‎. ‎Additionally‎, ‎we present two duals of the Mond-Weir type for the problem and provide (weak and strong) duality results for them‎. ‎All of the results are given in terms of Clarke subdifferential‎.

The paper deals with the mathematical programming problems with switching constraints that are defined with continuously differentiable functions. The main results are the upper approximations of the Fr`echet Normal Cone of the feasible set for the problem. As applications of the main results, we present some stationary conditions of the considered problem.

We consider the multiobjective semi-infinite programming problems with feasible sets defined by equality and inequality constraints, in which the objective and the constraints functions are locally Lipschitz. First, we introduce an Arrow-Hurwitcz-Uzawa type constraint qualification which is based on the Clarke subdifferential. Then, we derive the strong Karush-Kuhn-Tucker type necessary optimality condition for properly efficient solutions of the considered problems.

In this paper‎, ‎some constraint qualifications ‎of‎ the Guignard type are defined for optimization problems with continuously differentiable objective functions and locally Lipschitz switching constraints‎. ‎Then‎, ‎a new type of stationary condition‎, ‎named parametric stationary condition‎, ‎is presented for the problem‎, ‎and it is shown that all the stationarity conditions in various papers can be deduced from it‎. ‎This paper can be considered as an extension of a recent article (see Kanzow, et al.) to the nonsmooth case‎. ‎Finally‎, ‎the article ends with two important examples‎. ‎The results of the article are formulated according to Clark subdifferential and using nonsmooth analysis methods.

‎In this paper‎, ‎we formulate and study the duality problems in Wolfe type for the mathematical programs with vanishing constraints in nonsmooth case‎, ‎whereas Mishra \emph{et al.} (Ann Oper Res 243(1):249–272‎, ‎2016) investigated it in smooth case‎. ‎Also‎, ‎we derive the weak‎, ‎strong‎, ‎strict converse duality results for the problems with Lipschitzian data utilizing Clarke subdifferential‎.

We consider a nonsmooth optimization problem with a feasible set defined by vanishing constraints. First, we introduce a constraint qualification for the problem, named NNAMCQ. Then, NNAMCQ is applied to obtain a necessary M-stationary condition. Finally, we present a sufficient condition for M-stationarity, under generalized convexity assumption. Our results are formulated in terms of Mordukhovich subdifferential.

This paper studies the convex multiobjective optimization problem with vanishing constraints‎. ‎We introduce a new constraint qualification for these problems‎, ‎and then a necessary optimality condition for properly efficient solutions is presented‎. ‎Finally by imposing some assumptions‎, ‎we show that our necessary condition is also sufficient for proper efficiency‎. ‎Our results are formulated in terms of convex subdifferential.

For a nonsmooth multiobjective mathematical programming problem governed by infinitely many constraints‎, ‎we define a new gap function that generalizes the definitions of this concept in other articles‎. ‎Then‎, ‎we characterize the efficient‎, ‎weakly efficient‎, ‎and properly efficient solutions of the problem utilizing this new gap function‎. ‎Our results are based on $(Phi,rho)-$invexity‎, ‎defined by Clarke subdifferential.

In this paper, we consider a nonsmooth multiobjective programming problem with equilibrium constraints. We present three constraint qualications (CQs) and investigate their relations. Furthermore, we derive two types of necessary optimality conditions under these CQs. In addition, some examples are given to clarify our result

This paper deals with a class of vector semi-infinite optimization problems with differentiable data and arbitrary index set of inequality constraints. A suitable constraint qualification and a new extension of invexity are introduced, and the weak and strong KarushKuhn-Tucker type optimality conditions are investigated

This paper is devoted to the study of semi-infinite optimization with nonsmooth data. We introduce the Arrow-HurwitczUzawa constraint qualification which is based on the Clarke subdifferential. Then, we derive a suitable Karush-Kuhn-Tucker type necessary optimality condition

‎This paper proposes a new form of optimization problem which is a two-level programming problem with infinitely many lower level constraints‎. ‎Firstly‎, ‎we consider some lower level constraint qualifications (CQs) for this problem‎. ‎Then‎, ‎under these CQs‎, ‎we derive formula for estimating the subdifferential of its valued function‎. ‎Finally‎, ‎we present some necessary optimality conditions as Fritz-John type for the problem.

In this paper we study optimization problems with infinite many inequality constraints on a Banach space where the objective function and the binding constraints are Lipschitz near the optimal solution. Necessary optimality conditions and constraint qualifications in terms of Michel-Penot subdifferential are given.