switching constraints

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.

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.

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.