Weak sharp efficiency in multiobjective optimization

Abstract

By using the generalized Fermat rule, the Mordukhovich subdifferential for maximum functions, the fuzzy sum rule for Fréchet subdifferentials and the sum rule for Mordukhovich subdifferentials, we establish a necessary optimality condition for the local weak sharp efficient solution of a constrained multiobjective optimization problem. Moreover, by employing the approximate projection theorem, and some appropriate convexity and affineness conditions, we also obtain some sufficient optimality conditions respectively for the local and global weak sharp efficient solutions of such a multiobjective optimization problem.

Local Sharp Vector Variational Type Inequality and Optimization Problems

“…The results of our research in this paper are generalized, extended, and improved studies of concepts of -efficient solutions for vector minimization problems [13], -optimality for scalar problems to vector maximization problems, or efficiency problems [14], weak sharp minima for scalar optimization problem [15], weak local sharp efficient solution of a constrained multi-objective optimization, and the local and global weak sharp efficient solutions of such a multi-objective optimization problem [16].…”

“…Recently, in 2016, Zhu [16] suggested the necessary optimal conditions for the weak local sharp efficient solution of a constrained multi-objective optimization problem by using the generalized Fermat formula, the Mordukhovich subdifferential for maximum functions, the fuzzy sum rule for Fréchet subdifferentials, and the sum rule for Mordukhovich subdifferentials, and also got the some sufficient optimal conditions respectively for the local and global weak sharp efficient solutions of such a multi-objective optimization problem, by applying the approximate projection method, and some appropriate convexity and affineness conditions.…”

Characterizing robust weak sharp solution sets of convex optimization problems with uncertainty

“…Now, we recall the following concepts of robust weak sharp weakly efficient solutions in multi-objective optimization, which can be found in the literature; see e.g., [20] and [30].…”

“…In the context of optimization, much attention has been paid to concerning sufficient and/or necessary conditions for weak sharp minimizers/solutions and characterizing weak sharp solution sets (of such weak sharp minimizers) in various types of problems. Particularly, the study of characterizations of the weak sharp solution sets covers both single-objective and multi-objective optimization problems (see, [10,11,12,30] and references therein) and, recently, is extended to mathematical programs with inequality constraints and semi-infinite programs (see, e.g.,…”

On weak sharp solutions in $(\rho , \mathbf{b}, \mathbf{d})$-variational inequalities

“…Based on the works of Burke and Ferris [3], Patriksson [11] and following Marcotte and Zhu [10], the concept of weak sharp solution associated with variational-type inequalities has attracted the attention of many researchers (see, for instance, Hu and Song [7], Liu and Wu [9], Zhu [17] and Jayswal and Singh [8]). Recently, by using gap-type functions, in accordance with Ferris and Mangasarian [5] and following Hiriart-Urruty and Lemaréchal [6], Alshahrani et al [1] studied the minimum and maximum principle sufficiency properties associated with nonsmooth variational inequalities.…”