Lagrange Duality and Saddle-Point Optimality Conditions for Nonsmooth Interval-Valued Multiobjective Semi-Infinite Programming Problems with Vanishing Constraints

Abstract

This article deals with a class of nonsmooth interval-valued multiobjective semi-infinite programming problems with vanishing constraints (NIMSIPVC). We introduce the VC-Abadie constraint qualification (VC-ACQ) for NIMSIPVC and employ it to establish Karush–Kuhn–Tucker (KKT)-type necessary optimality conditions for the considered problem. Regarding NIMSIPVC, we formulate interval-valued weak vector, as well as interval-valued vector Lagrange-type dual and scalarized Lagrange-type dual problems. Subsequently, we establish the weak, strong, and converse duality results relating to the primal problem, NIMSIPVC, and the corresponding dual problems. Moreover, we introduce the notion of saddle points for the interval-valued vector Lagrangian and scalarized Lagrangian of NIMSIPVC. Furthermore, we derive the saddle-point optimality criteria for NIMSIPVC by establishing the relationships between the solutions of NIMSIPVC and the saddle points of the corresponding Lagrangians of NIMSIPVC, under convexity assumptions. Non-trivial illustrative examples are provided to demonstrate the validity of the established results. The results presented in this paper extend the corresponding results derived in the existing literature from smooth to nonsmooth optimization problems, and we further generalize them for interval-valued multiobjective semi-infinite programming problems with vanishing constraints.