On relations between nonsmooth interval-valued multiobjective programming problems and generalized Stampacchia vector variational inequalities

Abstract

In this article, we study nonsmooth interval-valued multiobjective programming problem and generalized Stampacchia vector variational inequality with its weak form for interval-valued functions. Using the tools of Mordukhovich subdifferential, we define some new classes of generalized approximate LU-convex functions. These functions are then employed to establish the relations between the solutions of generalized vector variational inequalities and the approximate LU-efficient solutions of the nonsmooth interval-valued multiobjective programming problem. Moreover, we identify the Kuhn–Tucker vector critical points of the considered nonsmooth interval-valued multiobjective programming problem. Under suitable constraint qualification, we establish the equivalence among local approximate LU-efficient points, Kuhn–Tucker vector critical points and the solutions of generalized Stampacchia vector variational inequalities. The results of this paper extend and sharpen the corresponding results of [Giannessi F. On Minty variational principle. In: Giannessi F, Komlósi S, Rapcśak T. editors. New trends in mathematical programming. Dordrecht: Kluwer Academic Publishers; 1997. p. 93–99], [Golestani M, Sadeghi H, Tavan Y. Nonsmooth multiobjective problems and generalized vector variational inequalities using quasi-efficiency. J Optim Theory Appl. 2018;179(3):896–916], [Lee GM, Lee KB. Vector variational inequalities for nondifferentiable convex vector optimization problems. J Global Optim. 2005;32(4):597–612], [Mishra SK, Upadhyay BB. Some relations between vector variational inequality problems and nonsmooth vector optimization problems using quasi efficiency. Positivity. 2013;17(4):1071–1083], [Upadhyay BB, Mohapatra RN, Mishra SK. On relationships between vector variational inequality and nonsmooth vector optimization problems via strict minimizers. Adv Nonlinear Var Inequal. 2017;20(2):1–12], [Zhang J, Zheng Q,Ma X, Li L. Relationships between interval-valued vector optimization problems and vector variational inequalities. Fuzzy Optim Decis Mak. 2016;15(1):33–55] for nonsmooth interval-valued multiobjective programming problem by using the powerful tool of Mordukhovich subdifferential.