Construction of Vector-Valued Weak Separation Functions with Applications to Conjugate Duality in Vector Optimization

In our previous paper (J. Nonlinear Var. Anal. 7: 859–896, 2023), collections of scalar weak separation functions for image space analysis were proposed, while, this paper is concerned with vector-valued weak separation functions. Applying vector-valued Gerstewitz and topical functions, some vector-valued nonlinear weak separation functions are constructed and investigated. Then, with the aid of that, another framework of conjugate duality for constrained vector optimization problems is established. Using vector-valued separation functions, the dual model is given for the vector optimization problem directly, unlike the previous work, where the primal problem was scalarized. In our new approach to duality, we avoid a scalarization in the formulation of the dual problem. We study a pair of a primal vector-valued problem and a dual set-valued problem. Simultaneously, the zero duality gap and strong duality are studied by certain concepts of subdifferentials, separation and saddle points in the vector-valued sense.