Vector-valued separation functions and constrained vector optimization problems: optimality and saddle points

Abstract

In this paper, we consider a class of constrained vector optimization problems by using image space analysis. A class of vector-valued separation functions and a C-solution notion are proposed for the constrained vector optimization problems, respectively. Moreover, existence of a saddle point for the vector-valued separation function is characterized by the (regular) separation of two suitable subsets of the image space. By employing the separation function, we introduce a class of generalized vector-valued Lagrangian functions without involving any elements of the feasible set of constrained vector optimization problems. The relationships between the type-I(II) saddle points of the generalized Lagrangian functions and that of the function corresponding to the separation function are also established. Finally, optimality conditions for C-solutions of constrained vector optimization problems are derived by the saddle-point conditions.

Image space analysis for uncertain multiobjective optimization problems: Robust optimality conditions

“…Definition 3.1. [13,10] The class of all functions w : R k+m × Π → R p such that (i) lev ≥ 0 w(•; π) ⊇ H for all π ∈ Π, (ii) π∈Π lev > 0 0 w(•; π) ⊆ H, is called the class of weak separation functions and it is denoted by W (Π).…”

“…(c) The nonlinear functions w 3 and w 4 can be proved to be weak separation functions by using similar argument as in [29]. (d) The vector-valued function w 5 can be proved to be weak separation function by using similar argument as in [10]. Remark 2.…”