Asymptotic conditions for weak and proper optimality in infinite dimensional convex vector optimization

In this paper, we establish necessary and sufficient dual conditions for weak and proper minimality of infinite dimensional vector convex programming problems without any regularity conditions. The optimality conditions are given in asymptotic forms using epigraphs of conjugate functions and subdifferentials. It is shown how these asymptotic conditions yield standard Lagrangian conditions under appropriate regularity conditions. The main tool, used to obtain these results, is a new solvability result of Motzkin type for cone convex systems. We also provide local Lagrangian necessary conditions for certain non-convex problems using convex approximations.