Abstract: Optimality conditions and duality are of great importance in vector optimization with set-valued mappings. The aim of this paper is to establish the sufficient optimality condition and duality theorems for a kind of generalized convex set-valued optimization problems. Based upon the concept of invexity in terms of cone-approximating multifunction for a set-valued map, a new kind of generalized invexities, termed subinvex set-valued mappings, is introduced, and optimality conditions and duality theorems are investigated for its constraint set-valued optimization. It also presents an example to illustrate their existence. By employing the analytic method, a sufficient optimality condition and weak, strong, converse duality theorems between Mond-Weir and Wolfe dual problems and the primal constraint set-valued optimization problems are proposed in sense of weakly approximate minimizers. The results  obtained in this note enrich and deepen the theory and applications of set-valued optimization.

Key words: set-valued optimization, invexity, optimality conditions, cone-approximating multifunction, duality