Abstract. The purpose of this paper is to establish necessary and sufficient conditions for a point to be solution of a vector equilibrium problem with cone and affine constraints. Using a separation theorem, which involves the quasi-interior of a convex set, we obtain optimality conditions for solutions of the vector equilibrium problem. Then, the main result is applied to vector optimization problems with cone and affine constraints and to duality theory.1. Introduction. In this paper we study a vector equilibrium problem which generalize the famous Ky Fan inequality introduced by Fan in [20]. Later on, this inequality was called equilibrium problem by Blum and Oettli [11]. The vector equilibrium problems include vector optimization problems, vector variational inequality problems, vector complementarity problems, and cone saddle point problems, as particular cases.A large number of results for different vector equilibrium problems are established, such as existence of solutions (see, for instance, [2,3,4,5,6,7,8,10,16,21]), well-posedness (see, for instance, [1, 9, 31]), sensitivity analysis (see, for instance, [24,25]). But, as far as we know, there are only a few papers concerned with optimality conditions for solutions. Recently, Gong [22] and Ma and Gong [27] have obtained optimality conditions for weakly efficient solutions, Henig efficient solutions, and Henig globally efficient solutions of a vector equilibrium problem, using Slater type conditions and different differentiability notions for vector functions. In [28] and [29] the authors obtained optimality conditions for Henig globally efficient solutions under weaker convexity assumptions than those stated in [22]. The results existent in the literature deal with necessary and sufficient conditions for weak efficient solutions, Henig efficient solutions, superefficient solutions and Henig globally efficient solutions, in the hypothesis of a cone with nonempty interior. Motivated by the fact that important practical spaces admit cones with empty interiors, in what follows we give optimality conditions for a point to be weakly efficient solution, by using the notion of quasi-relative interior of a convex set. Our results generalize corresponding results established in [22].The paper is organized as follows. In Section 2 we recall some notions and auxiliary results. In Section 3 we give optimality conditions for solutions of problem
