## Preprint 2011-10

## Fabián Flores-Bazán, Fernando Flores-Bazán, Cristian Vera:

### Gordan-type alternative theorems and vector optimization revisited

### Abstract:

Theorems of the alternative has proved to be one of the most powerful tools in optimization theory. They provide existence of Lagrange multipliers, (strong) duality results, linear scalarizations of various classes of solutions to vector optimization problems. This chapter is devoted to this last part of applications. The chapter starts by recalling the (1957) Fan-Glicksberg-Hoffman alternative theorem for convex functions. Then, many equivalent formulations to a general Gordan-type alternative theorem valid for (not necessarily pointed) convex cones with possibly empty interior, are established. They will be expressed in terms of quasi relative interior. Several classes of generalized convexity for sets and for vector valued mappings, are revisited. Applications to linear characterizations of weakly efficient, (Benson) proper efficient solutions, and to characterize the Fritz-John type optimality condition in vector optimization, are discussed. Finally, we also present some recent developments about proper efficiency.