ABSTRACT
Abstract convexity provides global information of functions under consideration and presents some approaches to establish necessary and sufficient conditions for their global optimums. In this chapter we present the basic notions of abstract convex analysis and study three classes of abstract convex functions: Increasing and Positively Homogeneous (IPH) functions, Increasing and Co-Radiant (ICR) functions and increasing and plus-homogeneous (topical) functions. The solvability theorems from abstract convex analysis, also known as Farkas-type theorems, are instrumental to study global behavior of functions. Some versions of solvability theorems for systems of inequalities consisting of real-valued IPH, ICR and topical functions are established by means of generalized Fenchel-Moreau conjugate and polar functions. Vector-valued IPH, ICR and topical functions are introduced and solvability theorems for systems involving them are established as well. As applications, we apply these results to solve constrained IPH, ICR and topical optimization problems with unconstrained counterparts. In particular, linear programming problems with nonnegative coefficients in both objective function and constraints are transformed to unconstrained concave problems, without adding extra variables.
