The Sufficiency of Solutions for Non-smooth Minimax Fractional Semi-Infinite Programming with (<i>B_K,ρ</i>)−Invexity

Abstract

Minimax fractional semi-infinite programming is an important research direction for semi-infinite programming, and has a wide range of applications, such as military allocation problems, economic theory, cooperative games, and other fields. Convexity theory plays a key role in many aspects of mathematical programming and is the foundation of mathematical programming research. The relevant theories of semi-infinite programming based on different types of convex functions have their own applicable scope and limitations. It is of great value to study semi-infinite programming on the basis of more generalized convex functions and obtain more general results. In this paper, we defined a new type of generalized convex function, based on the concept of the K−directional derivative, that is, uniform (BK,ρ)−invex, strictly uniform (BK,ρ)−invex, uniform (BK,ρ)−pseudoinvex, strictly uniform (BK,ρ)−pseudoinvex, uniform (BK,ρ)−quasiinvex and weakly uniform (BK,ρ)−quasiinvex function. Then, we studied a class of non-smooth minimax fractional semi-infinite programming problems involving this generalized convexity and obtained sufficient optimality conditions.

Keywords

• non-smooth programming
• fractional semi-infinite programming
• <i>K</i>−directional derivative
• uniform (<i>B_K</i>,<i>ρ</i>)−invexity
• optimality conditions