Convex Extensions in Combinatorial Optimization and Their Applications

Abstract

This research focuses on problems of combinatorial optimization necessary for mapping combinatorial sets into arithmetic Euclidean space. The analysis shows that there is a class of vertex-located sets that coincide with the set of vertices within their convex hull. The author has proved the theorems on the existence of convex, strongly convex, and differentiable extensions for functions defined on vertex-located sets. An equivalent problem of mathematical programming with convex objective function and functional constraints has been formulated. The author has studied the properties of convex function extremes on vertex-located sets. The research contains the examples of vertex-located combinatorial sets and algorithms for constructing convex, strongly convex, and differentiable extensions for functions defined on these sets. The conditions have been formulated that are sufficient for a minimum value of functions, as well as lower bounds of functions have been defined on the permutation set. The results obtained can be well used for developing new methods of combinatorial optimization.