Euclidean Combinatorial Configurations: Continuous Representations and Convex Extensions

Abstract

This paper presents general approaches to continuous functional representations (f-representations) of sets of Euclidean combinatorial configurations underlying the possibility of applying continuous programming to optimization on them. A concept of Euclidean combinatorial configurations (e-configurations) is offered. Applications of f-representations in optimization and reformulations of extreme combinatorial problems as global optimization problems are outlined. A typology of f-representations is presented and approaches to construct them for classes that were singled out are described and applied in forming a number of polynomial f-representations of basic sets of e-configurations related to permutation and Boolean configurations. The paper’s results can be applied in solving numerous real-world problems formulated as permutation-based, binary or Boolean.