On modeling and global solutions for d.c. optimization problems by canonical duality theory

TL;DR

This paper introduces a canonical duality theory approach to model and solve complex nonconvex optimization problems by transforming them into concave maximization problems, providing a unified framework for global solutions.

Contribution

It develops a canonical duality framework that converts a broad class of nonconvex problems into easier-to-solve concave maximization problems, including proof of triality theory.

Findings

Nonconvex minimization problems can be transformed into concave maximization problems.

The canonical duality approach simplifies solving complex global optimization problems.

Proofs of triality theory help identify local extremal solutions.

Abstract

This paper presents a canonical d.c. (difference of canonical and convex functions) programming problem, which can be used to model general global optimization problems in complex systems. It shows that by using the canonical duality theory, a large class of nonconvex minimization problems can be equivalently converted to a unified concave maximization problem over a convex domain, which can be solved easily under certain conditions. Additionally, a detailed proof for triality theory is provided, which can be used to identify local extremal solutions. Applications are illustrated and open problems are presented.