Four strategies to develop canonical dual algorithms for global optimization problems

TL;DR

This paper introduces four strategies based on unconstrained approaches to develop canonical dual algorithms for global optimization, leveraging the equivalence to semi-definite programming to solve challenging polynomial problems.

Contribution

It proposes four novel strategies to develop canonical dual algorithms that avoid SDP difficulties, expanding the applicability of duality theory in global optimization.

Findings

Strategies successfully applied to fourth-order polynomial benchmarks.

Canonical dual problems shown to be equivalent to SDP under certain conditions.

Algorithms improve solution efficiency for challenging nonconvex problems.

Abstract

The canonical duality theory has provided with a unified analytic solution to a range of discrete and continuous problems in global optimization, which can transform a nonconvex primal problem to a concave maximization dual problem over a convex domain without duality gap. This paper shows that under certain conditions, this canonical dual problem is equivalent to the standard semi-definite programming (SDP) problem, which can be solved by well-developed software packages. In order to avoid certain difficulties of using the SDP method, four strategies are proposed based on unconstrained approaches, which can be used to develop algorithms for solving some challenging problems. Applications are illustrated by fourth-order polynomials benchmark optimization problems.