Canonical dual method for mixed integer fourth-order polynomial minimization problems with fixed cost terms

TL;DR

This paper introduces a canonical duality approach to solve complex mixed-integer nonconvex fourth-order polynomial minimization problems, transforming them into a dual problem that allows for global optimality analysis and solutions.

Contribution

The paper develops a novel canonical duality framework that converts challenging mixed-integer nonconvex problems into continuous concave dual problems, enabling global optimality conditions and solutions.

Findings

The dual problem has no duality gap.

Global optimality conditions are established.

Analytic solutions are obtained for specific cases.

Abstract

We study a canonical duality method to solve a mixed-integer nonconvex fourth-order polynomial minimization problem with fixed cost terms. This constrained nonconvex problem can be transformed into a continuous concave maximization dual problem without duality gap. The global optimality conditions are proposed and the existence and uniqueness criteria are discussed. Application to a decoupled mixed-integer problem is illustrated and analytic solution for a global minimum is obtained under some suitable conditions. Several examples are given to show the method is effective.