Global Solutions to Nonconvex Optimization of 4th-Order Polynomial and Log-Sum-Exp Functions

TL;DR

This paper introduces a canonical duality-triality framework for solving complex nonconvex optimization problems involving fourth-order polynomials and log-sum-exp functions, enabling analytical solutions for global and local extrema.

Contribution

It develops a novel canonical dual approach that transforms challenging nonconvex problems into solvable dual problems, providing explicit solutions and classification criteria.

Findings

Global minimizers and local extrema can be obtained analytically from dual solutions.

Existence conditions help classify problem difficulty.

Applications demonstrate effectiveness on nonconvex, nonsmooth examples.

Abstract

This paper presents a canonical dual approach for solving a nonconvex global optimization problem governed by a sum of fourth-order polynomial and a log-sum-exp function. Such a problem arises extensively in engineering and sciences. Based on the canonical duality-triality theory, this nonconvex problem is transformed to an equivalent dual problem, which can be solved easily under certain conditions. We proved that both global minimizer and the biggest local extrema of the primal problem can be obtained analytically from the canonical dual solutions. As two special cases, a quartic polynomial minimization and a minimax problem are discussed. Existence conditions are derived, which can be used to classify easy and relative hard instances. Applications are illustrated by several nonconvex and nonsmooth examples.