On the Triality Theory for a Quartic Polynomial Optimization Problem

TL;DR

This paper rigorously proves the triality theorem for fourth-order polynomial optimization problems, clarifying conditions under which different duality forms hold and demonstrating practical applications through numerical examples.

Contribution

It provides a complete proof of the triality theorem for quartic polynomial problems and resolves an open problem from 2003 regarding double-min duality.

Findings

Triality theory holds strongly when primal and dual have same dimension.

Double-min duality is weakly valid when dimensions differ.

Numerical examples show the theory can identify global and local extrema.

Abstract

This paper presents a detailed proof of the triality theorem for a class of fourth-order polynomial optimization problems. The method is based on linear algebra but it solves an open problem on the double-min duality left in 2003. Results show that the triality theory holds strongly in a tri-duality form if the primal problem and its canonical dual have the same dimension; otherwise, both the canonical min-max duality and the double-max duality still hold strongly, but the double-min duality holds weakly in a symmetrical form. Four numerical examples are presented to illustrate that this theory can be used to identify not only the global minimum, but also the largest local minimum and local maximum.