Completely Positive Reformulations of Polynomial Optimization Problems with Linear Inequality Constraints

TL;DR

This paper introduces new convex reformulations of polynomial optimization problems with linear inequalities using completely positive tensors, enabling more tractable solutions and duality analysis.

Contribution

It presents novel completely positive reformulations for POPs with linear inequalities, including dual formulations and conditions for strong duality.

Findings

Reformulations as conic programs over the CPT cone.

Dual formulations with strict feasibility under mild conditions.

Strong duality results for the proposed reformulations.

Abstract

Polynomial optimization encompasses a broad class of problems in which both the objective function and constraints are polynomial functions of the decision variables. In recent years, a substantial body of research has focused on reformulating polynomial optimization problems (POPs) as conic programs over the cone of completely positive tensors (CPTs). In this article, we propose several new completely positive reformulations for a class of POPs with linear inequality constraints. Our approach begins by lifting these problems into a novel convex optimization framework, wherein the variables are represented as combinations of symmetric rank-one tensors. Based on this lifted formulation, we present a general characterization of POPs with linear inequality constraints that can be reformulated as conic programs over the CPT cone. Additionally, we construct the dual formulations of the resulting completely positive programs. Under mild assumptions, we prove that these dual problems are strictly feasible and strong duality holds.