A Boosted-DCA with Power-Sum-DC Decomposition for Linearly Constrained Polynomial Programs

TL;DR

This paper introduces a new DC decomposition for polynomials using power-sum representation and develops an efficient boosted DCA method with exact line search for linearly constrained polynomial programs, validated through numerical experiments.

Contribution

The paper presents a novel power-sum-based DC decomposition and a boosted DCA with exact line search, improving solution efficiency for constrained polynomial optimization.

Findings

BDCAe converges to critical points with proven subsequential convergence.

The exact line search reduces to finding roots of a univariate polynomial.

Numerical experiments show BDCAe outperforms existing methods and standard solvers.

Abstract

This paper proposes a novel Difference-of-Convex (DC) decomposition for polynomials using a power-sum representation, achieved by solving a sparse linear system. We introduce the Boosted DCA with Exact Line Search (BDCAe) for addressing linearly constrained polynomial programs within the DC framework. Notably, we demonstrate that the exact line search equates to determining the roots of a univariate polynomial in an interval, with coefficients being computed explicitly based on the power-sum DC decompositions. The subsequential convergence of BDCAe to critical points is proven, and its convergence rate under the Kurdyka-Lojasiewicz property is established. To efficiently tackle the convex subproblems, we integrate the Fast Dual Proximal Gradient (FDPG) method by exploiting the separable block structure of the power-sum DC decompositions. We validate our approach through numerical experiments on the Mean-Variance-Skewness-Kurtosis (MVSK) portfolio optimization model and box-constrained polynomial optimization problems. Comparative analysis of BDCAe against DCA, BDCA with Armijo line search, UDCA, and UBDCA with projective DC decomposition, alongside standard nonlinear optimization solvers FMINCON and FILTERSD, substantiates the efficiency of our proposed approach.