[Paper Review] On Difference-of-SOS and Difference-of-Convex-SOS Decompositions for Polynomials

Yi-Shuai Niu, Thi, Hoai An Le+1 more|arXiv (Cornell University)|Mar 27, 2018

Advanced Optimization Algorithms Research38 references4 citations

TL;DR

This paper introduces Difference-of-Sums-of-Squares (D-SOS) and Difference-of-Convex-Sums-of-Squares (DC-SOS) decompositions for multivariate polynomials, enabling reformulation of nonconvex polynomial optimization problems into DC programs. The proposed methods allow efficient solution via semidefinite programming or direct algebraic algorithms, with applications in portfolio optimization, matrix completion, and Boolean programming.

ABSTRACT

In this article, we are interested in developing polynomial decomposition techniques based on sums-of-squares (SOS), namely the difference-of-sums-of-squares (D-SOS) and the difference-of-convex-sums-of-squares (DC-SOS). In particular, the DC-SOS decomposition is very useful for difference-of-convex (DC) programming formulation of polynomial optimization problems. First, we introduce the cone of convex-sums-of-squares (CSOS) polynomials and discuss its relationship to the sums-of-squares (SOS) polynomials, the non-negative polynomials and the SOS-convex polynomials. Then, we propose the set of D-SOS and DC-SOS polynomials, and prove that any polynomial can be formulated as D-SOS and DC-SOS. The problem of finding D-SOS and DC-SOS decompositions can be formulated as a semi-definite program and solved for any desired precision in polynomial time using interior point methods. Some algebraic properties of CSOS, D-SOS and DC-SOS are established. Second, we focus on establishing several practical algorithms for exact D-SOS and DC-SOS polynomial decompositions without solving any SDP. The numerical performance of the proposed D-SOS and DC-SOS decomposition algorithms and their parallel versions, tested on a dataset of 1750 randomly generated polynomials, is reported.

Motivation & Objective

• To develop algebraic decomposition techniques that express any real-valued multivariate polynomial as a difference of sums-of-squares (D-SOS) or difference of convex-sums-of-squares (DC-SOS).
• To establish that D-SOS and DC-SOS polynomials form vector spaces equivalent to the space of all real polynomials.
• To provide practical, non-SDP-based algorithms for constructing D-SOS and DC-SOS decompositions with minimal-degree components.
• To enable reformulation of polynomial optimization problems into the DC programming framework for efficient numerical solution.
• To demonstrate the utility of these decompositions in real-world applications such as moment portfolio optimization, eigenvalue complementarity, and Boolean polynomial programs.

Proposed method

• Prove that the sets of D-SOS and DC-SOS polynomials are vector spaces and equivalent to the space of all real polynomials.
• Show that finding D-SOS and DC-SOS decompositions is equivalent to solving semidefinite programs (SDPs), which can be solved to arbitrary precision in polynomial time.
• Propose practical, non-SDP algorithms—MD-DCSOS and FMD-DCSOS—for constructing D-SOS and DC-SOS decompositions without solving SDPs.
• Use spectral decomposition techniques to generate undominated D-SOS decompositions, improving convergence in DC algorithms.
• Construct CSOS polynomial bases for even-degree homogeneous polynomials using power-product matrices, enabling efficient DC-SOS representations with sub-exponential growth in the number of squares.
• Integrate DC-SOS decomposition with DCA and Lasserre’s SDP relaxation to improve local minimizer quality in polynomial optimization.

Experimental results

Research questions

• RQ1 Can any real-valued multivariate polynomial be decomposed into a difference of sums-of-squares (D-SOS) polynomials, and is this decomposition unique or structurally constrained?
• RQ2 Can D-SOS and DC-SOS decompositions be constructed without solving semidefinite programs, and what are the computational advantages of such algebraic methods?
• RQ3 What is the relationship between the sets of SOS, PSD, convex-SOS, SOS-convex, D-SOS, and DC-SOS polynomials, and how do they relate to polynomial optimization?
• RQ4 Can undominated DC-SOS decompositions be systematically generated, and how does this affect the convergence of DCA in polynomial optimization?
• RQ5 Can CSOS bases be constructed for homogeneous polynomials of even degree such that the number of squares grows sub-exponentially, enabling efficient DC-SOS representations?

Key findings

• The sets of D-SOS and DC-SOS polynomials are vector spaces and are equivalent to the space of all real-valued polynomials.
• D-SOS and DC-SOS decomposition problems are equivalent to semidefinite programs (SDPs), solvable in polynomial time to any desired accuracy.
• Practical, non-SDP-based algorithms—MD-DCSOS and FMD-DCSOS—were developed for constructing D-SOS and DC-SOS decompositions with minimal-degree components.
• Numerical experiments on 1750 synthetic polynomials (sparse and dense, large and small) confirmed the efficiency and scalability of the proposed decomposition algorithms and their parallelized variants.
• The spectral D-SOS decomposition yields undominated D-SOS decompositions, which are beneficial for convergence in DCA-based solvers.
• A CSOS basis construction using power-product matrices was proposed, showing potential for sub-exponential growth in the number of squares, enabling efficient DC-SOS representations for even-degree homogeneous polynomials.