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On Difference-of-SOS and Difference-of-Convex-SOS Decompositions for Polynomials

Yi-Shuai Niu, Hoai An Le Thi, Dinh Tao Pham

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.

On Difference-of-SOS and Difference-of-Convex-SOS Decompositions for Polynomials

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.

Paper Structure

This paper contains 30 sections, 20 theorems, 57 equations, 2 figures, 2 tables, 9 algorithms.

Table of Contents

  1. Introduction
  2. Polynomial decompositions based on SOS
  3. The proper cone of CSOS polynomials
  4. Algebraic properties of CSOS polynomials
  5. Complexity of CSOS certification
  6. Relationships (CSOS, SOS-convexity and PSD convexity)
  7. The vector spaces of D-SOS and DC-SOS polynomials
  8. Algebraic properties of D-SOS and DC-SOS
  9. Complexity of D-SOS and DC-SOS decompositions
  10. Spectral D-SOS decomposition techniques
  11. General spectral D-SOS decomposition
  12. Direct basis spectral D-SOS decomposition
  13. Minimal basis spectral D-SOS decomposition
  14. Undominated D-SOS decomposition
  15. Comparisons of D-SOS decomposition algorithms
  16. ...and 15 more sections

Key Result

Lemma 2.3

(See Paper_Kojima2014) \newlabellemma:sosmatrix0 A polynomial matrix $P(x)\in \mathop{\mathrm{\mathbb{R}}}\nolimits[x]^{m\times m}$ is an SOS-matrix if and only if $y^{\mkern-1.5mu\mathsf{T}} P(x) y$ is an SOS polynomial in $\mathop{\mathrm{\mathbb{R}}}\nolimits[(x,y)]$.

Figures (2)

  • Figure 1: Numerical performance of D-SOS and DC-SOS decomposition algorithms
  • Figure 2: Speedup ratios of the parallel (IP-DCSOS) and the parallel (MD-DCSOS) for full-basis polynomials using up to $60$ processors

Theorems & Definitions (44)

  • Definition 2.1: CSOS polynomial
  • Definition 2.2: SOS-matrix and SOS-convex polynomial, see Paper_Helton2010
  • Lemma 2.3
  • Proposition 2.4
  • Proof 1
  • Proposition 2.5
  • Proposition 2.6
  • Proof 2
  • Proposition 2.7
  • Proof 3
  • ...and 34 more