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

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

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arXiv · 1803.09900

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

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

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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.

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BibTeX RIS

Yi-Shuai Niu, Hoai An Le Thi, Dinh Tao Pham. 2018-03-27. On Difference-of-SOS and Difference-of-Convex-SOS Decompositions for Polynomials. https://arxiv.org/abs/1803.09900

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