Block SOS Decomposition

Awidely usedmethod for determiningwhether amultivariate polynomial is a sum of squares of polynomials (SOS), called SOS decomposition, is to decide the feasibility of corresponding semi-definite programming (SDP) problem which can be efficiently solved in theory. In practice, although existing SDP solvers can work out some problems of big scale, the efficiency and reliability of such method decrease greatly while the input size increases. Recently, by exploiting the sparsity of the input SOS decomposition problem, some preprocessing algorithms were proposed [5, 17], which first divide the input problem satisfying special properties into smaller SDP problems and then pass the smaller ones to SDP solvers to obtain reliable results efficiently. A natural question is that to what extent the above mentioned preprocessing algorithms work. That is, how many polynomials satisfying those properties are there in the SOS polynomials? In this paper, we define a concept of block SOS decomposable polynomials which is a generalization of those special classes of polynomials in [5] and [17]. Roughly speaking, it is a class of polynomials whose SOS decomposition problem can be transformed into smaller ones (in other words, the corresponding SDP matrices can be block-diagnolized) by considering their supports only (coefficients are not considered). Then we prove that the set of block SOS decomposable polynomials has measure zero in the set of SOS polynomials. That means if we only consider supports (not with coefficients) of polynomials, such algorithms decreasing the size of SDPs for those SDP-based SOS solvers can only work on very few polynomials. ACM Reference format: Haokun Li and Bican Xia. 2016. Block SOS Decomposition. In Proceedings of ACM Conference, Washington, DC, USA, July 2017 (Conference’17), 6 pages. DOI: 10.1145/nnnnnnn.nnnnnnn