Smaller SDP for SOS Decomposition

A popular numerical method to compute SOS (sum of squares of polynomials) decompositions for polynomials is to transform the problem into semi-definite programming (SDP) problems and then solve them by SDP solvers. In this paper, we focus on reducing the sizes of inputs to SDP solvers to improve the efficiency and reliability of those SDP based methods. Two types of polynomials, convex cover polynomials and split polynomials, are defined. A convex cover polynomial or a split polynomial can be decomposed into several smaller sub-polynomials such that the original polynomial is SOS if and only if the sub-polynomials are all SOS. Thus the original SOS problem can be decomposed equivalently into smaller sub-problems. It is proved that convex cover polynomials are split polynomials and it is quite possible that sparse polynomials with many variables are split polynomials, which can be efficiently detected in practice. Some necessary conditions for polynomials to be SOS are also given, which can help refute quickly those polynomials which have no SOS representations so that SDP solvers are not called in this case. All the new results lead to a new SDP based method to compute SOS decompositions, which improves this kind of methods by passing smaller inputs to SDP solvers in some cases. Experiments show that the number of monomials obtained by our program is often smaller than that by other SDP based software, especially for polynomials with many variables and high degrees. Numerical results on various tests are reported to show the performance of our program.Comment: 18 page

Constructing Fewer Open Cells by GCD Computation in CAD Projection

A new projection operator based on cylindrical algebraic decomposition (CAD) is proposed. The new operator computes the intersection of projection factor sets produced by different CAD projection orders. In other words, it computes the gcd of projection polynomials in the same variables produced by different CAD projection orders. We prove that the new operator still guarantees obtaining at least one sample point from every connected component of the highest dimension, and therefore, can be used for testing semi-definiteness of polynomials. Although the complexity of the new method is still doubly exponential, in many cases, the new operator does produce smaller projection factor sets and fewer open cells. Some examples of testing semi-definiteness of polynomials, which are difficult to be solved by existing tools, have been worked out efficiently by our program based on the new method.Comment: Accepted by ISSAC 2014 (July 23--25, 2014, Kobe, Japan