Sparsity in sums of squares of polynomials

Abstract

Abstract.Representation of a given nonnegative multivariate polynomial in terms of a sum of squares of polynomials has become an essential subject in recent developments of sums of squares optimization and SDP (semidefinite programming) relaxation of polynomial optimization problems. We discuss effective methods to obtain a simpler representation of a "sparse" polynomial as a sum of squares of sparse polynomials by eliminating redundancy.

A facial reduction algorithm for finding sparse SOS representations

“…As a by-procut of this result, we prove that a computationally heavy part of EMSSOSP proposed in [3] is redundant for finding a set of unnecessary monomials for an SOS representation of a sparse SOS polynomial. This part enumerates all integer vectors in the convex hull of a set, and the authors in [3] reported that the part has much more computational cost than the other part.…”

“…(2), then stop and return G i . Step 3:: Otherwise set G i+1 = H i and i = i + 1, and go back to Step 2. We remark that EMSSOSP(Ḡ) is EMSSOSP proposed in [3].…”

“…In [3], the authors reported that before executing EMSSOSP(Ḡ), we need to enumerate all integer points ofḠ and that we need much computational cost for this part. The following theorem guarantees that we can obtain the same set G * of monomials as EMSSOSP(Ḡ) even if we start EMSSOSP(G) from an arbitrary set G includingḠ.…”

“…sparsity and/or symmetry, for reducing the size of the SDP. For this, Kojima et al [3] proposed a method to reduce the size of the SDP obtained from a sparse SOS polynomial. This method was also discussed in [6].…”

“…(ii) if G ∈ Γ and there exist nonempty sets B, H ⊆ G such that the triplet (B, H, G) satisfies (2), H ∈ Γ. In [3], the authors proved that if, G, G ∈ Γ, then G ∩G ∈ Γ. This implies the existence of the smallest finite setǦ in Γ in the sense thatǦ ⊆ G for any G ∈ Γ.…”