1,338 research outputs found
Sparse sum-of-squares (SOS) optimization: A bridge between DSOS/SDSOS and SOS optimization for sparse polynomials
Optimization over non-negative polynomials is fundamental for nonlinear
systems analysis and control. We investigate the relation between three
tractable relaxations for optimizing over sparse non-negative polynomials:
sparse sum-of-squares (SSOS) optimization, diagonally dominant sum-of-squares
(DSOS) optimization, and scaled diagonally dominant sum-of-squares (SDSOS)
optimization. We prove that the set of SSOS polynomials, an inner approximation
of the cone of SOS polynomials, strictly contains the spaces of sparse
DSOS/SDSOS polynomials. When applicable, therefore, SSOS optimization is less
conservative than its DSOS/SDSOS counterparts. Numerical results for
large-scale sparse polynomial optimization problems demonstrate this fact, and
also that SSOS optimization can be faster than DSOS/SDSOS methods despite
requiring the solution of semidefinite programs instead of less expensive
linear/second-order cone programs.Comment: 9 pages, 3 figure
Sparse sum-of-squares certificates on finite abelian groups
Let G be a finite abelian group. This paper is concerned with nonnegative
functions on G that are sparse with respect to the Fourier basis. We establish
combinatorial conditions on subsets S and T of Fourier basis elements under
which nonnegative functions with Fourier support S are sums of squares of
functions with Fourier support T. Our combinatorial condition involves
constructing a chordal cover of a graph related to G and S (the Cayley graph
Cay(,S)) with maximal cliques related to T. Our result relies on two
main ingredients: the decomposition of sparse positive semidefinite matrices
with a chordal sparsity pattern, as well as a simple but key observation
exploiting the structure of the Fourier basis elements of G.
We apply our general result to two examples. First, in the case where , by constructing a particular chordal cover of the half-cube
graph, we prove that any nonnegative quadratic form in n binary variables is a
sum of squares of functions of degree at most , establishing
a conjecture of Laurent. Second, we consider nonnegative functions of degree d
on (when d divides N). By constructing a particular chordal
cover of the d'th power of the N-cycle, we prove that any such function is a
sum of squares of functions with at most nonzero Fourier
coefficients. Dually this shows that a certain cyclic polytope in
with N vertices can be expressed as a projection of a section
of the cone of psd matrices of size . Putting gives a
family of polytopes with LP extension complexity
and SDP extension complexity
. To the best of our knowledge, this is the
first explicit family of polytopes in increasing dimensions where
.Comment: 34 page
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