401 research outputs found
Improving Efficiency and Scalability of Sum of Squares Optimization: Recent Advances and Limitations
It is well-known that any sum of squares (SOS) program can be cast as a
semidefinite program (SDP) of a particular structure and that therein lies the
computational bottleneck for SOS programs, as the SDPs generated by this
procedure are large and costly to solve when the polynomials involved in the
SOS programs have a large number of variables and degree. In this paper, we
review SOS optimization techniques and present two new methods for improving
their computational efficiency. The first method leverages the sparsity of the
underlying SDP to obtain computational speed-ups. Further improvements can be
obtained if the coefficients of the polynomials that describe the problem have
a particular sparsity pattern, called chordal sparsity. The second method
bypasses semidefinite programming altogether and relies instead on solving a
sequence of more tractable convex programs, namely linear and second order cone
programs. This opens up the question as to how well one can approximate the
cone of SOS polynomials by second order representable cones. In the last part
of the paper, we present some recent negative results related to this question.Comment: Tutorial for CDC 201
Bilinearity rank of the cone of positive polynomials and related cones
For a proper cone K β Rn and its dual cone K the complementary slackness condition xT s = 0 defines an n-dimensional manifold C(K) in the space { (x, s) | x β K, s β K^* }. When K is a symmetric cone, this manifold can be described by a set of n bilinear equalities. When K is a symmetric cone, this fact translates to a set of n linearly independent bilinear identities (optimality conditions) satisfied by every (x, s) β C(K). This proves to be very useful when
optimizing over such cones, therefore it is natural to look for similar optimality conditions for non-symmetric cones. In this paper we define the bilinearity rank of a cone, which is the number of linearly independent bilinear identities valid for the cone, and describe a linear algebraic technique to bound this quantity. We examine several well-known cones, in particular
the cone of positive polynomials P2n+1 and its dual, the closure of the moment cone M2n+1, and compute their bilinearity ranks. We show that there are exactly four linearly independent bilinear identities which hold for all (x,s) β C(P2n+1), regardless of the dimension of the cones. For nonnegative polynomials over an interval or half-line there are only two linearly independent bilinear identities. These results are extended to trigonometric and exponential
polynomials
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