19,557 research outputs found
A Note on the Hyperbolicity Cone of the Specialized V\'amos Polynomial
The specialized V\'amos polynomial is a hyperbolic polynomial of degree four
in four variables with the property that none of its powers admits a definite
determinantal representation. We will use a heuristical method to prove that
its hyperbolicity cone is a spectrahedron.Comment: Notable easier arguments and minor correction
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
A Tensor Analogy of Yuan's Theorem of the Alternative and Polynomial Optimization with Sign structure
Yuan's theorem of the alternative is an important theoretical tool in
optimization, which provides a checkable certificate for the infeasibility of a
strict inequality system involving two homogeneous quadratic functions. In this
paper, we provide a tractable extension of Yuan's theorem of the alternative to
the symmetric tensor setting. As an application, we establish that the optimal
value of a class of nonconvex polynomial optimization problems with suitable
sign structure (or more explicitly, with essentially non-positive coefficients)
can be computed by a related convex conic programming problem, and the optimal
solution of these nonconvex polynomial optimization problems can be recovered
from the corresponding solution of the convex conic programming problem.
Moreover, we obtain that this class of nonconvex polynomial optimization
problems enjoy exact sum-of-squares relaxation, and so, can be solved via a
single semidefinite programming problem.Comment: acceted by Journal of Optimization Theory and its application, UNSW
preprint, 22 page
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