2,023 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
The Computational Complexity of the Restricted Isometry Property, the Nullspace Property, and Related Concepts in Compressed Sensing
This paper deals with the computational complexity of conditions which
guarantee that the NP-hard problem of finding the sparsest solution to an
underdetermined linear system can be solved by efficient algorithms. In the
literature, several such conditions have been introduced. The most well-known
ones are the mutual coherence, the restricted isometry property (RIP), and the
nullspace property (NSP). While evaluating the mutual coherence of a given
matrix is easy, it has been suspected for some time that evaluating RIP and NSP
is computationally intractable in general. We confirm these conjectures by
showing that for a given matrix A and positive integer k, computing the best
constants for which the RIP or NSP hold is, in general, NP-hard. These results
are based on the fact that determining the spark of a matrix is NP-hard, which
is also established in this paper. Furthermore, we also give several complexity
statements about problems related to the above concepts.Comment: 13 pages; accepted for publication in IEEE Trans. Inf. Theor
Exponential Lower Bounds for Polytopes in Combinatorial Optimization
We solve a 20-year old problem posed by Yannakakis and prove that there
exists no polynomial-size linear program (LP) whose associated polytope
projects to the traveling salesman polytope, even if the LP is not required to
be symmetric. Moreover, we prove that this holds also for the cut polytope and
the stable set polytope. These results were discovered through a new connection
that we make between one-way quantum communication protocols and semidefinite
programming reformulations of LPs.Comment: 19 pages, 4 figures. This version of the paper will appear in the
Journal of the ACM. The earlier conference version in STOC'12 had the title
"Linear vs. Semidefinite Extended Formulations: Exponential Separation and
Strong Lower Bounds
Block Factor-width-two Matrices and Their Applications to Semidefinite and Sum-of-squares Optimization
Semidefinite and sum-of-squares (SOS) optimization are fundamental
computational tools in many areas, including linear and nonlinear systems
theory. However, the scale of problems that can be addressed reliably and
efficiently is still limited. In this paper, we introduce a new notion of
\emph{block factor-width-two matrices} and build a new hierarchy of inner and
outer approximations of the cone of positive semidefinite (PSD) matrices. This
notion is a block extension of the standard factor-width-two matrices, and
allows for an improved inner-approximation of the PSD cone. In the context of
SOS optimization, this leads to a block extension of the \emph{scaled
diagonally dominant sum-of-squares (SDSOS)} polynomials. By varying a matrix
partition, the notion of block factor-width-two matrices can balance a
trade-off between the computation scalability and solution quality for solving
semidefinite and SOS optimization. Numerical experiments on large-scale
instances confirm our theoretical findings.Comment: 26 pages, 5 figures. Added a new section on the approximation quality
analysis using block factor-width-two matrices. Code is available through
https://github.com/zhengy09/SDPf
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