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Finding Low-rank Solutions of Sparse Linear Matrix Inequalities using Convex Optimization
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A mathematical framework to relate the rank of the minimum-rank solution of the LMI problem to the sparsity of its underlying graph is developed and three graph-theoretic convex programs to obtain a low- rank solution are proposed.
Graph-theoretic algorithms for polynomial optimization problems
- Computer Science, Mathematics53rd IEEE Conference on Decision and Control
- 2014
It is shown that every polynomial optimization problem admits a sparse representation whose SDP relaxation has a rank 1 or 2 solution.
Polynomial Optimization via Penalized Conic Relaxation
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- 2018
This paper revisits conic programming relaxations for the class of quadratically-constrained quadratic programs (QCQPs). We present penalty terms, whose incorporation into the objective of convex…
Grothendieck‐Type Inequalities in Combinatorial Optimization
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- 2011
We survey connections of the Grothendieck inequality and its variants to combinatorial optimization and computational complexity. © 2011 Wiley Periodicals, Inc.
Rank-2 Matrix Solution for Semidefinite Relaxations of Arbitrary Polynomial Optimization Problems
- Computer Science
- 2014
This work shows that an arbitrary polynomial optimization has an equivalent formulation in the form of a sparse quadraticallyconstrained quadratic program (QCQP) whose SDP relaxation possesses a matrix solution with rank at most 2.
Combinatorial conditions for low rank solutions in semidefinite programming
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This thesis investigates combinatorial conditions that guarantee the existence of low-rank optimal solutions to semidefinite programs and introduces a graph parameter called the extreme Gram dimension of a graph, which is upper bounded by the Gram dimension and closely related to the rank-constrained Grothendieck constant.
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