One often encounters numerical difficulties in solving linear matrix inequality (LMI) problems obtained from $H_\infty$ control problems. We discuss the reason from the viewpoint of optimization, and provide necessary and sufficient conditions for LMI problem and its dual not to be strongly feasible. Moreover, we interpret them in terms of control system. In this analysis, … Read more
We consider semidefinite optimization problems that include constraints that G(x) and H(x) are positive semidefinite (PSD), where the components of the symmetric matrices G(x) and H(x) are affine functions of an n-vector x. In such a case we obtain a new constraint that a matrix K(x,X) is PSD, where the components of K(x,X) are affine … Read more
A framework, PhaseLift, was recently proposed to solve the phase retrieval problem. In this framework, the problem is solved by optimizing a cost function over the set of complex Hermitian positive semidefinite matrices. This approach to phase retrieval motivates a more general consideration of optimizing cost functions on semidefinite Hermitian matrices where the desired minimizers … Read more
We derive sharp probability bounds on the tails of a product of symmetric non-negative random variables using only information about their first two moments. If the covariance matrix of the random variables is known exactly, these bounds can be computed numerically using semidefinite programming. If only an upper bound on the covariance matrix is available, … Read more
The bounded degree sum-of-squares (BSOS) hierarchy of Lasserre, Toh, and Yang [EURO J. Comput. Optim., 2015] constructs lower bounds for a general polynomial optimization problem with compact feasible set, by solving a sequence of semi-definite programming (SDP) problems. Lasserre, Toh, and Yang prove that these lower bounds converge to the optimal value of the original … Read more
Mixed-integer semidefinite programs arise in many applications and several problem-specific solution approaches have been studied recently. In this paper, we investigate a generic branch-and-bound framework for solving such problems. We first show that strict duality of the semidefinite relaxations is inherited to the subproblems. Then solver components like dual fixing, branching rules, and primal heuristics … Read more
One of the fundamental tasks in compressed sensing is finding the sparsest solution to an underdetermined system of linear equations. It is well known that although this problem is NP-hard, under certain conditions it can be solved by using a linear program which minimizes the 1-norm. The restricted isometry property has been one of the … Read more
We consider the maximum k-cut problem that consists in partitioning the vertex set of a graph into k subsets such that the sum of the weights of edges joining vertices in different subsets is maximized. We focus on identifying effective classes of inequalities to tighten the semidefinite programming relaxation. We carry out an experimental study … Read more
We consider the problem of minimizing a given $n$-variate polynomial $f$ over the hypercube $[-1,1]^n$. An idea introduced by Lasserre, is to find a probability distribution on $[-1,1]^n$ with polynomial density function $h$ (of given degree $r$) that minimizes the expectation $\int_{[-1,1]^n} f(x)h(x)d\mu(x)$, where $d\mu(x)$ is a fixed, finite Borel measure supported on $[-1,1]^n$. It … Read more
We present a coordinate ascent method for a class of semidefinite programming problems that arise in non-convex quadratic integer optimization. These semidefinite programs are characterized by a small total number of active constraints and by low-rank constraint matrices. We exploit this special structure by solving the dual problem, using a barrier method in combination with … Read more