Related papers: Semidefinite relaxations for semi-infinite polynom…
We consider optimization problems containing nonconvex quadratic functions for which semidefinite programming (SDP) relaxations often yield strong bounds. We investigate linear inequalities that outer approximate the positive semidefinite…
This paper develops a new storage-optimal algorithm that provably solves generic semidefinite programs (SDPs) in standard form. This method is particularly effective for weakly constrained SDPs. The key idea is to formulate an approximate…
This paper considers the optimization problem of minimizing a rational function. We reformulate this problem as polynomial optimization by the technique of homogenization. These two problems are shown to be equivalent under some generic…
We propose a homogeneous primal-dual interior-point method to solve sum-of-squares optimization problems by combining non-symmetric conic optimization techniques and polynomial interpolation. The approach optimizes directly over the…
Semidefinite programs (SDPs) often arise in relaxations of some NP-hard problems, and if the solution of the SDP obeys certain rank constraints, the relaxation will be tight. Decomposition methods based on chordal sparsity have already been…
Semidefinite relaxations are a powerful tool for approximately solving combinatorial optimization problems such as MAX-CUT and the Grothendieck problem. By exploiting a bounded rank property of extreme points in the semidefinite cone, we…
The problem of sparse approximation and the closely related compressed sensing have received tremendous attention in the past decade. Primarily studied from the viewpoint of applied harmonic analysis and signal processing, there have been…
The min-knapsack problem with compactness constraints extends the classical knapsack problem, in the case of ordered items, by introducing a restriction ensuring that they cannot be too far apart. This problem has applications in…
We consider a new semidefinite programming (SDP) relaxation of the symmetric traveling salesman problem (TSP) that may be obtained via an SDP relaxation of the more general quadratic assignment problem (QAP). We show that the new relaxation…
Semidefinite programming (SDP) provides a powerful relaxation for the maximum cut problem. For a graph with rational weights, the decision problem of whether the SDP relaxation for the maximum cut problem is exact is known to be $NP$-hard;…
Probabilistic inference in pairwise Markov Random Fields (MRFs), i.e. computing the partition function or computing a MAP estimate of the variables, is a foundational problem in probabilistic graphical models. Semidefinite programming…
We give a quantum speedup for solving the canonical semidefinite programming relaxation for binary quadratic optimization. This class of relaxations for combinatorial optimization has so far eluded quantum speedups. Our methods combine…
In this study, we investigate the application of Semidefinite Programming (SDP) to phylogenetics. SDP is a powerful optimization framework that seeks to optimize a linear objective function over the cone of positive semidefinite matrices.…
We consider the class of polynomial optimization problems $\inf \{f(x):x\in K\}$ for which the quadratic module generated by the polynomials that define $K$ and the polynomial $c-f$ (for some scalar $c$) is Archimedean. For such problems,…
We introduce a generic technique to obtain linear relaxations of semidefinite programs with provable guarantees based on the commutativity of the constraint and the objective matrices. We study conditions under which the optimal value of…
The pooling problem is an important industrial problem in the class of network flow problems for allocating gas flow in pipeline transportation networks. For P-formulation of the pooling problem with time discretization, we propose second…
We approximate the backward reachable set of discrete-time autonomous polynomial systems using the recently developed occupation measure approach. We formulate the problem as an infinite-dimensional linear programming (LP) problem on…
We propose a manifold optimization approach to solve linear semidefinite programs (SDP) with low-rank solutions, with an emphasis on SDP relaxations for polynomial optimization problems. This approach incorporates the inexact augmented…
This thesis explores algorithmic applications and limitations of convex relaxation hierarchies for approximating some discrete and continuous optimization problems. - We show a dichotomy of approximability of constraint satisfaction…
In this paper, we propose a low-rank coordinate descent approach to structured semidefinite programming with diagonal constraints. The approach, which we call the Mixing method, is extremely simple to implement, has no free parameters, and…