Diego Cifuentes We present a Julia-based interface to the precompiled HALLaR and cuHALLaR binaries for large-scale semidefinite programs (SDPs). Both solvers are established as fast and numerically stable, and accept problem data in formats compatible with SDPA and a new enhanced data format taking advantage of Hybrid Sparse Low-Rank (HSLR) structure. The interface allows users to load … Read more
This paper introduces cuHALLaR, a GPU-accelerated implementation of the HALLaR method proposed in Monteiro et al. 2024 for solving large-scale semidefinite programming (SDP) problems. We demonstrate how our Julia-based implementation efficiently uses GPU parallelism through optimization of simple, but key, operations, including linear maps, adjoints, and gradient evaluations. Extensive numerical experiments across three problem classes—maximum … Read more
The Lovasz theta function is a semidefinite programming (SDP) relaxation for the maximum weighted stable set problem, which is tight for perfect graphs. However, even for perfect graphs, there is no known rounding method guaranteed to extract an optimal stable set from the SDP solution. In this paper, we develop a novel rounding scheme for … Read more
This paper introduces HALLaR, a new first-order method for solving large-scale semidefinite programs (SDPs) with bounded domain. HALLaR is an inexact augmented Lagrangian (AL) method where the AL subproblems are solved by a novel hybrid low-rank (HLR) method. The recipe behind HLR is based on two key ingredients: 1) an adaptive inexact proximal point method … Read more
The Burer-Monteiro method is one of the most widely used techniques for solving large-scale semidefinite programs (SDP). The basic idea is to solve a nonconvex program in $Y$, where $Y$ is an $n \times p$ matrix such that $X = Y Y^T$. In this paper, we show that this method can solve SDPs in polynomial … Read more
Given an affine space of matrices L and a matrix \theta in L, consider the problem of finding the closest rank deficient matrix to \theta on L with respect to the Frobenius norm. This is a nonconvex problem with several applications in estimation problems. We introduce a novel semidefinite programming (SDP) relaxation, and we show … Read more
Consider a semidefinite program (SDP) involving an $n\times n$ positive semidefinite matrix $X$. The Burer-Monteiro method consists in solving a nonconvex program in $Y$, where $Y$ is an $n\times p$ matrix such that $X = Y Y^T$. Despite nonconvexity, Boumal et al. showed that the method provably solves generic equality-constrained SDP’s when $p > \sqrt{2m}$, … Read more
In this paper we consider a parametric family of polynomial optimization problems over algebraic sets. Although these problems are typically nonconvex, tractable convex relaxations via semidefinite programming (SDP) have been proposed. Often times in applications there is a natural value of the parameters for which the relaxation will solve the problem exactly. We study conditions … Read more