Scalable Semidefinite Programming

TL;DR

This paper introduces a scalable randomized algorithm for large semidefinite programming problems, significantly reducing computational costs and enabling solutions for extremely large matrices in data science applications.

Contribution

It presents a provably correct, memory-efficient randomized method for solving large, weakly constrained SDPs, expanding the practical applicability of SDP in data science.

Findings

Effective for relaxations of MaxCut, phase retrieval, and quadratic assignment

Handles SDP matrices with over 10^14 entries on a standard laptop

Demonstrates computational efficiency and accuracy in numerical experiments

Abstract

Semidefinite programming (SDP) is a powerful framework from convex optimization that has striking potential for data science applications. This paper develops a provably correct randomized algorithm for solving large, weakly constrained SDP problems by economizing on the storage and arithmetic costs. Numerical evidence shows that the method is effective for a range of applications, including relaxations of MaxCut, abstract phase retrieval, and quadratic assignment. Running on a laptop equivalent, the algorithm can handle SDP instances where the matrix variable has over $10^{14}$ entries.