A Projection Method for Metric-Constrained Optimization

TL;DR

This paper introduces a new projection-based approach for efficiently solving large-scale metric-constrained optimization problems, with applications in machine learning and graph clustering, overcoming high memory challenges of traditional solvers.

Contribution

The authors develop a generalized projection algorithm for metric-constrained linear and quadratic programs, improving scalability and providing novel approximation guarantees.

Findings

Able to solve problems with up to 10^8 variables and 10^11 constraints

Established equivalence between metric-constrained LP relaxation and metric nearness

Provided new approximation guarantees for graph clustering lower bounds

Abstract

We outline a new approach for solving optimization problems which enforce triangle inequalities on output variables. We refer to this as metric-constrained optimization, and give several examples where problems of this form arise in machine learning applications and theoretical approximation algorithms for graph clustering. Although these problem are interesting from a theoretical perspective, they are challenging to solve in practice due to the high memory requirement of black-box solvers. In order to address this challenge we first prove that the metric-constrained linear program relaxation of correlation clustering is equivalent to a special case of the metric nearness problem. We then developed a general solver for metric-constrained linear and quadratic programs by generalizing and improving a simple projection algorithm originally developed for metric nearness. We give several novel approximation guarantees for using our framework to find lower bounds for optimal solutions to several challenging graph clustering problems. We also demonstrate the power of our framework by solving optimizing problems involving up to 10^{8} variables and 10^{11} constraints.