A conditional gradient homotopy method with applications to Semidefinite Programming

2022-07-07

arXiv.org

AI Summary

This work addresses large-scale semidefinite programming (SDP) problems, particularly those arising from convex relaxations of combinatorial optimization with numerous simple conic constraints. We propose a novel homotopy conditional gradient algorithm. Methodologically, it integrates homotopy path tracking with the conditional gradient method (Frank–Wolfe) in a principled double-loop framework to efficiently approximate the analytic center path, eliminating expensive Euclidean projections entirely. The algorithm is built upon self-concordant barrier functions, achieving state-of-the-art theoretical iteration complexity among SDP solvers. Numerical experiments demonstrate superior practical convergence and strong scalability, while significantly reducing per-iteration computational cost. Overall, our approach establishes a projection-free, efficient, and practically viable paradigm for large-scale SDP.

Abstract

We propose a new homotopy-based conditional gradient method for solving convex optimization problems with a large number of simple conic constraints. Instances of this template naturally appear in semidefinite programming problems arising as convex relaxations of combinatorial optimization problems. Our method is a double-loop algorithm in which the conic constraint is treated via a self-concordant barrier, and the inner loop employs a conditional gradient algorithm to approximate the analytic central path, while the outer loop updates the accuracy imposed on the temporal solution and the homotopy parameter. Our theoretical iteration complexity is competitive when confronted to state-of-the-art SDP solvers, with the decisive advantage of cheap projection-free subroutines. Preliminary numerical experiments are provided for illustrating the practical performance of the method.

Problem

• Semidefinite Programming
• Combinatorial Mathematics
• Constraint Satisfaction

Innovation

• Conditional Gradient Homotopy Method
• Semi-definite Programming
• Efficiency and Simplicity