Momentum-inspired Low-Rank Coordinate Descent for Diagonally Constrained SDPs

TL;DR

This paper introduces a simple, scalable, and provably convergent non-convex optimization algorithm for diagonally constrained SDPs, achieving significant speedups in practical applications like MaxCut, MaxSAT, and MIMO detection.

Contribution

It proposes a momentum-inspired low-rank coordinate descent method that combines acceleration, coordinate power iteration, and matrix factorization for efficient SDP solving.

Findings

Achieves 5X speedup over state-of-the-art non-convex SDP solvers.

Provides 9 to 1000 times faster solutions than convex SDP solvers.

Maintains comparable or better solution quality across applications.

Abstract

We present a novel, practical, and provable approach for solving diagonally constrained semi-definite programming (SDP) problems at scale using accelerated non-convex programming. Our algorithm non-trivially combines acceleration motions from convex optimization with coordinate power iteration and matrix factorization techniques. The algorithm is extremely simple to implement, and adds only a single extra hyperparameter -- momentum. We prove that our method admits local linear convergence in the neighborhood of the optimum and always converges to a first-order critical point. Experimentally, we showcase the merits of our method on three major application domains: MaxCut, MaxSAT, and MIMO signal detection. In all cases, our methodology provides significant speedups over non-convex and convex SDP solvers -- 5X faster than state-of-the-art non-convex solvers, and 9 to 10^3 X faster than convex SDP solvers -- with comparable or improved solution quality.