Inexact PDHG for Large-Scale Semidefinite Programming

Abstract

First-order methods, such as the primal-dual hybrid gradient (PDHG) method, have recently been explored for large-scale semidefinite programming (SDP). However, exact projections can be computationally prohibitive, motivating inexact projections using conditional gradient methods. Although computationally cheap and projection-free, these methods typically converge sublinearly. Recent variants, such as Spectrahedron Frank-Wolfe (Garber, 2026), achieve linear convergence in expectation near the optimal face of the spectrahedron. We develop an inexact PDHG framework with tractable spectrahedral projections using two techniques: (i) adaptive error tolerances that avoid unnecessary inner iterations while guaranteeing convergence, and (ii) a heuristic spectral warm start based on the Lanczos algorithm that accelerates Spectrahedron Frank-Wolfe by initializing near the optimal face. We establish an O(1/t) convergence rate for PDHG in the presence of projection errors and estimate the total number of inner iterations required for convergence. Experiments on large-scale SDP instances demonstrate that combining spectral warm starts with Spectrahedron Frank-Wolfe outperforms Lanczos-based inexact projections.