A squared smoothing Newton method for semidefinite programming

TL;DR

This paper introduces a novel squared smoothing Newton method using the Huber function for semidefinite programming, providing rigorous convergence analysis and demonstrating superior practical performance over existing solvers.

Contribution

The paper develops a new Newton-type algorithm for SDPs based on Huber smoothing, with proven global and superlinear convergence, and shows its efficiency through extensive numerical experiments.

Findings

The method is globally convergent.

Under regularity conditions, it achieves superlinear convergence.

Numerical results outperform existing SDP solvers.

Abstract

This paper proposes a squared smoothing Newton method via the Huber smoothing function for solving semidefinite programming problems (SDPs). We first study the fundamental properties of the matrix-valued mapping defined upon the Huber function. Using these results and existing ones in the literature, we then conduct rigorous convergence analysis and establish convergence properties for the proposed algorithm. In particular, we show that the proposed method is well-defined and admits global convergence. Moreover, under suitable regularity conditions, i.e., the primal and dual constraint nondegenerate conditions, the proposed method is shown to have a superlinear convergence rate. To evaluate the practical performance of the algorithm, we conduct extensive numerical experiments for solving various classes of SDPs. Comparison with the state-of-the-art SDP solvers demonstrates that our method is also efficient for computing accurate solutions of SDPs.