A Squared Smoothing Newton Method for Semidefinite Programming

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, that is, the primal and dual constraint nondegenerate conditions, the proposed method is shown to have a super-linear 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. Funding: The research of D. Sun was supported in part by the Hong Kong Research Grants Council under Grant 15307523, and the research of K.-C. Toh was supported by the Ministry of Education, Singapore, under its Academic Research Fund Tier 3 grant call [MOE-2019-T3-1-010].