A Semi-smooth Newton Method for Solving Semidefinite Programs in Electronic Structure Calculations

TL;DR

This paper introduces a semi-smooth Newton method for efficiently solving semidefinite programming problems in electronic structure calculations, improving accuracy and speed over existing methods.

Contribution

The paper presents a novel second-order semi-smooth Newton method tailored for large-scale SDP problems in electronic structure calculations, with techniques for enhanced efficiency and convergence.

Findings

Method achieves high accuracy in fewer iterations

Computational efficiency surpasses first-order methods

Numerical experiments demonstrate competitive performance

Abstract

The ground state energy of a many-electron system can be approximated by an variational approach in which the total energy of the system is minimized with respect to one and two-body reduced density matrices (RDM) instead of many-electron wavefunctions. This problem can be formulated as a semidefinite programming problem. Due the large size of the problem, the well-known interior point method can only be used to tackle problems with a few atoms. First-order methods such as the the alternating direction method of multipliers (ADMM) have much lower computational cost per iteration. However, their convergence can be slow, especially for obtaining highly accurate approximations. In this paper, we present a practical and efficient second-order semi-smooth Newton type method for solving the SDP formulation of the energy minimization problem. We discuss a number of techniques that can be used to improve the computational efficiency of the method and achieve global convergence. Extensive numerical experiments show that our approach is competitive to the state-of-the-art methods in terms of both accuracy and speed.