An Accelerated DC Programming Approach with Exact Line Search for The Symmetric Eigenvalue Complementarity Problem

TL;DR

This paper introduces an accelerated DC programming algorithm with exact line search for efficiently solving symmetric eigenvalue complementarity problems, demonstrating significant speed and accuracy improvements over existing solvers.

Contribution

It develops a novel accelerated DC algorithm (BDCA) with exact line search for SEiCP and SQEiCP, extending the approach to large-scale problems with superior performance.

Findings

BDCA significantly accelerates convergence compared to DCA.

BDCA outperforms KNITRO, FILTERED, and FMINCON in speed and precision.

The method is effective on both synthetic and real datasets.

Abstract

In this paper, we are interested in developing an accelerated Difference-of-Convex (DC) programming algorithm based on the exact line search for efficiently solving the Symmetric Eigenvalue Complementarity Problem (SEiCP) and Symmetric Quadratic Eigenvalue Complementarity Problem (SQEiCP). We first proved that any SEiCP is equivalent to SEiCP with symmetric positive definite matrices only. Then, we established DC programming formulations for two equivalent formulations of SEiCP (namely, the logarithmic formulation and the quadratic formulation), and proposed the accelerated DC algorithm (BDCA) by combining the classical DCA with inexpensive exact line search by finding real roots of a binomial for acceleration. We demonstrated the equivalence between SQEiCP and SEiCP, and extended BDCA to SQEiCP. Numerical simulations of the proposed BDCA and DCA against KNITRO, FILTERED and MATLAB FMINCON for SEiCP and SQEiCP on both synthetic datasets and Matrix Market NEP Repository are reported. BDCA demonstrated dramatic acceleration to the convergence of DCA to get better numerical solutions, and outperformed KNITRO, FILTERED, and FMINCON solvers in terms of the average CPU time and average solution precision, especially for large-scale cases.