Spectral projected gradient methods for generalized tensor eigenvalue complementarity problem

TL;DR

This paper introduces two spectral projected gradient methods and an improved shifted scaling-and-projection algorithm for solving the tensor eigenvalue complementarity problem, relevant in mechanical stability analysis and polynomial optimization.

Contribution

The paper proposes novel monotone ascent spectral projected gradient methods and an enhanced shifted SPA algorithm for TEiCP, demonstrating improved efficiency over existing methods.

Findings

Numerical experiments show the proposed methods outperform existing gradient methods.

The shifted SPA algorithm significantly improves convergence speed.

The methods are effective for stability analysis and polynomial optimization applications.

Abstract

This paper looks at the tensor eigenvalue complementarity problem (TEiCP) which arises from the stability analysis of finite dimensional mechanical systems and is closely related to the optimality conditions for polynomial optimization. We investigate two monotone ascent spectral projected gradient (SPG) methods for TEiCP. We also present a shifted scaling-and-projection algorithm (SPA), which is a great improvement of the original SPA method proposed by Ling, He and Qi [Comput. Optim. Appl., DOI 10.1007/s10589-015-9767-z]. Numerical comparisons with some existed gradient methods in the literature are reported to illustrate the efficiency of the proposed methods.