A Semismooth Newton Method for Tensor Eigenvalue Complementarity Problem

TL;DR

This paper introduces a damped semismooth Newton method for solving the tensor eigenvalue complementarity problem, reformulating it via an NCP-function and demonstrating its efficiency through numerical experiments.

Contribution

The paper develops a novel damped semismooth Newton method tailored for tensor eigenvalue complementarity problems, including a new generalized Jacobian evaluation procedure.

Findings

Method is efficient and promising based on numerical experiments.

Convergence is guaranteed under mild assumptions.

Reformulation via NCP-function enables the application of semismooth Newton methods.

Abstract

In this paper, we consider the tensor eigenvalue complementarity problem which is closely related to the optimality conditions for polynomial optimization, as well as a class of differential inclusions with nonconvex processes. By introducing an NCP-function, we reformulate the tensor eigenvalue complementarity problem as a system of nonlinear equations. We show that this function is strongly semismooth but not differentiable, in which case the classical smoothing methods cannot apply. Furthermore, we propose a damped semismooth Newton method for tensor eigenvalue complementarity problem. A new procedure to evaluate an element of the generalized Jocobian is given, which turns out to be an element of the B-subdifferential under mild assumptions. As a result, the convergence of the damped semismooth Newton method is guaranteed by existing results. The numerical experiments also show that our method is efficient and promising.