Inexact Regularized Quasi-Newton Algorithm for Solving Monotone Variational Inequality Problems

TL;DR

This paper introduces an inexact regularized quasi-Newton method with a merit function and hyperplane globalization for monotone variational inequalities, ensuring global convergence without regularity assumptions.

Contribution

It proposes a novel inexact quasi-Newton algorithm with a merit function and hyperplane technique, achieving global convergence for monotone variational inequalities without regularity.

Findings

The method guarantees global convergence to a solution.

Subproblems have unique solutions at each iteration.

Numerical results demonstrate high efficiency.

Abstract

Newton's method has been an important approach for solving variational inequalities, quasi-Newton method is a good alternative choice to save computational cost. In this paper, we propose a new method for solving monotone variational inequalities where we introduce a merit function based on the merit function. With the help of the merit function, we can locally accepts unit step size. And a globalization technique based on the hyperplane is applied to the method. The proposed method applied to monotone variational inequality problems is globally convergent in the sense that subproblems always have unique solutions, and the whole sequence of iterates converges to a solution of the problem without any regularity assumptions. We also provide extensive numerical results to demonstrate the efficiency of the proposed algorithm.