Inexact Regularized Quasi-Newton Algorithm for Solving Monotone Variational Inequality Problems
Yuge Ye, Qingna Li, Deren Han
TL;DR
The paper develops an inexact regularized quasi-Newton method (IRQN) for monotone variational inequality problems, integrating a merit-function to promote unit steps and a hyperplane-based globalization to guarantee global convergence without strong regularity assumptions. By combining a BFGS-based quasi-Newton update with an inexact subproblem solve and a projection-corrected hyperplane step, IRQN achieves robust global behavior and superior numerical performance in many VIP instances. Theoretical results establish global and local convergence under standard monotonicity and Lipschitz conditions, while extensive numerical experiments demonstrate reduced iterations and computation time relative to the inexact Newton method (INM), especially on large-scale and initialization-sensitive problems. Overall, IRQN offers a scalable, reliable alternative for solving monotone VIPs with practical impact in computational optimization and related fields.
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.