A search-free $O(1/k^{3/2})$ homotopy inexact proximal-Newton extragradient algorithm for monotone variational inequalities

TL;DR

This paper introduces a new homotopy-based inexact proximal-Newton extragradient algorithm for smooth monotone variational inequalities, achieving improved iteration complexity and practical performance without a search procedure.

Contribution

The paper proposes a novel search-free homotopy approach for proximal-Newton methods, reducing complexity and enhancing practical efficiency in solving variational inequalities.

Findings

Achieves pointwise $O(1/\rho)$ iteration complexity.

Achieves ergodic $O(1/\rho^{2/3})$ iteration complexity.

Preliminary experiments show improved practical performance.

Abstract

We present and study the iteration-complexity of a relative-error inexact proximal-Newton extragradient algorithm for solving smooth monotone variational inequality problems in real Hilbert spaces. We removed a search procedure from Monteiro and Svaiter (2012) by introducing a novel approach based on homotopy, which requires the resolution (at each iteration) of a single strongly monotone linear variational inequality. For a given tolerance $\rho>0$, our main algorithm exhibits pointwise $O\left(\frac{1}{\rho}\right)$ and ergodic $O\left(\frac{1}{\rho^{2/3}}\right)$ iteration-complexities. From a practical perspective, preliminary numerical experiments indicate that our main algorithm outperforms some previous proximal-Newton schemes.