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arXiv · 2308.05887

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

M. Marques Alves·João M. Pereira·Benar F. Svaiter
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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.

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M. Marques Alves, João M. Pereira, Benar F. Svaiter. 2023-08-11. A search-free $O(1/k^{3/2})$ homotopy inexact proximal-Newton extragradient algorithm for monotone variational inequalities. https://arxiv.org/abs/2308.05887

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