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Golden Ratio Algorithms for Variational Inequalities

Yura Malitsky(y.malitsky***at***gmail.com)

Abstract: The paper presents a fully explicit algorithm for monotone variational inequalities. The method uses variable stepsizes that are computed using two previous iterates as an approximation of the local Lipschitz constant without running a linesearch. Thus, each iteration of the method requires only one evaluation of a monotone operator $F$ and a proximal mapping $g$. The operator $F$ need not be Lipschitz-continuous, which also makes the algorithm interesting in the area of composite minimization where one cannot use the descent lemma. The method exhibits an ergodic $O(1/k)$ convergence rate and $R$-linear rate, if $F, g$ satisfy the error bound condition. We discuss possible applications of the method to fixed point problems. Furthermore, we show theoretically that the method still converges under a new relaxed monotonicity condition and confirm numerically that it can robustly work even for some highly nonmonotone/nonconvex problems.

Keywords: variational inequality, first-order methods, linesearch, saddle point problem, composite minimization, fixed point problem

Category 1: Complementarity and Variational Inequalities

Category 2: Convex and Nonsmooth Optimization (Convex Optimization )

Category 3: Convex and Nonsmooth Optimization (Nonsmooth Optimization )

Citation:

Download: [PDF]

Entry Submitted: 04/27/2018
Entry Accepted: 05/01/2018
Entry Last Modified: 04/27/2018

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