Golden ratio algorithms for variational inequalities

• Published in: Mathematical programming Vol. 184; no. 1-2; pp. 383 - 410
• Main Author: Malitsky, Yura
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.11.2020; Springer Nature B.V
• Subjects: Adaptive algorithms; Algorithms; Applied mathematics; Calculus of Variations and Optimal Control; Optimization; Combinatorics; Control theory; Full Length Paper; Inequalities; Mapping; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematical programming; Mathematics; Mathematics and Statistics; Mathematics of Computing; Methods; Numerical Analysis; Optimization; Theoretical; Fixed point problem; 65K15; 47J20; Variational inequality; Composite minimization; 90C33; 65K10; Saddle point problem; 65Y20; Linesearch; First-order methods
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-019-01416-w

The paper presents a fully adaptive algorithm for monotone variational inequalities. In each iteration the method uses two previous iterates for an approximation of the local Lipschitz constant without running a linesearch. Thus, every 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. The method exhibits an ergodic O (1 /  k ) convergence rate and R -linear rate under an error bound condition. We discuss possible applications of the method to fixed point problems as well as its different generalizations.