Golden ratio algorithms for variational inequalities - Malitsky, Yura (Mathematical programming, 2020)
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.