Golden ratio algorithms for variational inequalities

Abstract

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.Keywords. variational inequality · first-order methods · linesearch · saddle point problem · composite minimization · fixed point problem MSC2010. 47J20, 65K10, 65K15, 65Y20, 90C33

Golden ratio algorithms with new stepsize rules for variational inequalities

“…Recall that the proximal operator prox g of g is defined by prog g (x) = arg min We are also interested here in the problem (VIP) where the proximal mapping of g is computable. The problem (VIP) is known as a central problem in nonlinear analysis, especially in optimization, control theory, games theory [1][2][3][4][5][6][7][8] and other fields. [9][10][11][12][13][14][15][16][17] Considering the problem (VIP) in a special case, when g = C , the indicator operator of a nonempty closed convex set C in ℜ m , this problem reduces to the classical VIP 5,18 :…”

“…Several models arising naturally can be formulated as the problem (VIP), see, eg, in previous studies. 3,5,6 We restrict our interest in the following two problems. The first basic problem is a convex-concave saddle point problem:…”

“…Moreover, in the case F is even Lipschitz continuous, but in general the Lipschitz constant is often unknown and in nonlinear problems, it can be difficult to approximate. Recently, without any linesearch procedure, some interesting methods for solving the classical variational inequalities can be found, for instance, in Malitsky, 6 Khobotov, 43 and Tinti, 44 where stepsizes are updated over iteration by some cheap computations.…”

“…An especially interesting idea has been developed recently by Malitsky. 6 He has proposed a new algorithm, named the explicit golden ratio algorithm (EGRAAL), for solving problem (VIP) when F is locally Lipschitz continuous. The EGRAAL has a simple and elegant structure and only requires the computations of one proximal mapping value (with function g) and one value of F. His algorithm is explicit in the sense that it does not require any linesearch procedure.…”

“…The stepsize strategies here are simpler than those ones in Malitsky. 6 The variable stepsizes in the new algorithms are chosen previously or computed easily. Also, the resulting algorithms work without any information of Lipschitz constant of operator, ie, the Lipschitz constant must not be the input parameters of the algorithms.…”