Methods for Variational Inequality Problem Over the Intersection of Fixed Point Sets of Quasi-Nonexpansive Operators
Andrzej Cegielski, Rafał Zalas · Numerical Functional Analysis and Optimization · 2013
Many convex optimization problems in a Hilbert space ℋ can be written as the following variational inequality problem VIP (ℱ, C): Find such that for all z ∈ C, where C ⊂ ℋ is closed convex and ℱ: ℋ → ℋ is monotone. We consider a special case of VIP (ℱ, C), where and U i : ℋ → ℋ are quasi-nonexpansive operators having a common fixed point, i ∈ I: = {1, 2,…, m}. A standard method for VIP (ℱ, C) is the projected gradient method u k+1 = P C (u k − μℱu k ) which generates sequences converging to a unique solution of VIP (ℱ, C) if ℱ is strongly monotone and Lipschitz continuous. Unfortunately, the method cannot be applied for , because, in general, P C u cannot be computed explicitly, u ∈ ℋ. Lions in 1977 and Bauschke in 1996 considered a special case of , where ℱ = Id −a, for some a ∈ ℋ, U i are firmly nonexpansive or nonexpansive, respectively, and studied the convergence properties of the following method: u k+1 = U i k u k − λ k ℱU i k u k , where λ k ↓ 0 and is a cyclic control, i.e., i k = k(mod m) +1 for all k ≥ 0 (see [1 H. H. Bauschke ( 1996 ). The approximation of fixed points of composition of nonexpansive mapping in Hilbert space . J. Math. Anal. Appl. 202 : 150 – 159 .[Crossref], [Web of Science ®] , [Google Scholar], 22 P.-L. Lions ( 1977 ). Approximation de points fixes de contractions . C. R. Acad. Sci. Paris Sér. A 284 : 1357 – 1359 . [Google Scholar]]). We apply this method in case ℱ is strongly monotone and Lipschitz continuous, U i are quasi-nonexpansive and is almost-cyclic. We present the method in a more general form where T k : ℋ → ℋ, k ≥ 0, are quasi-nonexpansive, and Fix T k approximate Fix T in some sense. A special case of the method with T k = T for all k ≥ 0 was studied in Yamada and Ogura [32 I. Yamada and N. Ogura ( 2004 ). Hybrid steepest descent method for variational inequality problem over the fixed point set of certain quasi-nonexpansive mappings . Numer. Funct. Anal. Optimiz. 25 : 619 – 655 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]] and by Yamada in [31 I. Yamada ( 2001 ). The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings . In: Inherently Parallel Algorithms in Feasibility and Optimization and their Applications (D. Butnariu , Y. Censor and S. Reich , eds.). Elsevier , Amsterdam , pp. 473 – 504 .[Crossref] , [Google Scholar]], (in the latter paper, T was supposed to be nonexpansive). We give sufficient conditions for the convergence of (1) as well as present examples of methods which satisfy these conditions.