PROPERTIES OF A CLASS OF APPROXIMATELY SHRINKING OPERATORS AND THEIR APPLICATIONS

In this paper we present an application of a class of quasi-nonexpansive operators to iterative methods for solving the following variational inequality problem VIP(F,C): Find ū ∈ C such that 〈F ū, z−ū〉 ≥ 0 for all z ∈ C, where C is a closed and convex subset of a Hilbert space H and F : H → H is strongly monotone and Lipschitz continuous. A classical method for VIP(F,C) is the gradient projection (GP) method x = PC(x k − μFx) which generates sequences converging to the unique solution of VIP(F,C) if μ > 0 is sufficiently small. Unfortunately, in many optimization problems the GP method cannot be applied, because it requires an explicit computation of PCu k in each iteration, where u = x − μFx. To overcome this disadvantage of the GP method, one can replace the operator PC employed in the k-th iteration of the method by a quasi-nonexpansive operator Tk and a constant μ by λk ≥ 0, k ≥ 0, satisfying ⋂ ∞ k=0 FixTk ⊇ FixT and limk λk = 0. The new method can be presented equivalently as a so called general hybrid steepest descent (GHSD) method in the form u = Tku k − λkFTku . One should, however, suppose something more on the operators Tk in order to guarantee the convergence of u k to the solution of VIP(F,C). In this paper we introduce a class of approximately shrinking operators, prove the closedness of this class with respect to compositions and convex combinations and apply the operators from this class to a general hybrid steepest descent method for solving VIP(F,C). We give sufficient conditions for the convergence of the GHSD method as well as present several examples of methods which satisfy these conditions. In particular, we apply the results in the case, when C = ⋂ m i=1 FixUi and Ui : H → H are quasi-nonexpansive operators having a common fixed point, i = 1, 2, ...,m.

[1]  D. Kinderlehrer,et al.  An introduction to variational inequalities and their applications , 1980 .

[2]  P. Lions.  Approximation de Points Fixes de Contractions , 1977 .

[3]  Heinz H. Bauschke,et al.  On the convergence of von Neumann's alternating projection algorithm for two sets , 1993 .

[4]  W. A. Kirk,et al.  Topics in Metric Fixed Point Theory , 1990 .

[5]  I. Yamada,et al.  NON-STRICTLY CONVEX MINIMIZATION OVER THE FIXED POINT SET OF AN ASYMPTOTICALLY SHRINKING NONEXPANSIVE MAPPING , 2002 .

[6]  A. Cegielski.  Iterative Methods for Fixed Point Problems in Hilbert Spaces , 2012 .

[7]  P. L. Combettes,et al.  Hilbertian convex feasibility problem: Convergence of projection methods , 1997 .

[8]  Yair Censor,et al.  Averaging Strings of Sequential Iterations for Convex Feasibility Problems , 2001 .

[9]  Heinz H. Bauschke,et al.  On Projection Algorithms for Solving Convex Feasibility Problems , 1996, SIAM Rev..

[10]  D. Schoot.  A general iterative scheme with applications to convex optimization and related fields , 1991 .

[11]  Boris Polyak,et al.  The method of projections for finding the common point of convex sets , 1967 .

[12]  Yair Censor,et al.  Convergence of String-Averaging Projection Schemes for Inconsistent Convex Feasibility Problems , 2003, Optim. Methods Softw..

[13]  E. Zeidler.  Nonlinear Functional Analysis and its Applications: III: Variational Methods and Optimization , 1984 .

[14]  Andrzej Cegielski,et al.  An Algorithm for Solving the Variational Inequality Problem Over the Fixed Point Set of a Quasi-Nonexpansive Operator in Euclidean Space , 2013, 1304.0690.

[15]  Heinz H. Bauschke,et al.  A Weak-to-Strong Convergence Principle for Fejé-Monotone Methods in Hilbert Spaces , 2001, Math. Oper. Res..

[16]  Yair Censor,et al.  On the string averaging method for sparse common fixed-point problems , 2009, Int. Trans. Oper. Res..

[17]  C. Byrne,et al.  A unified treatment of some iterative algorithms in signal processing and image reconstruction , 2003 .

[18]  S. Hirstoaga.  Iterative selection methods for common fixed point problems , 2006 .

[19]  Andrzej Cegielski,et al.  A generalization of the Opial's theorem , 2007 .

[20]  Cristina Popirlan,et al.  On the Mann-type iteration and the convex feasibility problem , 2008 .

[21]  P. L. Combettes,et al.  Quasi-Fejérian Analysis of Some Optimization Algorithms , 2001 .

[22]  Z. Opial.  Weak convergence of the sequence of successive approximations for nonexpansive mappings , 1967 .

[23]  B. Halpern.  Fixed points of nonexpanding maps , 1967 .

[24]  Hong-Kun Xu,et al.  Convergence of Hybrid Steepest-Descent Methods for Variational Inequalities , 2003 .

[25]  Cristina Popîrlan,et al.  On the regularity condition in a convex feasibility problem , 2009 .

[26]  I. Ekeland,et al.  Convex analysis and variational problems , 1976 .

[27]  F. Facchinei,et al.  Finite-Dimensional Variational Inequalities and Complementarity Problems , 2003 .

[28]  I. Yamada.  The Hybrid Steepest Descent Method for the Variational Inequality Problem over the Intersection of Fixed Point Sets of Nonexpansive Mappings , 2001 .

[29]  Yair Censor,et al.  Convergence and perturbation resilience of dynamic string-averaging projection methods , 2012, Computational Optimization and Applications.

[30]  Andrzej Cegielski,et al.  Opial-Type Theorems and the Common Fixed Point Problem , 2011, Fixed-Point Algorithms for Inverse Problems in Science and Engineering.

[31]  Y. Censor.  Row-Action Methods for Huge and Sparse Systems and Their Applications , 1981 .

[32]  R. Wittmann.  Approximation of fixed points of nonexpansive mappings , 1992 .

[33]  E. Zeidler.  Nonlinear functional analysis and its applications , 1988 .

[34]  S. Reich,et al.  Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings , 1984 .

[35]  Yair Censor,et al.  String-averaging projected subgradient methods for constrained minimization , 2013, Optim. Methods Softw..

[36]  A. Cegielski,et al.  Methods for Variational Inequality Problem Over the Intersection of Fixed Point Sets of Quasi-Nonexpansive Operators , 2013 .

[37]  A. Goldstein.  Convex programming in Hilbert space , 1964 .

[38]  Karin Schwab,et al.  Best Approximation In Inner Product Spaces , 2016 .

[39]  I. Yamada,et al.  Hybrid Steepest Descent Method for Variational Inequality Problem over the Fixed Point Set of Certain Quasi-nonexpansive Mappings , 2005 .

[40]  Y. Censor,et al.  Sparse string-averaging and split common fixed points , 2008 .

[41]  Heinz H. Bauschke.  The Approximation of Fixed Points of Compositions of Nonexpansive Mappings in Hilbert Space , 1996 .