The Hybrid Steepest Descent Method for the Variational Inequality Problem Over the Intersection of Fixed Point Sets of Nonexpansive Mappings
Section snippets
INTRODUCTION
The Variational Inequality Problem 6., 52., 118., 119. has been and will continue to be one of the central problems in nonlinear analysis and is defined as follows: given monotone operator and closed convex set C ⊂ ℋ, where ℋ is a real Hilbert space with inner product 〈·,·〉 and induced norm ∥ · ∥, find x* ∈ C such that 〈x − x*, ℱ(x*)〉 ≥ 0 for all x ∈ C. This condition is the optimality condition of the convex optimization problem: min Θ over C when ℱ = Θ ′. The simplest iterative procedure for the
Fixed points, Nonexpansive mappings, and Convex projections
A fixed point of a mapping T : ℋ → ℋ is a point x ∈ ℋ such that T(x) = x. Fix(T) := {x ∈ ℋ | T(x) = x} denotes the set of all fixed points of T. A mapping T : ℋ → ℋ is called κ-Lipschitzian (or κ-Lipschitz continuous) over S ⊂ ℋ if there exists κ > 0 such that
In particular, a mapping T : ℋ → ℋ is called (i) strictly contractive if ‖T(x) − T(y)‖ ≤ κ‖x − y‖ for some κ ∈ (0,1) and all x, y ∈ ℋ [ The Banach-Picard fixed point theorem guarantees the unique existence of the fixed point, say x* ∈ Fix(T), of T and
HYBRID STEEPEST DESCENT METHOD
The hybrid steepest descent method for minimization of certain convex functions over the set of fixed points of nonexpansive mappings 108., 109., 47., 111., 112., 80. has been developed by generalizing the results for approximation of the fixed point of nonexpansive mappings. To demonstrate simply the underlying ideas of the hybrid steepest descent method, we present it as algorithmic solutions to the variational inequality problem (VIP) defined over the fixed point set of nonexpansive
Projection algorithms for best approximation and convex feasibility problems
The best approximation problem of finding the projection of a given point in a Hilbert space onto the (nonempty) intersection of a finite number of closed convex sets Ci (i = 1,2 …, m) arises in many branches of applied mathematics, the physical and computer sciences, and engineering. One frequently employed approach to solving this problem is algorithmic. The idea of this approach is to generate a sequence of points which converges to the solution of the problem by using projections
CONCLUDING REMARKS
In this paper, we have presented in a simple way the underlying ideas of the hybrid steepest descent method as an algorithmic solution to a VIP defined over the fixed point set of a nonexpansive mapping in a real Hilbert space. As seen from the discussion in section 4, the proposed hybrid steepest descent method plays notable roles in various inverse problems. Some straightforward applications of the hybrid steepest descent method to certain signal estimation or design problems have been shown
Acknowledgments
It is my great honor to thank Frank Deutsch, Patrick L. Combettes, Heinz H. Bauschke, Jonathan M. Borwein, Boris T. Polyak, Charles L. Byrne, Paul Tseng, M. Zuhair Nashed, K. Tanabe and U. M. Garcia-Palomares for their encouraging advice at the Haifa workshop (March 2000). I also wish to thank Dan Butnariu, Yair Censor and Simeon Reich for giving me this great opportunity and their helpful comments.
REFERENCES (119)
- et al.
Block-iterative projection methods for parallel computation of solutions to convex feasibility problems
Linear Algebra an Its Applications
(1989) - et al.
Dykstra's alternating projection algorithm for two sets
J. Approx. Theory
(1994) The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space
J. Math. Anal. Appl.
(1996)The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming
USSR Computational Mathematics and Mathematical Physics
(1967)- et al.
Construction of fixed points of nonlinear mappings in Hilbert space
J. Math. Anal. Appl.
(1967) - et al.
Strong convergence of almost simultaneous block-iterative projection methods in Hilbert spaces
Journal of Computational and Applied Mathematics
(1994) Foundation of set theoretic estimation
Proc. IEEE
(1993)On the Mann iterative process
Trans. Amer. Math. Soc.
(1970)A successive projection method
Math. Programming
(1988)- et al.
Constrained Lp Approximation
Constr. Approx.
(1985)
An adaptive regularized method for deconvolution of signal with edges by convex projections
IEEE Trans. Signal Processing
On the asymptotic behavior of nonexpansive mappings and semigroups in Banach spaces
Houston J. Math.
On projection algorithms for solving convex feasibility problems
SIAM Review
The method of cyclic projections for closed convex sets in Hilbert space
Contemp. Math.
Dykstra's algorithm with Bregman projections: a convergence proof
Optimization
A method for finding projections onto the intersection of convex sets in Hilbert spaces
The method of successive projection for finding a common point of convex sets
Soviet Math. Dokl.
Dykstra's algorithm as the nonlinear extension of Bregman's optimization method
J. Convex Analysis
Nonexpansive nonlinear operators in Banach space
Proc. Nat. Acad. Sci. USA
Convergence of approximants to fixed points of nonexpansive nonlinear mappings in Banach spaces
Arch. Rat. Mech. Anal.
On the behavior of a block-iterative projection method for solving convex feasibility problems
International Journal of Computer Mathematics
Iterative averaging of entropic projections for solving stochastic convex feasibility
Computational Optimization and Applications
Totally Convex Functions for fixed point computation and infinite dimensional optimization
Nonexpansive projections and resolvents of accretive operators in Banach spaces
Houston J. Math.
Iterative projection onto convex sets using multiple Bregman distances
Inverse Problems
Proximity function minimization for separable jointly convex Bregman distances, with applications
Technical Report
Proximity function minimization using multiple Bregman projections, with applications to split feasibility and Kullback-Leibler distance minimization
Technical Report
Row-action methods for huge and sparse systems and their applications
SIAM Review
An iterative row-action method for interval convex programming
Journal of Optimization Theory and Applications
Parallel application of block-iterative methods in medical imaging and radiation therapy
Math. Programming
A multiprojection algorithm using Bregman projections in a product space
Numerical Algorithms
The Dykstra algorithm with Bregman projections
Communications in Applied Analysis
An interior point method with Bregman functions for the variational inequality problem with paramonotone operators
Math. Programming
Proximity maps for convex sets
Proc. Amer. Math. Soc.
Constrained best approximation in Hilbert space
Constr. Approx.
Constrained best approximation in Hilbert space II
J. Approx. Theory
Inconsistent signal feasibility problems: least squares solutions in a product space
IEEE Trans. on Signal Processing
Construction d'un point fixe commun à une famille de contractions fermes
C.R. Acad. Sci. Paris Sèr. I Math.
Convex set theoretic image recovery by extrapolated iterations of parallel subgradient projections
IEEE Trans. Image Processing
Hard-constrained inconsistent signal feasibility problem
IEEE Trans. Signal Processing
A parallel constraint disintegration and approximation scheme for quadratic signal recovery
Strong convergence of block-iterative outer approximation methods for convex optimization
SIAM J. Control Optim.
Finding projections onto the intersection of convex sets in Hilbert spaces
Numer. Funct. Anal. Optim.
The rate of convergence of Dykstra's cyclic projections algorithms: the polyhedral case
Numer. Funct. Anal. Optim.
The rate of convergence for the method of alternating projections II
J. Math. Anal. Appl.
Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings
Numer. Funct. Anal. Optim.
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