The Hybrid Steepest Descent Method for the Variational Inequality Problem over the Intersection of Fixed Point Sets of Nonexpansive Mappings

By Isao Yamada. 2001

747 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/9f9646250b617455ee9cbf0884bb6f9ec5598b23

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In Computational Theory and Mathematics

In Geometry and Topology

In Numerical Analysis

Citations in the corpus

A General Iterative Method for Equilibrium Problems and Fixed Point Problems in Hilbert Spaces (2008, methodology): “Recently iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems; see, for example, [4–7] and the references therein.”
Linear and strong convergence of algorithms involving averaged nonexpansive operators (2014, neutral): “, [6], [7], [12], [13], [14], [20], [21], [27], and the references therein.”
A subgradient-type method for the equilibrium problem over the fixed point set and its applications (2009, methodology): “the hybrid steepest descent method [35] for the variational inequality problem and a scheme [16] to the equilibrium problem over a closed convex set C.”
Projection and proximal point methods: convergence results and counterexamples (2004, background): “In passing, we note that certain modifications of these algorithms are able to always generate norm convergent sequences; see, for instance, [11] and [53].”
Compositions and convex combinations of averaged nonexpansive operators (2014, neutral): “Keywords. averaged operator · fixed-point algorithm · forward-backward splitting · monotone operator · nonexpansive operator ∗Contact author: P. L. Combettes, plc@ljll.math.upmc.fr, phone: +33 1 4427 6319, fax: +33 1 4427 7200.”
Weak Convergence of an Iterative Method for Pseudomonotone Variational Inequalities and Fixed-Point Problems (2010, neutral): “, [2–7] and references therein.”
Fixed-Point Algorithms for Inverse Problems in Science and Engineering (2011, methodology): “Another approach to nonsmooth optimization problems arising in signal processing was proposed by Yamada [93], who employed the Moreau envelope to smooth the original problem and used a fixed point model to represent the constraints in the spirit of [94, 95].”
Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization (2008, methodology): “Most algorithms for solving VIs, however, require a projection operator that takes an arbitrary image and yields the closest (in the (cid:1) 2 sense) image that satisﬁes the constraint equations.”
A Conditional Gradient-based Method for Simple Bilevel Optimization with Convex Lower-level Problem (2022, methodology): “Along another line of research, several works (Yamada, 2001; Xu, 2004) have studied the more general problem of solving a variational inequality over the fixed-point set of a nonexpansive mapping, to which the problem in (1) is a special case.”