Convergence of Hybrid Steepest-Descent Methods for Variational Inequalities

By Hong-Kun Xu, T. Kim. 2003

341 citations recorded by the source corpus.

Subfields: Civil and Structural Engineering, Computational Theory and Mathematics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/cf1ef7d05de60f3bbb0aaa1d9e829438e420d6e3

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In Civil and Structural Engineering

In Computational Theory and Mathematics

In Numerical Analysis

Citations in the corpus

STRONG CONVERGENCE THEOREM BY AN EXTRAGRADIENT METHOD FOR FIXED POINT PROBLEMS AND VARIATIONAL INEQUALITY PROBLEMS (2006, background): “H. K. Xu and T. H. Kim, Convergence of hybrid steepest-descent methods for variational inequalities, J. Optim.”
Mann-Type Steepest-Descent and Modified Hybrid Steepest-Descent Methods for Variational Inequalities in Banach Spaces (2008, background): “J.S. Jung and S.S. Kim (1998).”
A hybrid extragradient method extended to fixed point problems and equilibrium problems (2013, background): “2 [22] Let {an} be a sequence of nonnegative numbers satisfying the conditions: anþ1 ð1 nÞan þ n n 8n 0, where { n} and { n} are sequences of real numbers such that { n} [0, 1] and P1 n1⁄40 n 1⁄4 1, or Q1 n1⁄40ð1 nÞ 1⁄4 0.”
Convergence Analysis of Modified Hybrid Steepest-Descent Methods with Variable Parameters for Variational Inequalities (2007, neutral): “Given an initial guess u0 ∈ H, let {un} be the sequence generated by algorithm un+1 = Tun − λn+1μn+1θ(Tun), n ≥ 0, (16)”
Outer approximation methods for solving variational inequalities in Hilbert space (2017, methodology): “Examples of such restrictions can be found in the papers by Halpern [16], Lions [17],Wittmann [18], Bauschke [19], Deutsch and Yamada [15], Yamada [21], Xu and Kim [22], Zeng et al. [25], Takahashi and Yamada [26], Aoyama and Kimura [27], and Zhang and He [28].”
Hybrid inertial subgradient extragradient methods for variational inequalities and fixed point problems involving asymptotically nonexpansive mappings (2019, methodology): “4 (see [37]): Let {an} be a sequence of nonnegative numbers satisfying the conditions: an+1 ≤ (1 − λn)an + λnγn ∀n ≥ 1, where {λn} and {γn} are sequences of real numbers such that (i) {λn} ⊂ [0, 1] and ∑∞ n=1 λn = ∞, and (ii) lim supn→∞ γn ≤ 0 or ∑∞ n=1 |λnγn| < ∞.”
Hybrid Steepest Descent Method for Variational Inequality Problem over the Fixed Point Set of Certain Quasi-nonexpansive Mappings (2005, background): “Recently the condition (L3) was relaxed to limn→ n n+1 = 1 (Xu and Kim, 2003), which allows the case n = 1n .”
Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization (2008, methodology): “An exception, however, is the hybrid steepest descent (HSD) algorithm developed by Yamada for solving VIs (Yamada 2001, Xu and Kim 2003).”
Two inertial subgradient extragradient algorithms for variational inequalities with fixed-point constraints (2020, neutral): “4 ([33]): Assume that { n} is a sequence in [0,+∞) satisfying the condition: n+1 ≤ τnbn + (1 − τn) n ∀n ≥ 1, where {τn} and {bn} lie in R := (−∞,∞) s.”