Multi-step implicit iterative methods with regularization for minimization problems and fixed point problems
TL;DR: In this article, a multi-step implicit iterative scheme with regularization was proposed for finding a common solution of the minimization problem (MP) for a convex and continuously Frechet differentiable functional and the common fixed point problem of an infinite family of nonexpansive mappings in the setting of Hilbert spaces.
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Abstract: In this paper we introduce a multi-step implicit iterative scheme with regularization for finding a common solution of the minimization problem (MP) for a convex and continuously Frechet differentiable functional and the common fixed point problem of an infinite family of nonexpansive mappings in the setting of Hilbert spaces. The multi-step implicit iterative method with regularization is based on three well-known methods: the extragradient method, approximate proximal method and gradient projection algorithm with regularization. We derive a weak convergence theorem for the sequences generated by the proposed scheme. On the other hand, we also establish a strong convergence result via an implicit hybrid method with regularization for solving these two problems. This implicit hybrid method with regularization is based on the CQ method, extragradient method and gradient projection algorithm with regularization. MSC: 49J30; 47H09; 47J20
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Citations
Topics in metric fixed point theory: Minimal displacement
Kazimierz Goebel,William A. Kirk +1 more
- 01 Jan 1990
68
Regularized gradient-projection methods for finding the minimum-norm solution of the constrained convex minimization problem.
Ming Tian,Hui-Fang Zhang +1 more
TL;DR: q=P_{U}(0)$ is the minimum-norm solution of the constrained convex minimization problem, which also solves the variational inequality 〈−q,p−q〉≤0$\langle-q, p-q\rangle\leq0$, ∀p∈U$\forall p\in U$.
Iterative algorithms with regularization for hierarchical variational inequality problems and convex minimization problems
TL;DR: In this paper, a variational inequality problem is defined over the set of intersections of the sets of fixed points of a δ-strictly pseudocontractive mapping and the set the solutions of a minimization problem, and an iterative algorithm with regularization is proposed.
On an open question of moudafi for convex feasibility problems in hilbert spaces
TL;DR: In this paper, a strongly convergent iterative sequence of Halpern-type to a solution of convex feasibility problems in real Hilbert spaces has been obtained and an affirmative answer to an open question posed by Moudafi in his recent work is given.
Regularized gradient-projection methods for the constrained convex minimization problem and the zero points of maximal monotone operator
Ming Tian,Si-Wen Jiao +1 more
TL;DR: In this article, based on the viscosity approximation method and the regularized gradient projection algorithm, a common element of the solution set of a constrained convex minimization problem and the set of zero points of the maximal monotone operator problem is found.
References
Monotone Operators and the Proximal Point Algorithm
TL;DR: In this paper, the proximal point algorithm in exact form is investigated in a more general form where the requirement for exact minimization at each iteration is weakened, and the subdifferential $\partial f$ is replaced by an arbitrary maximal monotone operator T.
3.9K
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Topics in metric fixed point theory
Kazimierz Goebel,William A. Kirk +1 more
- 28 Sep 1990
TL;DR: In this paper, the basic fixed point theorems for non-pansive mappings are discussed and weak sequential approximations are proposed for linear mappings with normal structure and smoothness.
2.3K