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A framework for analyzing local convergence properties with applications to proximal-point algorithms

JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2006年第1期131卷 53-68页

作者： Dong, Y. D. Fischer, A. Zhengzhou Univ Dept Math Zhengzhou Peoples R China Tech Univ Dresden Inst Numer Math D-8027 Dresden Germany

A general algorithmic scheme for solving inclusions in a Banach space is investigated in respect to its local convergence behavior. Particular emphasis is placed on applications to certain proximal-point-type algorithms in Hilbert spaces. The assumptions do not necessarily require that a solution be isolated. In this way, results existing for the case of a locally unique solution can be extended to cases with nonisolated solutions. Besides the convergence rates for the distance of the iterates to the solution set, strong convergence to a sole solution is shown as well. As one particular application of the framework, an improved convergence rate for an important case of the inexact proximal-point algorithm is derived.

关键词： inclusions generalized equations local convergence upper Lipschitz continuity nonisolated solutions proximal-point algorithms

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A Variational Approach to the Design of Multivariable Discrete-Time Supertwisting-Like algorithms

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL 2025年第4期70卷 2495-2506页

作者： Miranda-Villatoro, Felix A. Univ Grenoble Alpes CNRS Grenoble INP Lab Jean KuntzmannCtr Inria F-38000 Grenoble France

In this article, a family of discrete-time, supertwisting-like algorithms is presented. The algorithms are naturally vector-valued and are described in an implicit fashion, reminiscent of backward-Euler discretization schemes. The well-posedness of the closed-loop is established and the robust stability, against a family of external disturbances, is thoroughly studied. Implementation strategies, involving splitting-algorithms from convex optimization, are also discussed and compared. Finally, numerical simulations show the performance of the proposed schemes.

关键词： Stability analysis Emulation Robustness Digital computers Thermal stability Numerical stability Context modeling Asymptotic stability Wind energy conversion Trajectory tracking Convex optimization discrete-time systems proximal-point algorithms robustness sliding-mode control splitting algorithms

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On Gabay's algorithms for mixed variational inequalities

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APPLIED MATHEMATICS AND OPTIMIZATION 1997年第1期35卷 21-44页

作者： Alduncin, G Instituto de Geofisica UNAM 04510-Mexico D.F. Mexico

Iterative numerical algorithms for variational inequalities are systematically constructed from fixed-point problem characterizations in terms of resolvent operators. The applied method is the one introduced by Gabay in [Ga], used here in the context of discrete variational inequalities, and with the emphasis on mixed finite element models. The algorithms apply to nonnecessarily potential problems, generalizing primal and mixed Uzawa and augmented Lagrangian-type algorithms. They are also identified with Euler and operator splitting methods for the time discretization of evolution first-order problems.

关键词： proximal-point algorithms operator-splitting methods variational inequalities initial and boundary-value constrained problems mixed finite elements

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Macro-hybrid variational formulations of constrained boundary value problems

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NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION 2007年第7-8期28卷 751-774页

作者： Alduncin, Gonzalo Univ Nacl Autonoma Mexico Inst Geophys Coyoacan 04510 DF Mexico

In the context of convex analysis, macro-hybrid variational formulations of constrained boundary value problems are presented. Monotone mixed variational inclusions are macro-hybridized on the basis of nonoverlapping domain decompositions, and corresponding three-field versions are derived. Then, for regularization purposes, augmented formulations are established via preconditioned exact penalizations and expressed in terms of proximation operators. Optimization interpretations are given for potential problems, recovering the classic two- and three field augmented Lagrangian formulations. Furthermore, associated parallel two and three-field proximal-point algorithms are discussed for numerical resolution of finite element discretizations. Applications to dual mixed variational formulations of problems from mechanics illustrate the theory.

关键词： augmented variational formulations composition duality methods constrained boundary value problem hybrid domain decomposition proximal-point algorithms subdifferential equations

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Using the banach contraction principle to implement the proximal point method for multivalued monotone variational inequalities

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2005年第2期124卷 285-306页

作者： Anh, P Muu, LD Nguyen, VH Strodiot, JJ Post & Telecommun Inst Technol Hanoi Vietnam Inst Math Hanoi 10000 Vietnam Fac Univ Notre Dame Paix Dept Math Unite Optimisat B-5000 Namur Belgium

We apply the Banach contraction-mapping fixed-point principle for solving multivalued strongly monotone variational inequalities. Then, we couple this algorithm with the proximal-point method for solving monotone multivalued variational inequalities. We prove the convergence rate of this algorithm and report some computational results.

关键词： multivalued monotone variational inequalities proximal-point algorithms Banach contraction-mapping fixed-point methods convergence rates

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Composition duality methods for mixed variational inclusions

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APPLIED MATHEMATICS AND OPTIMIZATION 2005年第3期52卷 311-348页

作者： Alduncin, G Univ Nacl Autonoma Mexico Inst Geophys Mexico City 04510 DF Mexico

Composition duality methods for mixed variational inclusions are studied in a functional framework of reflexive Banach spaces. On the basis of duality principles, the solvability of maximal monotone and subdifferential mixed variational inclusions is established. For computational purposes, mass-preconditioned augmented formulations are introduced for regularization, as well as three-field and macro-hybrid variational versions. At a finite-dimensional level, corresponding discrete mixed and macro-hybrid internal approximations are discussed, as well as proximal-point iterative algorithms. Primal and dual mixed variational inclusions from contact mechanics illustrate the theory.

关键词： mixed variational inclusions duality methods subdifferential equations augmented and macro-hybrid formulations internal approximations proximal-point algorithms

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New self-concordant barrier for the hypercube

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2007年第3期135卷 475-490页

作者： Quiroz, E. A. Papa Oliveira, P. R. Univ Fed Rio de Janeiro PESC COPPE Dept Syst Engn & Comp Sci Rio De Janeiro Brazil

We introduce a new barrier function to build new interior-point algorithms to solve optimization problems with bounded variables. First, we show that this function is a (3/2)n-self-concordant barrier for the unitary hypercube [0,1]n, assuring thus the polynomial property of related algorithms. Second, using the Hessian metric of that barrier, we present new explicit algorithms from the point of view of Riemannian geometry applications. Third, we prove that the central path defined by the new barrier to solve a certain class of linearly constrained convex problems maintains most of the properties of the central path defined by the usual logarithmic barrier. We present also a primal long-step path-following algorithm with similar complexity to the classical barrier. Finally, we introduce a new proximal-point Bregman type algorithm to solve linear problems in [0,1]n and prove its convergence.

关键词： interior-point methods self-concordant barrier Newton methods geodesic algorithms proximal-point algorithms

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Modified proximal-point algorithm for maximal monotone operators in banach spaces

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2008年第1期138卷 45-64页

作者： Li, L. Song, W. Harbin Normal Univ Dept Math Harbin 150080 Peoples R China NE Normal Univ Dept Math Changchun 130024 Peoples R China

We introduce an iterative sequence for finding the solution to 0 is an element of T(v), where T : E paired right arrows E* is a maximal monotone operator in a smooth and uniformly convex Banach space E. This iterative procedure is a combination of iterative algorithms proposed by Kohsaka and Takahashi (Abstr. Appl. Anal. 3:239-249, 2004) and Kamamura, Kohsaka and Takahashi (Set-Valued Anal. 12:417-429, 2004). We prove a strong convergence theorem and a weak convergence theorem under different conditions respectively and give an estimate of the convergence rate of the algorithm. An application to minimization problems is given.

关键词： proximal-point algorithms uniformly convex and smooth Banach spaces maximal monotone operators strong and weak convergence

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On the proximal point algorithm

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2008年第2期137卷 411-417页

作者： Rouhani, B. Djafari Khatibzadeh, H. Univ Texas El Paso Dept Math Sci El Paso TX 79968 USA Tarbiat Modares Univ Dept Math Tehran Iran

Let A be a maximal monotone operator in a real Hilbert space H and let {u(n)} be the sequence in H given by the proximal point algorithm, defined by u (n) =(I+c(n) A)(-1)(u(n-1)-f(n) ), for all n >= 1, with u(0) = z, where c(n) > 0 and f(n) is an element of H. We show, among other things, that under suitable conditions, u(n) converges weakly or strongly to a zero of A if and only if lim inf(n ->+infinity) vertical bar w(n)vertical bar +infinity, where w(n) = (Sigma(n)(k=1) c(k))(-1) Sigma(n)(k=1) c(k)u(k). Our results extend previous results by several authors who obtained similar results by assuming A(-1)(0) not equal phi.

关键词： proximal-point algorithms variational inequalities ergodic theorems maximal monotone operators asymptotic centers

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