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Generic linear convergence through metric subregularity in a variable-metric extension of the proximal point algorithm

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2023年第3期86卷 1327-1346页

作者： Rockafellar, R. Tyrrell Univ Washington Dept Math Box 354350 Seattle WA 98195 USA

The proximal point algorithm finds a zero of a maximal monotone mapping by iterations in which the mapping is made strongly monotone by the addition of a proximal term. Here it is articulated with the norm behind the proximal term possibly shifting from one iteration to the next, but under conditions that eventually make the metric settle down. Despite the varying geometry, the sequence generated by the algorithm is shown to converge to a particular solution. Although this is not the first variable-metric extension of proximal point algorithm, it is the first to retain the flexibility needed for applications to augmented Lagrangian methodology and progressive decoupling. Moreover, in a generic sense, the convergence it generates is Q-linear at a rate that depends in a simple way on the modulus of metric subregularity of the mapping at that solution. This is a tighter rate than previously identified and reveals for the first time the definitive role of metric subregularity in how the proximal point algorithm performs, even in fixed-metric mode.

关键词： Maximal monotone mappings proximal point algorithm Variable-metric implementation Localized executability Linear convergence guarantees Metric subregularity Convex optimization

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The proximal point algorithm for pseudomonotone variational inequalities on Hadamard manifolds

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OPTIMIZATION LETTERS 2013年第4期7卷 779-790页

作者： Tang, Guo-ji Zhou, Li-wen Huang, Nan-jing Sichuan Univ Dept Math Chengdu 610064 Peoples R China

In this paper, we investigate the proximal point algorithm (in short PPA) for variational inequalities with pseudomonotone vector fields on Hadamard manifolds. Under weaker assumptions than monotonicity, we show that the sequence generated by PPA is well defined and prove that the sequence converges to a solution of variational inequality, whenever it exists. The results presented in this paper generalize and improve some corresponding known results given in literatures.

关键词： Pseudomonotone vector field Variational inequality proximal point algorithm Hadamard manifold

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A two-phase-like proximal point algorithm in domains of positivity

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APPLIED MATHEMATICS AND COMPUTATION 2019年 343卷 67-89页

作者： Gregorio, R. M. Oliveira, P. R. Alves, C. D. S. Univ Fed Rural Rio de Janeiro Inst Multidisciplinar Dept Tecnol & Linguagens Inst Ciencias ExatasPrograma Posgrad Modelgem Ma Ave Governador Roberto da Silveira S-NBloco Adm BR-2602074 Nova Iguacu RJ Brazil Univ Fed Rio de Janeiro Inst Alberto Luiz Coimbra Posgrad & Pesquisa Engn Programa Engn Sistemas & Comp Ctr Tecnol Ave Horacio Macedo 2030Bloco HSala 319 BR-21941914 Rio De Janeiro RJ Brazil

This paper improves a decomposition-like proximal point algorithm, developed for computing minima of nonsmooth convex functions within a framework of symmetric positive semidefinite matrices, and extends it to domains of positivity of reducible type, in a nonlinear sense and in a Riemannian setting. Several computational experiments with weighted L-P (p = 1, 2) centers of mass are performed to demonstrate the practical feasibility of the method. (C) 2018 Elsevier Inc. All rights reserved.

关键词： proximal point algorithm Homogeneous domain of positivity Hadamard manifold Compact Lie group Weighted L(P)center of mass

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TIGHT SUBLINEAR CONVERGENCE RATE OF THE proximal point algorithm FOR MAXIMAL MONOTONE INCLUSION PROBLEMS

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SIAM JOURNAL ON OPTIMIZATION 2020年第3期30卷 1905-1921页

作者： Gu, Guoyong Yang, Junfeng Nanjing Univ Dept Math Nanjing 210093 Peoples R China

The tight sublinear convergence rate of the proximal point algorithm for maximal monotone inclusion problems is established based on the squared fixed point residual. By using the performance estimation framework, the tight sublinear convergence rate problem is written as an infinite dimensional nonconvex optimization problem, which is then equivalently reformulated as a finite dimensional semidefinite programming (SDP) problem. By solving the SDP, the exact sublinear rate is computed numerically. Theoretically, by constructing a feasible solution to the dual SDP, an upper bound is obtained for the tight sublinear rate. On the other hand, an example in two dimensional space is constructed to provide a lower bound. The lower bound matches exactly the upper bound obtained from the dual SDP, which also coincides with the numerical rate computed. Hence, we have established the worst case sublinear convergence rate, which is tight in terms of both the order and the constants involved.

关键词： proximal point algorithm maximal monotone inclusion tight sublinear convergence rate performance estimation framework semidefinite programming

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Strong convergence of a proximal point algorithm with general errors

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OPTIMIZATION LETTERS 2012年第4期6卷 621-628页

作者： Yao, Yonghong Shahzad, Naseer King Abdulaziz Univ Dept Math Jeddah 21589 Saudi Arabia Tianjin Polytech Univ Dept Math Tianjin 300160 Peoples R China

In this paper we construct a proximal point algorithm for maximal monotone operators with appropriate regularization parameters. We obtain the strong convergence of the proposed algorithm, which affirmatively answer t... 详细信息

关键词： Maximal monotone operator proximal point algorithm Regularization Resolvent identity

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Pseudomonotone operators and the Bregman proximal point algorithm

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JOURNAL OF GLOBAL OPTIMIZATION 2010年第4期47卷 537-555页

作者： Langenberg, Nils Univ Trier Dept Math D-54286 Trier Germany

To permit the stable solution of ill-posed problems, the proximal point algorithm (PPA) was introduced by Martinet (RIRO 4:154-159, 1970) and further developed by Rockafellar (SIAM J Control Optim 14:877-898, 1976). Later on, the usual proximal distance function was replaced by the more general class of Bregman(-like) functions and related distances;see e.g. Chen and Teboulle (SIAM J Optim 3:538-543, 1993), Eckstein (Math Program 83:113-123, 1998), Kaplan and Tichatschke (Optimization 56(1-2):253-265, 2007), and Solodov and Svaiter (Math Oper Res 25:214-230, 2000). An adequate use of such generalized non-quadratic distance kernels admits to obtain an interior-point-effect, that is, the auxiliary problems may be treated as unconstrained ones. In the above mentioned works and nearly all other works related with this topic it was assumed that the operator of the considered variational inequality is a maximal monotone and paramonotone operator. The approaches of El-Farouq (JOTA 109:311-326, 2001), and Schaible et al. (Taiwan J Math 10(2):497-513, 2006) only need pseudomonotonicity (in the sense of Karamardian in JOTA 18:445-454, 1976);however, they make use of other restrictive assumptions which on the one hand contradict the desired interior-point-effect and on the other hand imply uniqueness of the solution of the problem. The present work points to the discussion of the Bregman algorithm under significantly weaker assumptions, namely pseudomonotonicity [and an additional assumption much less restrictive than the ones used by El-Farouq and Schaible et al. We will be able to show that convergence results known from the monotone case still hold true;some of them will be sharpened or are even new. An interior-point-effect is obtained, and for the generated subproblems we allow inexact solutions by means of a unified use of a summable-error-criterion and an error criterion of fixed-relative-error-type (this combination is also new in the literature).

关键词： Pseudomonotone operators Variational inequalities Bregman distances proximal point algorithm Interior-point-effect

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Local convergence of the proximal point algorithm and multiplier methods without monotonicity

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MATHEMATICS OF OPERATIONS RESEARCH 2002年第1期27卷 170-191页

作者： Pennanen, T Helsinki Sch Econ Dept Management Sci Helsinki 00101 Finland

This paper studies the convergence of the classical proximal point algorithm without assuming monotonicity of the underlying mapping. Practical conditions are given that guarantee the local convergence of the iterates to a solution of T(x) B 0, where T is an arbitrary set-valued mapping from a Hilbert space to itself. In particular, when the problem is "nonsingular" in the sense that T-1 has a Lipschitz localization around one of the solutions, local linear convergence is obtained. This kind of regularity property of variational inclusions has been extensively studied in the literature under the name of strong regularity, and it can be viewed as a natural generalization of classical constraint qualifications in nonlinear programming to more general problem classes. Combining the new convergence results with an abstract duality framework for variational inclusions, the author proves the local convergence of multiplier methods for a very general class of problems. This gives as special cases new convergence results for multiplier methods for nonmonotone variational inequalities and nonconvex nonlinear programming.

关键词： proximal point algorithm local convergence strong regularity

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On the identification of degenerate indices in the nonlinear complementarity problem with the proximal point algorithm

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MATHEMATICAL PROGRAMMING 2004年第2期99卷 377-397页

作者： Yamashita, N Dan, H Fukushima, M Kyoto Univ Grad Sch Inofrmat Dept Appl Math & Phys Kyoto 6068501 Japan Math Syst Inc Tokyo 1600022 Japan

In this paper we focus on the problem of identifying the index sets P(x) := {i \ x(i)>0}, N(x) := {i \ F-i(x)>0} and C(x) := {i \ x(i) = F-i(x)=0} for a solution x of the monotone nonlinear complementarity problem NCP(F). The correct identification of these sets is important from both theoretical and practical points of view. Such an identification enables us to remove complementarity conditions from the NCP and locally reduce the NCP to a system which can be dealt with more easily. We present a new technique that utilizes a sequence generated by the proximal point algorithm (PPA). Using the superlinear convergence property of PPA, we show that the proposed technique can identify the correct index sets without assuming the nondegeneracy and the local uniqueness of the solution.

关键词： nonlinear complementarity problem proximal point algorithm degeneracy

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A proximal point algorithm converging strongly for general errors

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OPTIMIZATION LETTERS 2010年第4期4卷 635-641页

作者： Boikanyo, O. A. Morosanu, G. Cent European Univ Dept Math & Its Applicat H-1051 Budapest Hungary

In this paper a proximal point algorithm (PPA) for maximal monotone operators with appropriate regularization parameters is considered. A strong convergence result for PPA is stated and proved under the general condition that the error sequence tends to zero in norm. Note that Rockafellar (SIAM J Control Optim 14: 877-898, 1976) assumed summability for the error sequence to derive weak convergence of PPA in its initial form, and this restrictive condition on errors has been extensively used so far for different versions of PPA. Thus this Note provides a solution to a long standing open problem and in particular offers new possibilities towards the approximation of the minimum points of convex functionals.

关键词： proximal point algorithm Prox-Tikhonov algorithm Monotone operator Regularization parameters Strong convergence

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A regularization method for the proximal point algorithm

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JOURNAL OF GLOBAL OPTIMIZATION 2006年第1期36卷 115-125页

作者： Xu, Hong-Kun Univ KawZulu Natal Sch Math Sci ZA-4000 Durban South Africa

A regularization method for the proximal point algorithm of finding a zero for a maximal monotone operator in a Hilbert space is proposed. Strong convergence of this algorithm is proved.

关键词： maximal monotone operator projection proximal point algorithm regularization method resolvent identity strong convergence

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