Inexact Proximal Point Methods in Metric Spaces

Zaslavski, Alexander J.

doi:10.1007/s11228-011-0185-9

Cited by 11 publications

(3 citation statements)

References 29 publications

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“…Since minimizers of convex lsc functionals in these spaces play an important role in analysis and geometry (see, for instance, Sections 1.2 and 1.3), we dare to believe that the PPA will prove useful. Also, we should like to mention Zaslavski's very recent paper [27] with a different approach to the PPA in metric spaces.…”

Section: Introductionmentioning

confidence: 99%

The proximal point algorithm in metric spaces

2012

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The proximal point algorithm, which is a well-known tool for finding minima of convex functions, is generalized from the classical Hilbert space framework into a nonlinear setting, namely, geodesic metric spaces of nonpositive curvature. We prove that the sequence generated by the proximal point algorithm weakly converges to a minimizer, and also discuss a related question: convergence of the gradient flow.

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Section: Introductionmentioning

confidence: 99%

The proximal point algorithm in metric spaces

2012

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“…It will be interesting and useful to extend the ideas and various assertions described in this paper to other settings. In particular, to allow (with a suitable caution due to the presence of error terms) in the inexact resolvent problem (4.1) functions f having effective domains which are subsets of the whole space, to allow enlargements of operators (here it seems reasonable to extend the theory of resolvents mentioned briefly in Section 2 and the references cited there to resolvents of enlargements, and [23,24,26] may be of some help in this direction), to consider also inexactness coming from ǫ-subdifferentials, to allow spaces more general than normed spaces such as Hadamard spaces and other metric spaces [1,7,8,59,95,100,105] (the theory of resolvents for Hadamard spaces described in [60] may help in this direction), to allow certain nonlinear modifications of (4.1) such as the one given in [3, p. 179] and [5, pp. 648, 650, 658] (and to extend the latter ones so they will allow general Bregman distances which may not be induced from Bregman functions [75]), to allow inducing functions f more general than fully Legendre such as zero-convex functions [36] (or at least special but important classes of zero-convex functions such as quasiconvex functions [70]), d.c. functions [94], and so on.…”

Section: Discussionmentioning

confidence: 99%

Solutions to inexact resolvent inclusion problems with applications to nonlinear analysis and optimization

Reem

Reich

2017

Rend. Circ. Mat. Palermo, II. Ser

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Many problems in nonlinear analysis and optimization, among them variational inequalities and minimization of convex functions, can be reduced to finding zeros (namely, roots) of set-valued operators. Hence numerous algorithms have been devised in order to achieve this task. A lot of these algorithms are inexact in the sense that they allow perturbations to appear during the iterative process, and hence they enable one to better deal with noise and computational errors, as well as superiorization. For many years a certain fundamental question has remained open regarding many of these known inexact algorithmic schemes in various finite and infinite dimensional settings, namely whether there exist sequences satisfying these inexact schemes when errors appear. We provide a positive answer to this question. Our results also show that various theorems discussing the convergence of these inexact schemes have a genuine merit beyond the exact case. As a by-product we solve the standard and the strongly implicit inexact resolvent inclusion problems, introduce a promising class of functions (fully Legendre functions), establish continuous dependence (stability) properties of the solution of the inexact resolvent inclusion problem and continuity properties of the protoresolvent, and generalize the notion of strong monotonicity.This problem is very useful because many other problems can be reduced to solving it (especially when the operator is maximally monotone), among them finding solutions of minmax problems, complementarity problems, variational inequalities, convex feasibility problems, equilibrium problems, and minimizing convex functions. For instance, given a proper, lower semicontinuous and convex function F : X → (−∞, ∞], if A = ∂F (namely, the subdifferential of F ), then, as is well known, A is maximally Date: August 22, 2017. 2010 Mathematics Subject Classification. 90C31, 47H05, 47J25, 90C30, 49M37, 65K05.

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“…where {λ k } , {ε k } are given sequences of nonnegative real numbers, and q is a quasi distance. In the particular case where the generalized perturbation term Γ[q(x, y)] = q(x, y) 2 and q(x, y) = d(x, y) is a distance, instead of a quasi distance, our "global epsilon inexact proximal" algorithm coincides with the case considered by Zaslavski [24].…”

Section: The Algorithmmentioning

confidence: 99%

Generalized Inexact Proximal Algorithms: Routine’s Formation with Resistance to Change, Following Worthwhile Changes

Bento

Soubeyran

2015

J Optim Theory Appl

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This paper shows how, in a quasi metric space, an inexact proximal algorithm with a generalized perturbation term appears to be a nice tool for Behavioral Sciences (Psychology, Economics, Management, Game theory,. . . ). More precisely, the new perturbation term represents an index of resistance to change, defined as a "curved enough" function of the quasi distance between two successive iterates.Using this behavioral point of view, the present paper shows how such a generalized inexact proximal algorithm can modelize the formation of habits and routines in a striking way. This idea comes from a recent "variational rationality approach" of human behavior which links a lot of different theories of stability (habits, routines, equilibrium, traps,. . . ) and changes (creations, innovations, learning and destructions,. . . ) in Behavioral Sciences and a lot of concepts and algorithms in Variational Analysis. In this variational context, the perturbation term represents a specific instance of the very general concept of resistance to change, which is the disutility of some inconvenients to change. Central to the analysis are the original variational concepts of "worthwhile changes" and "marginal worthwhile stays". At the behavioral level, this paper advocates that proximal algorithms are well suited to modelize the emergence of habituation/routinized human behaviors. We show when, and at which speed, a "worthwhile to change" process converges to a behavioral trap.keywords. Nonconvex optimizationKurdyka-Lojasiewicz inequalityinexact proximal algorithmshabitsroutinesworthwhile changes IntroductionThe main message of this paper is that, using the behavioral context of a recent "Variational rationality" approach of worthwhile stay and change dynamics proposed by Soubeyran [1, 2], a generalized proximal algorithm can modelize fairly well an habituation process as described in Psychology for an agent, or a routinization process, in Management Sciences, for an organization. This opens the door to a new vision of proximal algorithms. They are not only very nice mathematical tools in optimization theory, with striking computational aspects. They can also be nice tools to modelize the dynamics of human behaviors.Theories of stability and change consider successions of stays and changes. Stays refer to habits, routines, equilibrium, traps, rules, conventions,. . . . Changes represent creations, destructions, learning processes, innovations, attitudes as well as beliefs formation and revision, self regulation problems, including goal setting, goal striving and goal revision, the formation and break of habits and routines,. . . . In the interdisciplinary context which characterizes all these theories in Behavioral Sciences, the "Variational rationality approach" (see [1,2]), shows how to modelize the course of human activities as a succession of worthwhile temporary

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Inexact Proximal Point Methods in Metric Spaces

Cited by 11 publications

References 29 publications

The proximal point algorithm in metric spaces

The proximal point algorithm in metric spaces

Solutions to inexact resolvent inclusion problems with applications to nonlinear analysis and optimization

Generalized Inexact Proximal Algorithms: Routine’s Formation with Resistance to Change, Following Worthwhile Changes

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