Convergence Rate Analysis for Averaged Fixed Point Iterations in Common Fixed Point Problems

Borwein, Jonathan M.; Li, Guoyin; Tam, Matthew K.

doi:10.1137/15m1045223

Cited by 60 publications

(59 citation statements)

References 45 publications

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“…may be recognized as the form in which the relaxation was presented by Borwein, Li, and Tam for their damped Douglas-Rachford variant [20].…”

Section: Background and Preliminariesmentioning

confidence: 94%

“…Convergence rates may frequently be obtained through analysis of regularity conditions [38]. Additionally, semialgebraic structure admits further bounds on convergence rates for projection methods more generally [20,21,34] and for the Douglas-Rachford method in particular [40]. For a recent survey on the Douglas-Rachford method, see [41].The idea of replacing projections with their approximations, and specifically with the approximations constructed from the subdifferentials of the convex functions that describe the sets, was introduced by Fukushima [36].…”

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confidence: 99%

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Comparing Averaged Relaxed Cutters and Projection Methods: Theory and Examples

Millán

Lindstrom

Roshchina

2020

Springer Proceedings in Mathematics &Amp; Statistics

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We focus on the convergence analysis of averaged relaxations of cutters, specifically for variants that-depending upon how parameters are chosen-resemble alternating projections, the Douglas-Rachford method, relaxed reflect-reflect, or the Peaceman-Rachford method. Such methods are frequently used to solve convex feasibility problems. The standard convergence analyses of projection algorithms are based on the firm nonexpansivity property of the relevant operators. However if the projections onto the constraint sets are replaced by cutters (projections onto separating hyperplanes), the firm nonexpansivity is lost. We provide a proof of convergence for a family of related averaged relaxed cutter methods under reasonable assumptions, relying on a simple geometric argument. This allows us to clarify fine details related to the allowable choice of the relaxation parameters, highlighting the distinction between the exact (firmly nonexpansive) and approximate (strongly quasi-nonexpansive) settings. We provide illustrative examples and discuss practical implementations of the method. IntroductionProjection and reflection methods are used for solving the feasibility problem of finding a point in the intersection of a finite collection of closed, convex sets in a Hilbert space. Such problems have a wide range of application in variational analysis, optimisation, physics and mathematics in general. One of the most successful methods from this class is the Douglas-Rachford method that uses a combination of reflections and averaging on each iteration. The idea first appeared in [33] as a numerical scheme for solving differential equations, and the convergence of a more general scheme for finding a zero of the sum of two maximally monotone operators was framed in [43] (also see [12, Chapter 26] for a modern treatment). Aragón Artacho and Campoy have recently introduced a modification of the Douglas-Rachford method for finding closest feasible points [7].Convergence rates for such methods are the subject of extensive research; we provide a brief sampling. Under appropriate conditions, the Douglas-Rachford method converges in finitely many steps [12]. Convergence rates may frequently be obtained through analysis of regularity conditions [38]. Additionally, semialgebraic structure admits further bounds on convergence rates for projection methods more generally [20,21,34] and for the Douglas-Rachford method in particular [40]. For a recent survey on the Douglas-Rachford method, see [41].The idea of replacing projections with their approximations, and specifically with the approximations constructed from the subdifferentials of the convex functions that describe the sets, was introduced by Fukushima [36]. It has been used in various contexts recently, including the numerical solution of variational arXiv:1810.02463v1 [math.OC] 4 Oct 2018inequalities; see, for example, [16,17]. In particular, Combettes has used relaxation parameters together with subgradient projections in the construction of his extrapolation method of parallel sub...

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“…may be recognized as the form in which the relaxation was presented by Borwein, Li, and Tam for their damped Douglas-Rachford variant [20].…”

Section: Background and Preliminariesmentioning

confidence: 94%

mentioning

confidence: 99%

Comparing Averaged Relaxed Cutters and Projection Methods: Theory and Examples

Millán

Lindstrom

Roshchina

2020

Springer Proceedings in Mathematics &Amp; Statistics

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“…Borwein, Li, and Tam [48] attribute the first convergence rate results for DR to Hesse, Luke, and Patrick Neumann who in 2014 showed local linear convergence in the possibly nonconvex context of sparse affine feasibility problems [109]. Bauschke, Bello Cruz, Nghia, Phan, and Wang extended this work by showing that the rate of linear convergence of DR for subspaces is the cosine of the Friedrichs angle [17].…”

Section: Convergencementioning

confidence: 99%

“…Motivated by the recent local linear convergence results in the possibly nonconvex setting [108,109,126,115], Borwein, Li, and Tam asked whether a global convergence rate for DR in finite dimensions might be found for a reasonable class of convex sets even when the regularity condition riA ∩ riB = / 0 is potentially not satisfied. They provided some partial answers in the context of Hölder regularity with special attention given to convex semi-algebraic sets [48].…”

Section: Convergencementioning

confidence: 99%

Survey: Sixty Years of Douglas–rachford

Lindstrom¹,

Sims²

2020

J. Aust. Math. Soc.

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DedicationThis work is dedicated to the memory of Jonathan M. Borwein our greatly missed friend, mentor, and colleague. His influence on both the topic at hand, as well as his impact on the present authors personally, cannot be overstated. AbstractThe Douglas-Rachford method is a splitting method frequently employed for finding zeroes of sums of maximally monotone operators. When the operators in question are normal cones operators, the iterated process may be used to solve feasibility problems of the form: Find x ∈ N k=1 S k . The success of the method in the context of closed, convex, nonempty sets S 1 , . . . , S N is well-known and understood from a theoretical standpoint. However, its performance in the nonconvex context is less understood yet surprisingly impressive. This was particularly compelling to Jonathan M. Borwein who, intrigued by Elser, Rankenburg, and Thibault's success in applying the method for solving Sudoku Puzzles began an investigation of his own. We survey the current body of literature on the subject, and we summarize its history. We especially commemorate Professor Borwein's celebrated contributions to the area. arXiv:1809.07181v2 [math.OC]

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“…First-order methods for convex problems lead generically to globally averaged fixed point mappings T. Convergence for convex problems can be determined from the averaging property of T and existence of fixed points. Hence, to quantify convergence, the only thing to be determined is the gauge of metric regularity at the fixed points of T. In this context, see Borwein et al [21]. Example 2.4 illustrates how this can be done.…”

Section: Example 24 (A Line Tangent To a Circle)mentioning

confidence: 99%

Quantitative Convergence Analysis of Iterated Expansive, Set-Valued Mappings

2018

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Abstract. We develop a framework for quantitative convergence analysis of Picard iterations of expansive set-valued fixed point mappings. There are two key components of the analysis. The first is a natural generalization of single-valued averaged mappings to expansive set-valued mappings that characterizes a type of strong calmness of the fixed point mapping. The second component to this analysis is an extension of the well-established notion of metric subregularity-or inverse calmness-of the mapping at fixed points. Convergence of expansive fixed point iterations is proved using these two properties, and quantitative estimates are a natural by-product of the framework. To demonstrate the application of the theory, we prove, for the first time, a number of results showing local linear convergence of nonconvex cyclic projections for inconsistent (and consistent) feasibility problems, local linear convergence of the forward-backward algorithm for structured optimization without convexity, strong or otherwise, and local linear convergence of the Douglas-Rachford algorithm for structured nonconvex minimization. This theory includes earlier approaches for known results, convex and nonconvex, as special cases.

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Convergence Rate Analysis for Averaged Fixed Point Iterations in Common Fixed Point Problems

Cited by 60 publications

References 45 publications

Comparing Averaged Relaxed Cutters and Projection Methods: Theory and Examples

Comparing Averaged Relaxed Cutters and Projection Methods: Theory and Examples

Survey: Sixty Years of Douglas–rachford

Quantitative Convergence Analysis of Iterated Expansive, Set-Valued Mappings

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