Affine nonexpansive operators, Attouch-Théra duality and the Douglas-Rachford algorithm

Bauschke, Heinz H.; Lukens, Brett; Moursi, Walaa M.

doi:10.48550/arxiv.1603.09418

Cited by 2 publications

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“…In view of (11) one might conjecture that Corollary 3.4 holds when we replace S by Z × K. The following example gives a negative answer to this conjecture. It also illustrates that when X = R, the conclusion of Lemma 3.5 could fail.…”

Section: (I)])mentioning

confidence: 97%

“…Now, if w = 0, by [11,Proposition 5.3] we have Tx = T (par U,par V) (x − w) + w. Therefore, (30) 2 , where the last equality follows from [8, Proposition 3.17].…”

Section: Linear Relationsmentioning

confidence: 99%

“…(See [11] and [21] for details on this connection.) They showed that (T n x) n∈N converges weakly to a point in Fix T and that the bounded shadow sequence (J A T n x) n∈N has all its weak cluster points in Z provided that A + B was maximally monotone.…”

Section: A Proof Of the Lions-mercier-svaiter Theoremmentioning

confidence: 99%

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On the Douglas–Rachford algorithm

Bauschke

Moursi

2016

Math. Program.

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The Douglas-Rachford algorithm is a very popular splitting technique for finding a zero of the sum of two maximally monotone operators. However, the behaviour of the algorithm remains mysterious in the general inconsistent case, i.e., when the sum problem has no zeros. More than a decade ago, however, it was shown that in the (possibly inconsistent) convex feasibility setting, the shadow sequence remains bounded and it is weak cluster points solve a best approximation problem.In this paper, we advance the understanding of the inconsistent case significantly by providing a complete proof of the full weak convergence in the convex feasibility setting. In fact, a more general sufficient condition for the weak convergence in the general case is presented. Several examples illustrate the results.

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Section: (I)])mentioning

confidence: 97%

“…Now, if w = 0, by [11,Proposition 5.3] we have Tx = T (par U,par V) (x − w) + w. Therefore, (30) 2 , where the last equality follows from [8, Proposition 3.17].…”

Section: Linear Relationsmentioning

confidence: 99%

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On the Douglas–Rachford algorithm

Bauschke

Moursi

2016

Math. Program.

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“…Since zer(A + B) = zer(αA + αB), the assumption that A is firmly nonexpansive could be replaced by A is α-cocoercive 5 . In this case (16) and (17) can be applied with the ordered pair (A, B) is replaced by (αA, αB). Definition 3.2 (paramonotone and 3 * monotone operators).…”

Section: The Forward-backward Operator and Dualitymentioning

confidence: 99%

“…Therefore we only need to check the points in (0, β) β ∈ R . To 16 Let f : X → ]−∞, +∞] be convex, lower semicontinuous, and proper. We use f * to denote the convex conjugate (a.k.a.…”

Section: The Range Of the Displacement Operatormentioning

confidence: 99%

The Forward–Backward Algorithm and the Normal Problem

Moursi

2018

J Optim Theory Appl

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The forward-backward splitting technique is a popular method for solving monotone inclusions that has applications in optimization. In this paper we explore the behaviour of the algorithm when the inclusion problem has no solution. We present a new formula to define the normal solutions using the forward-backward operator. We also provide a formula for the range of the displacement map of the forward-backward operator. Several examples illustrate our theory.Thanks to the fact that the subdifferential operator associated with a convex lower semicontinuous proper function is a maximally monotone operator (see Fact 3.6 below), the notion of monotone operators becomes of significant importance in optimization and nonlinear analysis. For further discussion on monotone operator theory and its connection to optimization see, e.g., the books [8], [17], [19], [21], [44], [45], [49], [50], and [51].The problem of finding a zero of the sum of two maximally monotone operators A and B is to find x ∈ X such that x ∈ (A + B) −1 0. When specializing A and B to subdifferential operators of convex lower semicontinuous proper functions, the problem is equivalent to finding a minimizer of the sum of the two functions, which is a classical optimization problem.Suppose that A is firmly nonexpansive 1 (see Section 2). Let x 0 ∈ X and let T FB be the forwardbackward operator associated with the pair (A, B) (see Section 3). When (A + B) −1 0 = ∅ the sequence (T n FB x 0 ) n∈N produced by iterating the forward-backward operator converges weakly 2 to a point in (A [33] or [23]). Applications of this setting appear in convex optimization (see, e.g., [8, Section 27.3]), evolution inclusions (see, e.g., [2]) and inverse problems (see, e.g., [24] and [25]).The goal of this work is to examine the forward-backward operator in the inconsistent case, i.e., when (A + B) −1 0 = ∅, using the framework of the normal problem introduced in [12]. In this case Fix T FB = ∅, and the classical analysis, which uses the advantage of iterating an averaged operator (see Section 2 below) that has a fixed point, is no longer applicable.Let us summarize the main contributions of the paper: R1We provide a systematic study of the forward-backward operator when the sum problem is possibly inconsistent. This is mainly illustrated in Proposition 4.1 where we establish the connection between the perturbed problem introduced in [12] and the forward-backward operator. R2We prove that the range of the displacement operator associated with the forward-backward operator T FB coincides with that of the Douglas-Rachford operator T DR . Consequently, the minimal displacement vectors associated with T FB and T DR coincide (see Theorem 4.2). This gives an alternative approach to define the normal problem introduced in [12].R3 A significant consequence of R2 is that it allows to use the advantage of the self-duality of T DR (which does not hold for T FB as we illustrate in Example 4.11) to draw more conclusions about T FB . In particular, in Theorem 5.3 we provide a formula for th...

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Affine nonexpansive operators, Attouch-Théra duality and the Douglas-Rachford algorithm

Cited by 2 publications

References 0 publications

On the Douglas–Rachford algorithm

On the Douglas–Rachford algorithm

The Forward–Backward Algorithm and the Normal Problem

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