Affine Nonexpansive Operators, Attouch–Théra Duality and the Douglas–Rachford Algorithm

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

doi:10.1007/s11228-016-0399-y

Cited by 9 publications

(11 citation statements)

References 37 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Now let (a, b, a * , b * ) be a weak cluster point of ((J A T n x, J B R A T n x, J A −1 T n x, J B −1 R A T n x)) n∈N . By (v) we know that (a, a * ) ∈ gra A and (b, b * ) = (a, b * ) ∈ gra B, which in view of (iv) implies a * ∈ Aa and −a * = b * ∈ Bb = Ba, hence (a, a * ) ∈ S, as claimed (see (11)). Theorem 6.2.…”

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

confidence: 65%

On the Douglas–Rachford algorithm

Bauschke

Moursi

2016

Math. Program.

Self Cite

View full text Add to dashboard Cite

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.

show abstract

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

confidence: 65%

On the Douglas–Rachford algorithm

Bauschke

Moursi

2016

Math. Program.

Self Cite

View full text Add to dashboard Cite

show abstract

“…DRS has strong primal-dual symmetry, in the sense of Fenchel duality for convex optimization [41,70] and, more generally, Attouch-Théra duality for monotone operators [55, p. 40] and [4]. See [37, Lemma 3.6 p. 133] or [7,14,16] for in-depth studies on this subject. Naturally, our results also exhibit a degree of primal-dual symmetry, although we do not explicitly address it in the interest of space.…”

Section: Prior Workmentioning

confidence: 99%

Douglas–Rachford splitting and ADMM for pathological convex optimization

2019

View full text Add to dashboard Cite

Despite the vast literature on DRS and ADMM, there has been very little work analyzing their behavior under pathologies. Most analyses assume a primal solution exists, a dual solution exists, and strong duality holds. When these assumptions are not met, i.e., under pathologies, the theory often breaks down and the empirical performance may degrade significantly. In this paper, we establish that DRS only requires strong duality to work, in the sense that asymptotically iterates are approximately feasible and approximately optimal.Keywords Douglas-Rachford splitting · Strong Duality · Pathological convex programs Mathematics Subject Classification (2010) 90C46 · 49N15 · 90C25 1 Introduction Douglas-Rachford splitting (DRS) and alternating directions method of multipliers (ADMM) are classical methods originally presented in [61,35,49,47] and [44,46], respectively. DRS and ADMM are closely related. Over the last decade, these methods have enjoyed a resurgence of popularity, as the demand to solve ever larger problems grew.DRS and ADMM have strong theoretical guarantees and empirical performance, but such results are often limited to non-pathological problems; in Ernest K. Ryu UCLA

show abstract

“…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%

“…Proof. The proof uses the same techniques as in [16]. "(i)⇔(ii)": See [4,Proposition 4], [3, Theorem 1.1], [9,Theorem 2.2] or [8,Proposition 5.27].…”

Section: Then a Is Firmly Nonexpansive And B Is Maximally Monotone Mmentioning

confidence: 99%

See 1 more Smart Citation

The Forward–Backward Algorithm and the Normal Problem

Moursi

2018

J Optim Theory Appl

Self Cite

View full text Add to dashboard Cite

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...

show abstract

Affine Nonexpansive Operators, Attouch–Théra Duality and the Douglas–Rachford Algorithm

Cited by 9 publications

References 37 publications

On the Douglas–Rachford algorithm

On the Douglas–Rachford algorithm

Douglas–Rachford splitting and ADMM for pathological convex optimization

The Forward–Backward Algorithm and the Normal Problem

Contact Info

Product

Resources

About