Demiclosedness Principles for Generalized Nonexpansive Mappings

Bartz, Sedi; Campoy, Rubén; Phan, Hung M.

doi:10.1007/s10957-020-01734-6

Cited by 6 publications

(6 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Finally, we recall the demiclosedness principle for cocoercive operators developed in [3]. A fundamental result in the theory of nonexpansive mapping is Browder's celebrated demiclosedness principle [9].…”

Section: Proposition 23 (Single-valued and Full Domain)mentioning

confidence: 99%

“…A fundamental result in the theory of nonexpansive mapping is Browder's celebrated demiclosedness principle [9]. It was extended for finitely many firmly nonexpansive mappings in [5], and was later generalized in [3] for a finite family of conically averaged mappings or for a finite family of cocoercive mappings. An instant application of the demiclosedness principles is to provide a simple proof for the weak convergence of the shadow sequence of the Douglas-Rachford algorithm [5], and of the adaptive Douglas-Rachford algorithm [3].…”

Section: Proposition 23 (Single-valued and Full Domain)mentioning

confidence: 99%

See 1 more Smart Citation

Adaptive Douglas--Rachford Splitting Algorithm for the Sum of Two Operators

Dao¹,

Phan²

2019

SIAM J. Optim.

Self Cite

View full text Add to dashboard Cite

The Douglas-Rachford algorithm is a classical and powerful splitting method for minimizing the sum of two convex functions and, more generally, finding a zero of the sum of two maximally monotone operators. Although this algorithm has been well understood when the involved operators are monotone or strongly monotone, the convergence theory for weakly monotone settings is far from being complete. In this paper, we propose an adaptive Douglas-Rachford splitting algorithm for the sum of two operators, one of which is strongly monotone while the other one is weakly monotone. With appropriately chosen parameters, the algorithm converges globally to a fixed point from which we derive a solution of the problem. When one operator is Lipschitz continuous, we prove global linear convergence which sharpens recent known results. entire mathematical model. It is worth mentioning (see, e.g., [23]) that several splitting methods such as the method of partial inverses [35] and the alternating direction method of multipliers (ADMM) [25] can be written in the form of the DR algorithm, which itself can be transformed into the proximal point algorithm [34]. Other splitting schemes can be found in [13,14,16] and the references therein.When applied to two normal cone operators, the DR algorithm can be used to solve the feasibility problem of finding a common point of two sets. In this context, the DR algorithm possesses many good properties, for example, it finds a best approximation point when the intersection of sets is empty [3,5,8], it finds an exact solution after only a finite number of iterations under verifiable conditions [1,4,7], and it converges globally in some nonconvex settings [9,19] while converges locally with linear or sublinear rate under some regularity assumptions [11,29,33]. In the absence of constraint qualifications, [6] suggests that the DR algorithm outperforms the well-known method of alternating projections. In attempting to generalize the DR algorithm for feasibility problems, several parameters were added to its formulation [12,18,17,24]. In this case, one has the freedom to modify the parameters that are associated with the projections without giving up the solution. This approach is possible because the underlying normal cone operators have homogeneous values, which allows for scaling them independently. The situation changes completely when working with general problems where two involved operators may no longer have such homogeneity. In this case, a naive scaling may destroy the ability to solve the original problem. Therefore, we aim to overcome this hurdle by proposing an adaptive approach.The paper is devoted to the convergence analysis of the adaptive DR algorithm for finding a zero of the sum of α-and β-monotone operators, in which α-monotonicity is a unification of strong and weak monotonicity (see Definition 3.1). This situation arises in various important applications; see [27] for a brief discussion. The main contribution is summarized below.(R1) We incorporate parameters into the DR algorith...

show abstract

Section: Proposition 23 (Single-valued and Full Domain)mentioning

confidence: 99%

Section: Proposition 23 (Single-valued and Full Domain)mentioning

confidence: 99%

Adaptive Douglas--Rachford Splitting Algorithm for the Sum of Two Operators

Dao¹,

Phan²

2019

SIAM J. Optim.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Finally, we recall the demiclosedness principle for cocoercive operators developed in [2]. A fundamental result in the theory of nonexpansive mapping is Browder's celebrated demiclosedness principle [9].…”

Section: Proposition 23 (Single-valued and Full Domain)mentioning

confidence: 99%

“…A fundamental result in the theory of nonexpansive mapping is Browder's celebrated demiclosedness principle [9]. It was extended for finitely many firmly nonexpansive mappings in [5] and was later generalized in [2] for a finite family of conically averaged mappings or for a finite family of cocoercive mappings. An instant application of the demiclosedness principle is to provide a simple proof for the weak convergence of the shadow sequence of the Douglas-Rachford algorithm [5] and of the adaptive Douglas-Rachford algorithm [2].…”

Section: Proposition 23 (Single-valued and Full Domain)mentioning

confidence: 99%

An adaptive splitting algorithm for the sum of two generalized monotone operators and one cocoercive operator

Dao

Phan

2021

Fixed Point Theory Algorithms Sci Eng

Self Cite

View full text Add to dashboard Cite

Splitting algorithms for finding a zero of sum of operators often involve multiple steps which are referred to as forward or backward steps. Forward steps are the explicit use of the operators and backward steps involve the operators implicitly via their resolvents. In this paper, we study an adaptive splitting algorithm for finding a zero of the sum of three operators. We assume that two of the operators are generalized monotone and their resolvents are computable, while the other operator is cocoercive but its resolvent is missing or costly to compute. Our splitting algorithm adapts new parameters to the generalized monotonicity of the operators and, at the same time, combines appropriate forward and backward steps to guarantee convergence to a solution of the problem.

show abstract

“…An additional benefit of our approach is that we relax and improve previously imposed assumptions on the strongly-weakly convex scenario such as in [33,44] (see Remarks 5.1 and 5.2). Finally, although augmentation is a viable option in the strongly-weakly convex setting, application of the adaptive algorithms to the original problem has its own merit and by now was studied in a number of recent publications such as [2,3,16,17,25,27,32,44,45], to name a few.…”

Section: Introductionmentioning

confidence: 99%

An Adaptive Alternating Direction Method of Multipliers

Bartz

Campoy

Phan

2022

J Optim Theory Appl

Self Cite

View full text Add to dashboard Cite

The alternating direction method of multipliers (ADMM) is a powerful splitting algorithm for linearly constrained convex optimization problems. In view of its popularity and applicability, a growing attention is drawn toward the ADMM in nonconvex settings. Recent studies of minimization problems for nonconvex functions include various combinations of assumptions on the objective function including, in particular, a Lipschitz gradient assumption. We consider the case where the objective is the sum of a strongly convex function and a weakly convex function. To this end, we present and study an adaptive version of the ADMM which incorporates generalized notions of convexity and penalty parameters adapted to the convexity constants of the functions. We prove convergence of the scheme under natural assumptions. To this end, we employ the recent adaptive Douglas–Rachford algorithm by revisiting the well-known duality relation between the classical ADMM and the Douglas–Rachford splitting algorithm, generalizing this connection to our setting. We illustrate our approach by relating and comparing to alternatives, and by numerical experiments on a signal denoising problem.

show abstract

Demiclosedness Principles for Generalized Nonexpansive Mappings

Cited by 6 publications

References 13 publications

Adaptive Douglas--Rachford Splitting Algorithm for the Sum of Two Operators

Adaptive Douglas--Rachford Splitting Algorithm for the Sum of Two Operators

An adaptive splitting algorithm for the sum of two generalized monotone operators and one cocoercive operator

An Adaptive Alternating Direction Method of Multipliers

Contact Info

Product

Resources

About