Computing the resolvent of the sum of operators with application to best approximation problems

Dao, Minh N.; Phan, Hung M.

doi:10.1007/s11590-019-01432-x

Cited by 10 publications

(10 citation statements)

References 15 publications

(22 reference statements)

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“…Intuitively, in the case A and B are maximally monotone, one would expect the use of equal resolvent parameters γ = δ, and in other cases, γ are δ are no longer the same. This phenomenon was initially observed in [11,12]. Although the imbalance of monotonicity can be resolved by shifting the identity between the operators as in [11,Remark 4.15], our plan is to conduct the convergence analysis of the algorithm applied to the original operators.…”

Section: The Algorithmmentioning

confidence: 98%

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

Dao¹,

Phan²

2019

SIAM J. Optim.

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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...

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Section: The Algorithmmentioning

confidence: 98%

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

Dao¹,

Phan²

2019

SIAM J. Optim.

Self Cite

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“…Intuitively, in the case A and B are maximally monotone, one would expect the use of equal resolvent parameters γ = δ, and in other cases, γ and δ are no longer the same. This phenomenon was initially observed in [11,12]. Although the imbalance of monotonicity can be resolved by shifting the identity between the operators as in [11,Remark 4.15], our plan is to conduct the convergence analysis of the algorithm applied to the original operators.…”

Section: The Algorithmmentioning

confidence: 98%

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

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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.

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“…Therefore, every weak sequential cluster point of (x k ) k∈N is contained in Ω γ , and Proposition 2 implies that (x k ) k∈N is weakly convergent to a point x ∈ Ω γ . Then (15) shows that ū = J γA (x) and z = γT (ū) are the unique cluster points of (u k ) k∈N and (z k ) k∈N , respectively, and hence u k ū, v k ū and z k z. Moreover, since x was arbitrarily chosen in Ω γ , (11) and ( 12) also hold with u replaced by ū and x replaced by x.…”

Section: Davis-yin Splitting Algorithmmentioning

confidence: 99%

“…The latter inclusion is equivalent to ū = J γA (x), z = γT (ū) and ū = J γB (2ū − x − z), (15) which implies x ∈ Ω γ . Therefore, every weak sequential cluster point of (x k ) k∈N is contained in Ω γ , and Proposition 2 implies that (x k ) k∈N is weakly convergent to a point x ∈ Ω γ .…”

Section: Davis-yin Splitting Algorithmmentioning

confidence: 99%

A direct proof of convergence of Davis-Yin splitting algorithm allowing larger stepsizes

Artacho¹,

David²

2021

Preprint

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This note is devoted to the splitting algorithm proposed by Davis and Yin in 2017 for computing a zero of the sum of three maximally monotone operators, with one of them being cocoercive. We provide a direct proof that guarantees its convergence when the stepsizes are smaller than four times the cocoercivity constant, thus doubling the size of the interval established by Davis and Yin. As a by-product, the same conclusion applies to the forward-backward splitting algorithm. Further, we use the notion of "strengthening" of a set-valued operator to derive a new splitting algorithm for computing the resolvent of the sum. Last but not least, we provide some numerical experiments illustrating the importance of appropriately choosing the stepsize and relaxation parameters of the algorithms.

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Computing the resolvent of the sum of operators with application to best approximation problems

Cited by 10 publications

References 15 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

A direct proof of convergence of Davis-Yin splitting algorithm allowing larger stepsizes

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