A splitting algorithm for dual monotone inclusions involving cocoercive operators
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A splitting algorithm for dual monotone inclusions involving cocoercive operators `˘ Công Vu˜ Bang
Received: 13 October 2011 / Accepted: 4 November 2011 © Springer Science+Business Media, LLC 2011
Abstract We consider the problem of solving dual monotone inclusions involving sums of composite parallel-sum type operators. A feature of this work is to exploit explicitly the properties of the cocoercive operators appearing in the model. Several splitting algorithms recently proposed in the literature are recovered as special cases. Keywords Cocoercivity · Forward-backward algorithm · Composite operator · Duality · Monotone inclusion · Monotone operator · Operator splitting · Primal-dual algorithm Mathematics Subject Classifications (2010) 47H05 · 49M29 · 49M27 · 90C25
1 Introduction Monotone operator splitting methods have found many applications in applied mathematics, e.g., evolution inclusions [3], partial differential equations [2, 20, 23], mechanics [21], variational inequalities [6, 19], Nash equilibria [8], and various optimization problems [7, 9, 10, 14, 16, 17, 25, 29]. In such formulations,
Communicated by Lixin Shen. This work was supported by the Agence Nationale de la Recherche under grant ANR-08-BLAN-0294-02 and the Vietnam National Foundation for Science and Technology Development. B. C. Vu˜ (B) Laboratoire Jacques-Louis Lions—UMR CNRS 7598, UPMC Université Paris 06, 75005 Paris, France e-mail: [email protected]
B.C. Vu˜
cocoercivity often plays a central role; see for instance [3, 6, 11, 13, 19– 21, 23, 28–30]. Recall that an operator C : H → H is cocoercive with constant β ∈ ]0, +∞[ if its inverse is β-strongly monotone, that is, (∀x ∈ H)(∀y ∈ H)
x − y | Cx − Cy ≥ β Cx − Cy 2 .
(1.1)
In this paper, we revisit a general primal-dual splitting framework proposed in [15] in the presence Lipschitzian operators in the context of cocoercive operators. This will lead to a new type of splitting technique and provide a unifying framework for some algorithms recently proposed in the literature. The problem under investigation is the following, where the parallel sum operation is denoted by (see (2.4)). Problem 1.1 Let H be a real Hilbert space, let z ∈ H, let m mbe a strictly positive integer, let (ωi )1≤i≤m be real numbers in ]0, 1] such that i=1 ωi = 1, let A : H → 2H be maximally monotone, and let C : H → H be μ-cocoercive for some μ ∈ ]0, +∞[. For every i ∈ {1, . . . , m}, let Gi be a real Hilbert space, let ri ∈ Gi , let Bi : Gi → 2Gi be maximally monotone, let Di : Gi → 2Gi be maximally monotone and νi -strongly monotone for some νi ∈ ]0, +∞[, and suppose that Li : H → Gi is a nonzero bounded linear operator. The problem is to solve the primal inclusion find x ∈ H such that z ∈ Ax +
m
ωi Li∗ Bi
Di Li x − ri + Cx,
(1.2)
i=1
together with the dual inclusion
⎧ m ∗ ⎪ ⎨z − i=1 ωi Li v i ∈ Ax + Cx find v 1 ∈ G1 , ..., v m ∈ Gm such that (∃ x ∈ H) (∀i ∈ {1, ..., m}) v i ∈ Bi Di ⎪ ⎩ Li x − ri . (1.3) We denote by P and D the sets of solutions to (1.2) and (1.3), respect
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