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

Abstract

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...

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

“…Since A is maximally α-monotone,Ā is maximally (θα+σ)-monotone. Now, since 1+γ(θα+σ) > 0, [8,Proposition 3.4] implies the conclusion.…”

“…It is straightforward to see that A σ is Lipschitz continuous with constant (θℓ + |σ|). The conclusion follows from[8, Theorem 4.8] with λ = µ = 2 and δ = γ. Some remarks regarding Theorem 3.2 are in order.…”

“…On the other hand, it follows from, e.g., [8,Lemma 5.3] that f + g is (α + β)-convex. Since 1 + ω(α + β) > 0, we learn from [8,Lemma 5.2] that Prox ω(f +g) = J ω ∂(f +g) is single-valued and has full domain. Combining with (36) implies (34).…”

“…Proof. By assumption, [8,Lemma 5.2] implies that ∂f and ∂g are respectively maximally α-and β-monotone, and that…”

Survey: Sixty Years of Douglas–rachford

“…Recently, Dao and Phan [68] have introduced what they call an adaptive DR splitting algorithm in the context where one operator is strongly monotone and the other weakly monotone.…”