Source-linked AI summary
A splitting algorithm for dual monotone inclusions involving cocoercive operators
Bang Cong Vu
TL;DR
The paper addresses dual monotone inclusions involving sums of composite parallel-sum operators, emphasizing the role of cocoercivity. It introduces and analyzes a forward-backward-structured splitting algorithm within a Hilbert-space framework. The resulting sequence converges weakly under the stated conditions, and the framework recovers several existing splitting algorithms and applies to minimization problems.
Problem
The paper studies dual monotone inclusions involving sums of composite parallel-sum operators while explicitly accounting for cocoercive components.
Method
The paper introduces a primal-dual splitting algorithm for the stated monotone inclusion model and analyzes it through forward-backward splitting in a product Hilbert space.
Results
The generated sequence converges weakly under the theorem’s assumptions, with the forward operator shown to be (βρ)-cocoercive when 2βρ > 1.
Takeaways & Limitations
The framework recovers several recently proposed splitting algorithms as special cases and provides an application to minimization problems.
Takeaways & Limitations
The model is restricted to maximally monotone A and Bi, νi-strongly monotone Di, μ-cocoercive C, and nonzero bounded linear Li under the stated assumptions.
Abstract
from arXiv · showhide
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 cocoercivity of some of the operators appearing in the model. Several splitting algorithms recently proposed in the literature are recovered as special cases.
1 Introduction
The paper studies a general primal-dual framework for dual monotone inclusions with composite parallel-sum operators, explicitly exploiting cocoercivity to develop a unifying splitting approach. Problem 1.1 specifies the primal and dual solution sets and encompasses several established duality and splitting settings.
- Contribution: The framework revisits a primal-dual splitting method for cocoercive operators and yields a new splitting technique.It is presented as a unifying framework for algorithms previously proposed in the literature.
- Problem formulation: Problem 1.1 combines maximally monotone operators, strongly monotone operators, cocoercive operators, and bounded linear couplings.The model uses A and Bi as maximally monotone, Di as νi-strongly monotone, C as μ-cocoercive, and Li as nonzero bounded linear operators.
- Problem formulation: The formulation includes a primal inclusion and a corresponding dual inclusion, with P and D denoting their respective solution sets.The paper explicitly introduces both primal and dual problems within the same framework.
- Related settings: The framework contains classical Fenchel-Rockafellar and Mosco duality settings, as well as the setting of.Further special cases are identified through the framework studied in.
- Organization: The paper organizes its development around notation, algorithmic convergence, comparison with existing work, and minimization applications.Section 3 presents and analyzes the algorithm, while Section 4 connects it to minimization problems and state-of-the-art methods.
2 Notation and background
This section establishes Hilbert-space notation and core concepts from convex analysis and monotone operator theory. It defines functions, conjugates, subdifferentials, proximity operators, infimal convolution, resolvents, monotonicity, strong monotonicity, and parallel sums.
- Notation: The paper works in real Hilbert spaces H, G, and Gi, using inner products, norms, weak convergence ⇀, strong convergence →, and bounded linear operators.The notation also introduces domains, graphs, and ranges for set-valued operators.
- Monotone operator theory: It defines resolvents, monotone and maximally monotone operators, uniform and strong monotonicity, and the parallel sum of set-valued operators.These notions provide the operator-theoretic vocabulary for the paper’s splitting framework.
- Convex analysis: The class Γ0(H) consists of lower semicontinuous convex functions with nonempty effective domain.The supplied definition characterizes the function class used in the convex-analytic background.
- Convex analysis: For f ∈ Γ0(H), the conjugate f* and subdifferential are defined as standard convex-analytic objects, and the proximity operator is introduced.These constructions connect convex functions with maximally monotone operators.
- Convex analysis: The section introduces infimal convolution and relative interior for convex sets.Infimal convolution combines two extended-real-valued functions, while ri S denotes the relative interior of a convex subset S.
3 Algorithm and convergence
The paper introduces a cocoercivity-aware primal-dual splitting algorithm for dual monotone inclusions and proves its convergence under stated positivity and cocoercivity conditions. The framework recovers or connects with several existing splitting methods and yields convergence results for minimization applications.
- The main theorem introduces a splitting algorithm for the stated dual monotone inclusion and establishes its convergence.
- Under additional uniform monotonicity assumptions on C or one component operator, the theorem provides stronger convergence conclusions for the corresponding iterates.The supplied passages state uniform monotonicity conditions and convergence of the relevant component sequence.
- The transformed operator B is (βρ)-cocoercive, and the condition 2βρ > 1 enables weak convergence of the generated sequence.
- Special cases recover standard forward-backward splitting and connect the framework with other primal-dual and non-primal-dual algorithms.The paper also notes that alternative existing algorithms may have different structures while solving related inclusions.
4 Application to minimization problems
The paper applies its splitting algorithm to convex minimization problems with a smooth term and strongly convex component functions. The resulting iterations converge weakly under cocoercivity-based conditions, with stronger convergence in specified uniformly convex cases and recovery of existing methods as special cases.
- Problem formulation: The minimization model uses a convex lower-semicontinuous function f, a differentiable convex function h with µ^-1-Lipschitzian gradient, and strongly convex functions ℓi composed with bounded linear operators Li.The formulation also includes weights ωi summing to one, offsets ri, and convex functions gi on auxiliary Hilbert spaces.
- Convergence: Under 2ρ min{µ, ν1, . . . , νm} > 1, with ρ defined by the application’s operator parameters, the generated primal sequence xn converges weakly.The algorithm permits relaxation parameters λn in [ε, 1] and absolutely summable error sequences.
- Operator correspondence: The application identifies ∇h as µ-cocoercive and ∂ℓi as νi-strongly monotone, allowing Theorem 3.1(i) to instantiate the general splitting framework.The correspondence sets A = ∂f and JτA = proxτf, while the dual operators are obtained from the strongly convex ℓi.
- Convergence: The associated dual sequence (v1,n, . . . , vm,n) also converges weakly to a dual solution, while uniform convexity yields stronger convergence for the corresponding component or primal sequence.The text states separate strengthened conclusions when h or one of the relevant functions is uniformly convex.