A splitting algorithm for dual monotone inclusions involving cocoercive operators (1110.1697v1)

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 cocoercivity of some of the operators appearing in the model. Several splitting algorithms recently proposed in the literature are recovered as special cases.

• The paper presents a novel primal-dual splitting algorithm that leverages cocoercivity to solve dual monotone inclusions.
• It establishes convergence results under strong positivity and uniform monotonicity conditions using variational inequality principles.
• The framework enhances stability in convex minimization and optimization, paving the way for advanced computational applications.

A Splitting Algorithm for Dual Monotone Inclusions Involving Cocoercive Operators

The paper at hand proposes a novel algorithmic framework for solving dual monotone inclusions that involve sums of composite, parallel-sum type operators. This research focuses primarily on the exploitation of cocoercivity within these operators, which provides a unifying basis for previously proposed splitting algorithms. The paper provides both theoretical results and implications, offering a comprehensive approach to operator splitting which is pivotal for various applications in applied mathematics, such as optimization problems and partial differential equations.

Algorithmic Framework

The main contribution of the paper is the presentation of a primal-dual splitting framework that is grounded on cocoercive operators. This framework extends previous studies by providing a generalized splitting method that inherently manages composite operators and cocoercivity. Specifically, the algorithm addresses a problem characterized by primal-dual inclusions under the presence of maximally monotone operators and cocoercive maps. The primal problem involves finding a solution within a Hilbert space, denoted by $H$, and the associated dual problem requires an analogous search across several Hilbert spaces $G_i$.

The resulting splitting technique incorporates novel elements of variational inequality theory—such as the use of a cocoercive operator $C$—combined with a collection of Lipschitzian operators, which are accommodated through the parameterized parallel-sum operation.

Numerical Results and Theoretical Implications

The paper supports its theoretical claims through numerical results that demonstrate convergence under assumed conditions, such as strong positivity and cocoercivity. The algorithm is rigorously tested using a derived inequality framework that guarantees convergence to a solution over iterative steps. The implications of such findings bolster solver stability in complex systems involving non-linearities and highlight a robust expansion over classical methods, including the forward-backward and other primal-dual algorithms.

Key technical achievements include:

• The identification of sufficient conditions for convergence based on strong positivity and cocoercivity measures.
• Statements of convergence in both weak and strong senses, contingent on uniform monotinicity conditions within convex functions.

Practical Applications

The paper also highlights practical applications of the proposed algorithm in optimization problems, notably enhancing the efficiency of primal-dual strategies in convex minimization problems. This aspect extends the utility of cocoercivity across optimization disciplines, enabling better structured and convergent solutions for real-world problems.

Speculations on Future Work

This research stands as a testament to the growing importance of sophisticated operator splitting algorithms in computational mathematics. The theoretical implications of this paper propose potential for further exploration into primal-dual frameworks and enhance existing algorithmic strategies for complex variational problems.

Future developments could explore the integration of such splitting methods into machine learning frameworks, potentially facilitating advanced training methods for neural networks and improving convergence rates in non-convex domains. Additionally, adapting the algorithm to accommodate more general distributions of cocoercivity and examining broader classes of monotonicity could further generalize the applicability of the framework.

In conclusion, this paper presents a significant advancement in the field of algorithmic solutions for dual monotone inclusions, emphasizing the critical role of cocoercivity in obtaining effective convergence processes across various computationally intensive tasks.