[Paper Review] A new splitting method for solving composite monotone inclusions involving parallel-sum operators

Quoc Tran-Dinh, Bằng Công Vũ|arXiv (Cornell University)|May 29, 2015

Optimization and Variational Analysis24 references4 citations

TL;DR

This paper introduces a novel primal-dual splitting method for solving composite monotone inclusions involving Lipschitzian and parallel-sum-type monotone operators. By leveraging the Kuhn-Tucker set and separately computing resolvents of each monotone operator, the method ensures weak convergence to a solution, offering a more flexible and modular approach than existing methods, particularly for composite convex minimization via independent proximal operators.

ABSTRACT

We propose a new primal-dual splitting method for solving composite inclusions involving Lipschitzian, and parallel-sum-type monotone operators. Our approach extends the framework in \cite{Siopt4} to a more general class of monotone inclusions in a nontrivial fashion. The main idea is to represent the solution set of both the primal and dual problems using their associated Kuhn-Tucker set, and then develop a projected method to successively approximate a feasible point of the Kuhn-Tucker set. We propose a splitting algorithm based on the resolvent of each maximally monotone operator to construct a primal-dual sequence that weakly converges to a solution of the original problem. The key feature of our method is that it only employes the resolvent of each monotone operator separately, which is different from existing methods in the literature. As a byproduct, our algorithm can be specialized to solve composite convex minimization problems that uses the proximal-operator of each objective component independently, and is equipped with a weakly convergence guarantee.

Motivation & Objective

• To address composite monotone inclusions involving parallel-sum-type monotone operators, which generalize many structured monotone inclusion problems.
• To extend existing primal-dual methods to a broader class of monotone inclusions beyond the scope of prior frameworks.
• To develop a method that only requires the resolvent of each individual monotone operator, avoiding complex joint resolvent computations.
• To ensure weak convergence of the primal-dual sequence to a solution of the original problem.
• To provide a flexible algorithmic framework applicable to composite convex minimization via independent proximal operators.

Proposed method

• The method represents the solution set of both primal and dual problems via the Kuhn-Tucker set, which encapsulates the optimality conditions.
• A projected iterative scheme is developed to successively approximate a feasible point within the Kuhn-Tucker set.
• The algorithm constructs a primal-dual sequence using the resolvent of each maximally monotone operator separately, avoiding joint resolvent computation.
• Weak convergence of the sequence to a solution is established through the properties of the Kuhn-Tucker set and the projection mechanism.
• The approach is specialized to composite convex minimization by treating each objective component’s proximal operator independently.
• The method ensures convergence by relying solely on the resolvent of each operator, enhancing modularity and computational feasibility.

Experimental results

Research questions

• RQ1 How can primal-dual splitting methods be extended to handle composite monotone inclusions involving parallel-sum-type monotone operators?
• RQ2 Can a splitting method be designed that only requires the resolvent of each individual monotone operator, without requiring joint resolvent computation?
• RQ3 What is the convergence behavior of such a method when applied to composite convex minimization problems with separable objectives?
• RQ4 How does the Kuhn-Tucker set framework enable a unified treatment of both primal and dual feasibility in this context?
• RQ5 What conditions ensure weak convergence of the primal-dual sequence under this new splitting strategy?

Key findings

• The proposed method achieves weak convergence of the primal-dual sequence to a solution of the original composite monotone inclusion problem.
• The algorithm only requires the resolvent of each individual monotone operator, enabling modular and efficient implementation.
• The method generalizes prior frameworks by accommodating a broader class of monotone inclusions, including those with parallel-sum operators.
• The approach can be specialized to solve composite convex minimization problems using independent proximal operators of each objective component.
• The use of the Kuhn-Tucker set as a solution representation allows for a unified and structured convergence analysis.
• The method provides a new convergence guarantee under weaker assumptions than some existing approaches, particularly in the context of separable and structured monotone inclusions.