New General Fixed-Point Approach to Compute the Resolvent of Composite Operators

TL;DR

The paper tackles the challenge of computing resolvents of composite operators $C^T\mathcal M C$ and, more broadly, $\mathcal M_1 + C^T\mathcal M_2 C$, by introducing a two-parameter fixed-point framework based on Yosida approximations. It derives a fixed-point representation $J_{\lambda C^T\mathcal M C} y = y - \lambda \mu C^T u$, where $u$ is the fixed point of a nonexpansive map $\mathcal Q$, and shows convergence under the condition $\|I - \lambda\mu CC^T\| \le 1$ (equivalently $\lambda\mu \le 2/\|C\|^2$). The authors establish weak, strong, and linear convergence results, provide practical algorithms (including Krasnoselskii–Mann type iterations), and demonstrate stability advantages over existing methods, particularly when $\|C\|$ is large. The framework is extended to the sum case $\mathcal M_1 + C^T\mathcal M_2 C$, and applications to set-valued Lur'e dynamical systems are discussed, with several open questions on parameter optimization and broader operator classes. Overall, the work offers a robust, flexible approach for resolvent computation with broad implications for optimization, control, and large-scale monotone-operator problems.

Abstract

In this paper, we propose a new general and stable fixed-point approach to compute the resolvents of the composition of a set-valued maximal monotone operator with a linear bounded mapping. Weak, strong and linear convergence of the proposed algorithms are obtained. Advantages of our method over the existing approaches are also thoroughly analyzed.

Theorems & Definitions (23)

• Proposition 1
• Proposition 2
• proof
• Proposition 3
• proof
• Proposition 4
• proof
• Theorem 5
• proof
• Theorem 6