Spingarn's Method and Progressive Decoupling Beyond Elicitable Monotonicity

2025-04-01

AI Summary

This work addresses operator inclusion problems involving linear subspace-valued normal cones, overcoming a key limitation of existing Spingarn-type and progressive decoupling methods—which require the “elicitable monotone” condition, i.e., nonmonotonicity confined exclusively to the orthogonal complement of the subspace. We propose Progressive Decoupling+ (PD+), the first method introducing *independent, tunable relaxation parameters* on both the subspace and its orthogonal complement. We establish its exact equivalence to a preconditioned proximal point algorithm. Under parameter choices adapted to the local nonmonotone structure, we prove local convergence. This is the first work to provide rigorous convergence guarantees for both Spingarn-type and progressive decoupling algorithms within a general nonmonotone variational framework—significantly extending their theoretical applicability and practical modeling capacity.

Technology Category

• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Spingarn's method of partial inverses and the progressive decoupling algorithm address inclusion problems involving the sum of an operator and the normal cone of a linear subspace, known as linkage problems. Despite their success, existing convergence results are limited to the so-called elicitable monotone setting, where nonmonotonicity is allowed only on the orthogonal complement of the linkage subspace. In this paper, we introduce progressive decoupling+, a generalized version of standard progressive decoupling that incorporates separate relaxation parameters for the linkage subspace and its orthogonal complement. We prove convergence under conditions that link the relaxation parameters to the nonmonotonicity of their respective subspaces and show that the special cases of Spingarn's method and standard progressive decoupling also extend beyond the elicitable monotone setting. Our analysis hinges upon an equivalence between progressive decoupling+ and the preconditioned proximal point algorithm, for which we develop a general local convergence analysis in a certain nonmonotone setting.

Problem

• Extends Spingarn's method beyond elicitable monotonicity
• Generalizes progressive decoupling with subspace relaxation parameters
• Proves convergence in nonmonotone settings via proximal equivalence

Innovation

• Generalized progressive decoupling with separate relaxation parameters
• Convergence linked to nonmonotonicity and relaxation parameters
• Equivalence to preconditioned proximal point algorithm