Spingarn's Method and Progressive Decoupling Beyond Elicitable Monotonicity

TL;DR

The paper addresses linkage problems arising from solving inclusions that combine a normal cone to a subspace with a set-valued operator. It introduces progressive decoupling+ as a three-parameter generalization of Spingarn's method, cast as a relaxed PPPA with two matrix relaxations, and proves local convergence under a semimonotonicity framework expressed via oblique weak Minty solutions. By establishing a formal equivalence to PPPA and extending local convergence analysis to nonmonotone settings, the authors recover and extend convergence results for Spingarn's method, relaxed Douglas–Rachford, and standard progressive decoupling, while widening admissible nonmonotonicity through the parameters $\mu$ and $\rho$. The framework provides a unified algorithmic approach for solving structured linkage problems beyond elicitable monotone regimes and offers practical guidance on parameter choices, with potential impact on nonconvex optimization and multistage stochastic programs. The work also lays out a path for extending similar convergence analysis to other splitting schemes under local semimonotonicity notions.

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.

Figures (2)

• Figure 1: Illustration of the subsets of $\R^n$ involved in \ref{['ass:PPPA-local']}.
• Figure 2: Different methods applied to \ref{['ex:linear-system']}, starting from the pair $x_0 = (-2,-2,-2,-2) \in X$ and $y_0 = (1,1,-1,-1) \in X^\bot$. (left) \ref{['eq:progdec']} with $\gamma = \lambda_x = \lambda_y = 1$, i.e., Spingarn's method of partial inverses. In this setting, both the sequence $\seq{\lVert x^k - x^\star\rVert^2+ \lVert y^k - y^\star\rVert^2}$, representing the distance to the solution, and the sequence $\seq{\lVert\bar{x}^k - x^k\rVert^2 + \lVert\bar{y}^k - y^k\rVert^2}$ do not converge to zero. (middle and right) \ref{['eq:progdec']} with $\gamma = 10/9$, $\lambda_x = 9/5(1-\gamma/2) = 4/5$ and $\lambda_y = 9/5(1-1/\gamma) = 9/50$, which complies with the stepsize conditions from \ref{['it:ex:linear-system:3']}. As indicated by the convergence results from \ref{['it:progdec-local:v', 'it:progdec-local:rate']}, the sequences $\gamma\lambda_x^{-1}\lVert x^k - x^\star\rVert^2 + \gamma^{-1}\lambda_y^{-1}\lVert y^k - y^\star\rVert^2$ and $\gamma\lambda_x\lVert\bar{x}^k - x^k\rVert^2+\gamma^{-1}\lambda_y\lVert\bar{y}^k - y^k\rVert^2$ are indeed nonincreasing and converge to zero. Note that although the sequences $\seq{\lVert x^k - x^\star\rVert^2+ \lVert y^k - y^\star\rVert^2}$ and $\seq{\lVert\bar{x}^k - x^k\rVert^2 + \lVert\bar{y}^k - y^k\rVert^2}$ also converge to zero, they are not nonincreasing.

Theorems & Definitions (9)

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