Spingarn's Method and Progressive Decoupling Beyond Elicitable Monotonicity
This work addresses a theoretical gap in optimization algorithms for inclusion problems, offering incremental improvements for researchers in mathematical optimization.
The paper tackles the limitation of existing convergence results for Spingarn's method and progressive decoupling, which are restricted to the elicitable monotone setting, by introducing progressive decoupling+ with separate relaxation parameters for subspaces. It proves convergence under conditions linking these parameters to nonmonotonicity and shows extensions for the special cases.
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.