Spingarn's Method and Progressive Decoupling Beyond Elicitable Monotonicity

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