Curvature Recycling Douglas-Rachford Splitting: Transported Quasi-Newton Models for Expensive Smooth Proximal Subproblems

Abstract

We consider \[ \min_x f(x)+g(x), \] where $f$ is smooth but its value and gradient are expensive, while $g$ is nonsmooth and has a cheap proximal map. Douglas--Rachford splitting (DRS) then requires a sequence of smooth proximal solves. These solves have different centers but share the same nonlinear curvature. We introduce curvature-recycling DRS (CR-DRS). The method transports an old exact residual to the new proximal center, reuses a quasi-Newton model, and moves the old proximal state before the first new gradient is evaluated. We give paired direct and inverse BFGS and safeguarded symmetric-rank-one realizations. A certified variant accepts a transported quasi-Newton step only when it passes descent and residual tests; otherwise it uses a fixed number of safe gradient steps. Under strong convexity and smoothness, a small-gain condition gives global linear convergence with a constant number of new gradients per outer iteration. A transported Dennis--Moré condition gives local superlinear reduction of the new proximal error. We show that this condition is not automatic, derive full-model and active-subspace sufficient conditions, and relate the active subspace to the local DRS dynamics induced by the nonsmooth proximal map. Experiments on $\ell_1$- and total-variation-regularized problems show large reductions in expensive-gradient calls relative to restarted and curvature-only quasi-Newton proximal solves. For directly proximal $\ell_1$ models, we also compare with FISTA.