Restarted Reflected Halpern Acceleration for Augmented Primal-Dual Methods

Abstract

We study linearly constrained composite convex optimization with a smooth term and a proximable nonsmooth term. We develop a unified augmented primal-dual framework with primal-dual hybrid gradient-type and augmented Chambolle-Pock-type metric choices, including a fully augmented Chambolle-Pock-type family that retains the augmented quadratic term. The exact scheme admits a degenerate proximal-point form; the linearized scheme admits a preconditioned forward-backward form. These representations allow reflected Halpern acceleration to be analyzed directly in primal-dual variables. For the shadow iterates, we prove convergence to Karush-Kuhn-Tucker (KKT) points and nonergodic O(1/k) bounds for the KKT residual and objective gap, with a scalar worst-case example. We show that finite identification belongs to the shadow sequence rather than to the anchored Halpern state. After identification, an affine-face model yields an exact reduced residual identity and a local-sharpness criterion. Finally, we prove linear convergence of restart anchors under fixed-point sharpness on the visited restart set, with local or tail convergence when sharpness follows from local error bounds. Experiments on linear and convex quadratic programs illustrate augmentation and linearization.