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.
From: Benqi Liu [view email]
[v1]
Mon, 15 Jun 2026 10:55:59 UTC (82 KB)
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