Accuracy Certificates for Convex Minimization with Inexact Oracle

Egor Gladin (), Alexander Gasnikov () and Pavel Dvurechensky ()
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Egor Gladin: HSE University
Alexander Gasnikov: Innopolis University
Pavel Dvurechensky: Weierstrass Institute for Applied Analysis and Stochastics

Journal of Optimization Theory and Applications, 2025, vol. 204, issue 1, No 1, 23 pages

Abstract: Abstract Accuracy certificates for convex minimization problems allow for online verification of the accuracy of approximate solutions and provide a theoretically valid online stopping criterion. When solving the Lagrange dual problem, accuracy certificates produce a simple way to recover an approximate primal solution and estimate its accuracy. In this paper, we generalize accuracy certificates for the setting of an inexact first-order oracle, including the setting of primal and Lagrange dual pair of problems. We further propose an explicit way to construct accuracy certificates for a large class of cutting plane methods based on polytopes. As a by-product, we show that the considered cutting plane methods can be efficiently used with a noisy oracle even though they were originally designed to be equipped with an exact oracle. Finally, we illustrate the work of the proposed certificates in the numerical experiments highlighting that our certificates provide a tight upper bound on the objective residual.

Keywords: Cutting plane methods; Inexact subgradient; Accuracy certificate; Primal-dual algorithms; Convex optimization; 90C25; 90C30; 68Q25; 65K05; 65Y20 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s10957-024-02599-9

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