Convergence Rate of a Penalty Method for Strongly Convex Problems with Linear Constraints

Angelia Nedich; Tatiana Tatarenko

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arXiv · 2004.13417

Convergence Rate of a Penalty Method for Strongly Convex Problems with Linear Constraints

Angelia Nedich·Tatiana Tatarenko

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Abstract

We consider an optimization problem with strongly convex objective and linear inequalities constraints. To be able to deal with a large number of constraints we provide a penalty reformulation of the problem. As penalty functions we use a version of the one-sided Huber losses. The smoothness properties of these functions allow us to choose time-varying penalty parameters in such a way that the incremental procedure with the diminishing step-size converges to the exact solution with the rate $O(1/{\sqrt k})$. To the best of our knowledge, we present the first result on the convergence rate for the penalty-based gradient method, in which the penalty parameters vary with time.

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Angelia Nedich, Tatiana Tatarenko. 2020-04-28. Convergence Rate of a Penalty Method for Strongly Convex Problems with Linear Constraints. https://arxiv.org/abs/2004.13417

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