Perturbed Fenchel Duality and Primal-Dual Convergence of First-Order Methods

Tiantian Zhao

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arXiv · 2411.09503

Perturbed Fenchel Duality and Primal-Dual Convergence of First-Order Methods

Tiantian Zhao

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Abstract

It has been shown that many first-order methods satisfy the perturbed Fenchel duality inequality, which yields a unified derivation of convergence. More first-order methods are discussed in this paper, e.g., dual averaging and bundle method. We show primal-dual convergence of them on convex optimization by proving the perturbed Fenchel duality property. We also propose a single-cut bundle method for saddle problem, and prove its convergence in a similar manner.

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Tiantian Zhao. 2024-11-14. Perturbed Fenchel Duality and Primal-Dual Convergence of First-Order Methods. https://arxiv.org/abs/2411.09503

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