Primal-dual proximal bundle and conditional gradient methods for convex problems

Jiaming Liang

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arXiv · 2412.00585

Primal-dual proximal bundle and conditional gradient methods for convex problems

Jiaming Liang

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Abstract

This paper studies the primal-dual convergence and iteration-complexity of proximal bundle methods for solving nonsmooth problems with convex structures. More specifically, we develop a family of primal-dual proximal bundle methods for solving convex nonsmooth composite optimization problems and establish the iteration-complexity in terms of a primal-dual gap. We also propose a class of proximal bundle methods for solving convex-concave nonsmooth composite saddle-point problems and establish the iteration-complexity to find an approximate saddle-point. This paper places special emphasis on the primal-dual perspective of the proximal bundle method. In particular, we discover an interesting duality between the conditional gradient method and the cutting-plane scheme used within the proximal bundle method. Leveraging this duality, we further develop novel variants of both the conditional gradient method and the cutting-plane scheme. Additionally, we report numerical experiments to demonstrate the effectiveness and efficiency of the proposed proximal bundle methods in comparison with the subgradient method for solving a regularized matrix game.

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BibTeX RIS

Jiaming Liang. 2024-11-30. Primal-dual proximal bundle and conditional gradient methods for convex problems. https://arxiv.org/abs/2412.00585

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