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An Inexact Primal-Dual Smoothing Framework for Large-Scale Non-Bilinear Saddle Point Problems

Author

Listed:
  • Le Thi Khanh Hien

    (University of Mons)

  • Renbo Zhao

    (University of Iowa)

  • William B. Haskell

    (Purdue University)

Abstract

We develop an inexact primal-dual first-order smoothing framework to solve a class of non-bilinear saddle point problems with primal strong convexity. Compared with existing methods, our framework yields a significant improvement over the primal oracle complexity, while it has competitive dual oracle complexity. In addition, we consider the situation where the primal-dual coupling term has a large number of component functions. To efficiently handle this situation, we develop a randomized version of our smoothing framework, which allows the primal and dual sub-problems in each iteration to be inexactly solved by randomized algorithms in expectation. The convergence of this framework is analyzed both in expectation and with high probability. In terms of the primal and dual oracle complexities, this framework significantly improves over its deterministic counterpart. As an important application, we adapt both frameworks for solving convex optimization problems with many functional constraints. To obtain an $$\varepsilon $$ ε -optimal and $$\varepsilon $$ ε -feasible solution, both frameworks achieve the best-known oracle complexities.

Suggested Citation

  • Le Thi Khanh Hien & Renbo Zhao & William B. Haskell, 2024. "An Inexact Primal-Dual Smoothing Framework for Large-Scale Non-Bilinear Saddle Point Problems," Journal of Optimization Theory and Applications, Springer, vol. 200(1), pages 34-67, January.
    • Handle: RePEc:spr:joptap:v:200:y:2024:i:1:d:10.1007_s10957-023-02351-9
    DOI: 10.1007/s10957-023-02351-9
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    References listed on IDEAS

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    1. NESTEROV, Yurii, 2013. "Gradient methods for minimizing composite functions," LIDAM Reprints CORE 2510, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. Doikov, Nikita & Nesterov, Yurii, 2023. "Affine-invariant contracting-point methods for Convex Optimization," LIDAM Reprints CORE 3240, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    3. A. Nedić & A. Ozdaglar, 2009. "Subgradient Methods for Saddle-Point Problems," Journal of Optimization Theory and Applications, Springer, vol. 142(1), pages 205-228, July.
    4. Le Thi Khanh Hien & Cuong V. Nguyen & Huan Xu & Canyi Lu & Jiashi Feng, 2019. "Accelerated Randomized Mirror Descent Algorithms for Composite Non-strongly Convex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 181(2), pages 541-566, May.
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