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An accelerated minimax algorithm for convex-concave saddle point problems with nonsmooth coupling function

Author

Listed:
  • Radu Ioan Boţ

    (University of Vienna
    University of Vienna)

  • Ernö Robert Csetnek

    (University of Vienna)

  • Michael Sedlmayer

    (University of Vienna)

Abstract

In this work we aim to solve a convex-concave saddle point problem, where the convex-concave coupling function is smooth in one variable and nonsmooth in the other and not assumed to be linear in either. The problem is augmented by a nonsmooth regulariser in the smooth component. We propose and investigate a novel algorithm under the name of OGAProx, consisting of an optimistic gradient ascent step in the smooth variable coupled with a proximal step of the regulariser, and which is alternated with a proximal step in the nonsmooth component of the coupling function. We consider the situations convex-concave, convex-strongly concave and strongly convex-strongly concave related to the saddle point problem under investigation. Regarding iterates we obtain (weak) convergence, a convergence rate of order $$\mathcal {O}(\frac{1}{K})$$ O ( 1 K ) and linear convergence like $$\mathcal {O}(\theta ^{K})$$ O ( θ K ) with $$\theta

Suggested Citation

  • Radu Ioan Boţ & Ernö Robert Csetnek & Michael Sedlmayer, 2023. "An accelerated minimax algorithm for convex-concave saddle point problems with nonsmooth coupling function," Computational Optimization and Applications, Springer, vol. 86(3), pages 925-966, December.
    • Handle: RePEc:spr:coopap:v:86:y:2023:i:3:d:10.1007_s10589-022-00378-8
    DOI: 10.1007/s10589-022-00378-8
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