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• W3204117509 abstract "We consider the convex-concave saddle point problem $min_{mathbf{x}}max_{mathbf{y}}Phi(mathbf{x},mathbf{y})$, where the decision variables $mathbf{x}$ and/or $mathbf{y}$ subject to a multi-block structure and affine coupling constraints, and $Phi(mathbf{x},mathbf{y})$ possesses certain separable structure. Although the minimization counterpart of such problem has been widely studied under the topics of ADMM, this minimax problem is rarely investigated. In this paper, a convenient notion of $epsilon$-saddle point is proposed, under which the convergence rate of several proposed algorithms are analyzed. When only one of $mathbf{x}$ and $mathbf{y}$ has multiple blocks and affine constraint, several natural extensions of ADMM are proposed to solve the problem. Depending on the number of blocks and the level of smoothness, $mathcal{O}(1/T)$ or $mathcal{O}(1/sqrt{T})$ convergence rates are derived for our algorithms. When both $mathbf{x}$ and $mathbf{y}$ have multiple blocks and affine constraints, a new algorithm called ExtraGradient Method of Multipliers (EGMM) is proposed. Under desirable smoothness condition, an $mathcal{O}(1/T)$ rate of convergence can be guaranteed regardless of the number of blocks in $mathbf{x}$ and $mathbf{y}$. In depth comparison between EGMM (fully primal-dual method) and ADMM (approximate dual method) is made over the multi-block optimization problems to illustrate the advantage of the EGMM." @default.
• W3204117509 created "2021-10-11" @default.
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• W3204117509 date "2021-09-29" @default.
• W3204117509 modified "2023-09-23" @default.
• W3204117509 title "Primal-Dual First-Order Methods for Affinely Constrained Multi-Block Saddle Point Problems" @default.
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• W3204117509 doi "https://doi.org/10.48550/arxiv.2109.14212" @default.
• W3204117509 hasPublicationYear "2021" @default.
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