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We consider the convex-concave saddle point problem$\min_{\mathbf{x}}\max_{\mathbf{y}}\Phi(\mathbf{x},\mathbf{y})$, where thedecision variables $\mathbf{x}$ and/or $\mathbf{y}$ subject to a multi-blockstructure and affine coupling constraints, and $\Phi(\mathbf{x},\mathbf{y})$possesses certain separable structure. Although the minimization counterpart ofsuch problem has been widely studied under the topics of ADMM, this minimaxproblem is rarely investigated. In this paper, a convenient notion of$\epsilon$-saddle point is proposed, under which the convergence rate ofseveral proposed algorithms are analyzed. When only one of $\mathbf{x}$ and$\mathbf{y}$ has multiple blocks and affine constraint, several naturalextensions of ADMM are proposed to solve the problem. Depending on the numberof 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 affineconstraints, a new algorithm called ExtraGradient Method of Multipliers (EGMM)is proposed. Under desirable smoothness condition, an $\mathcal{O}(1/T)$ rateof convergence can be guaranteed regardless of the number of blocks in$\mathbf{x}$ and $\mathbf{y}$. In depth comparison between EGMM (fullyprimal-dual method) and ADMM (approximate dual method) is made over themulti-block optimization problems to illustrate the advantage of the EGMM.

Primal-Dual First-Order Methods for Affinely Constrained Multi-Block Saddle Point Problems