Saddle-point problems appear in various settings including machine learning, zero-sum stochastic games, and regression problems. We consider decomposable saddle-point problems and study an extension of the alternating direction method of multipliers to such saddle-point problems. Instead of solving the original saddle-point problem directly, this algorithm solves smaller saddle-point problems by exploiting the decomposable structure. We show the convergence of this algorithm for convex-concave saddle-point problems under a mild assumption. We also provide a sufficient condition for which the assumption holds. We demonstrate the convergence properties of the saddle-point alternating direction method of multipliers with numerical examples on a power allocation problem in communication channels and a network routing problem with adversarial costs.
Recently the primal-dual method of multipliers (PDMM), a novel distributed optimization method, was proposed for solving a general class of decomposable convex optimizations over graphic models. In this work, we first study the convergence properties of PDMM for decomposable quadratic optimizations over tree-structured graphs. We show that with proper parameter selection, PDMM converges to its optimal solution in finite number of iterations. We then apply PDMM for the causal estimation problem over a statistical linear state-space model. We show that PDMM and the Kalman filter have the same up…
In the present paper we propose a novel convergence analysis of the Alternating Direction Methods of Multipliers (ADMM), based on its equivalence with the overrelaxed Primal-Dual Hybrid Gradient (oPDHG) algorithm. We consider the smooth case, which correspond to the cas where the objective function can be decomposed into one differentiable with Lipschitz continuous gradient part and one strongly convex part. An accelerated variant of the ADMM is also proposed, which is shown to converge linearly with same rate as the oPDHG.
Phase retrieval can be expressed as one non-convex constrained optimization problem to identify one phase minimizer in the primal space. Many iterative transform techniques have been proposed to identify the minimizer, e.g., relaxed averaged alternating reflections(RAAR) algorithms. RAAR algorithm is one alternating direction method of multipliers(ADMM) with one penalty parameter. Pairing with multipliers (dual vectors), phase vectors are lifted to higher dimensional vectors, RAAR algorithms actually is one continuation algorithm, which searches for local Nash equilibria in a primal-dual space…
We present a novel framework, namely AADMM, for acceleration of linearized alternating direction method of multipliers (ADMM). The basic idea of AADMM is to incorporate a multi-step acceleration scheme into linearized ADMM. We demonstrate that for solving a class of convex composite optimization with linear constraints, the rate of convergence of AADMM is better than that of linearized ADMM, in terms of their dependence on the Lipschitz constant of the smooth component. Moreover, AADMM is capable to deal with the situation when the feasible region is unbounded, as long as the corresponding sad…
We consider the solution of linear saddle-point problems, using the alternating direction method-of-multipliers (ADMM) as a preconditioner for the generalized minimum residual method (GMRES). We show, using theoretical bounds and empirical results, that ADMM is made remarkably insensitive to the parameter choice with Krylov subspace acceleration. We prove that ADMM-GMRES can consistently converge, irrespective of the exact parameter choice, to an $ε$-accurate solution of a $κ$-conditioned problem in $O(κ^{2/3}\logε^{-1})$ iterations. The accelerated method is applied to randomly generated prob…
In this paper, we present novel randomized algorithms for solving saddle point problems whose dual feasible region is given by the direct product of many convex sets. Our algorithms can achieve an ${\cal O}(1/N)$ and ${\cal O}(1/N^2)$ rate of convergence, respectively, for general bilinear saddle point and smooth bilinear saddle point problems based on a new prima-dual termination criterion, and each iteration of these algorithms needs to solve only one randomly selected dual subproblem. Moreover, these algorithms do not require strongly convex assumptions on the objective function and/or the …
We present distributed subgradient methods for min-max problems with agreement constraints on a subset of the arguments of both the convex and concave parts. Applications include constrained minimization problems where each constraint is a sum of convex functions in the local variables of the agents. In the latter case, the proposed algorithm reduces to primal-dual updates using local subgradients and Laplacian averaging on local copies of the multipliers associated to the global constraints. For the case of general convex-concave saddle-point problems, our analysis establishes the convergence…
In this work, we consider the asynchronous distributed optimization problem in which each node has its own convex cost function and can communicate directly only with its neighbors, as determined by a directed communication topology (directed graph or digraph). First, we reformulate the optimization problem so that Alternating Direction Method of Multipliers (ADMM) can be utilized. Then, we propose an algorithm, herein called Asynchronous Approximate Distributed Alternating Direction Method of Multipliers (AsyAD-ADMM), using finite-time asynchronous approximate ratio consensus, to solve the mu…
We consider the convex-concave saddle point problem $\min_{\mathbf{x}}\max_{\mathbf{y}}Φ(\mathbf{x},\mathbf{y})$, where the decision variables $\mathbf{x}$ and/or $\mathbf{y}$ subject to a multi-block structure and affine coupling constraints, and $Φ(\mathbf{x},\mathbf{y})$ 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 $ε$-saddle point is proposed, under which the convergence rate of several proposed algorithms are a…
In this work, we consider the distributed optimization problem in which each node has its own convex cost function and can communicate directly only with its neighbors, as determined by a directed communication topology (directed graph or digraph). First, we reformulate the optimization problem so that Alternating Direction Method of Multipliers (ADMM) can be utilized. Then, we propose an algorithm, herein called Distributed Alternating Direction Method of Multipliers using Finite-Time Exact Ratio Consensus (D-ADMM-FTERC), to solve the multi-node convex optimization problem, in which every nod…
This paper presents a new primal-dual method for computing an equilibrium of generalized (continuous) Nash game (referred to as generalized Nash equilibrium problem (GNEP)) where each player's feasible strategy set depends on the other players' strategies. The method is based on a new form of Lagrangian function with a quadratic approximation. First, we reformulate a GNEP as a saddle point computation problem using the new Lagrangian and establish equivalence between a saddle point of the Lagrangian and an equilibrium of the GNEP. We then propose a simple algorithm that is convergent to the sa…
This paper considers decentralized consensus optimization problems where different summands of a global objective function are available at nodes of a network that can communicate with neighbors only. The proximal method of multipliers is considered as a powerful tool that relies on proximal primal descent and dual ascent updates on a suitably defined augmented Lagrangian. The structure of the augmented Lagrangian makes this problem non-decomposable, which precludes distributed implementations. This problem is regularly addressed by the use of the alternating direction method of multipliers. T…
The alternating direction method of multipliers (ADM or ADMM) breaks a complex optimization problem into much simpler subproblems. The ADM algorithms are typically short and easy to implement yet exhibit (nearly) state-of-the-art performance for large-scale optimization problems. To apply ADM, we first formulate a given problem into the "ADM-ready" form, so the final algorithm depends on the formulation. A problem like $\mbox{minimize}_\mathbf{x} u(\mathbf{x}) + v(\mathbf{C}\mathbf{x})$ has six different "ADM-ready" formulations. They can be in the primal or dual forms, and they differ by how …
In this paper, we introduce a novel iterative algorithm for the problem of phase-retrieval where the measurements consist of only the magnitude of linear function of the unknown signal, and the noise in the measurements follow Poisson distribution. The proposed algorithm is based on the principle of majorization-minimization (MM); however, the application of MM here is very novel and distinct from the way MM has been usually used to solve optimization problems in the literature. More precisely, we reformulate the original minimization problem into a saddle point problem by invoking Fenchel dua…
