The alternating direction method of multipliers (ADMM) has been proved to be effective for solving separableconvex optimization subject to linear constraints. In this paper, we propose a generalized symmetric ADMM (G...
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The alternating direction method of multipliers (ADMM) has been proved to be effective for solving separableconvex optimization subject to linear constraints. In this paper, we propose a generalized symmetric ADMM (GS-ADMM), which updates the Lagrange multiplier twice with suitable stepsizes, to solve the multi-block separable convex programming. This GS-ADMM partitions the data into two group variables so that one group consists of p block variables while the other has q block variables, where and are two integers. The two grouped variables are updated in a Gauss-Seidel scheme, while the variables within each group are updated in a Jacobi scheme, which would make it very attractive for a big data setting. By adding proper proximal terms to the subproblems, we specify the domain of the stepsizes to guarantee that GS-ADMM is globally convergent with a worst-case ergodic convergence rate. It turns out that our convergence domain of the stepsizes is significantly larger than other convergence domains in the literature. Hence, the GS-ADMM is more flexible and attractive on choosing and using larger stepsizes of the dual variable. Besides, two special cases of GS-ADMM, which allows using zero penalty terms, are also discussed and analyzed. Compared with several state-of-the-art methods, preliminary numerical experiments on solving a sparse matrix minimization problem in the statistical learning show that our proposed method is effective and promising.
The augmented Lagrangian method (ALM) is a well-regarded algorithm for solving convex optimization problems with linear constraints. Recently, in He et al. [On full Jacobian decomposition of the augmented Lagrangian m...
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The augmented Lagrangian method (ALM) is a well-regarded algorithm for solving convex optimization problems with linear constraints. Recently, in He et al. [On full Jacobian decomposition of the augmented Lagrangian method for separable convex programming, SIAM J. Optim. 25(4) (2015), pp. 2274-2312], it has been demonstrated that a straightforward Jacobian decomposition of ALM is not necessarily convergent when the objective function is the sum of functions without coupled variables. Then, Wang et al. [A note on augmented Lagrangian-based parallel splitting method, Optim. Lett. 9 (2015), pp. 1199-1212] proved the global convergence of the augmented Lagrangian-based parallel splitting method under the assumption that all objective functions are strongly convex. In this paper, we extend these results and derive the worst-case convergence rate of this method under both ergodic and non-ergodic conditions, where t represents the number of iterations. Furthermore, we show that the convergence rate can be improved from to , and finally, we also demonstrate that this method can achieve global linear convergence, when the involved functions satisfy some additional conditions.
Proximal point algorithm(PPA)is a useful algorithm framework and has good convergence *** difficulty is that the subproblems usually only have iterative *** this paper,we propose an inexact customized PPA framework fo...
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Proximal point algorithm(PPA)is a useful algorithm framework and has good convergence *** difficulty is that the subproblems usually only have iterative *** this paper,we propose an inexact customized PPA framework for twoblock separableconvex optimization problem with linear *** design two types of inexact error criteria for the *** first one is absolutely summable error criterion,under which both subproblems can be solved *** one of the two subproblems is easily solved,we propose another novel error criterion which is easier to implement,namely relative error *** relative error criterion only involves one parameter,which is more *** establish the global convergence and sub-linear convergence rate in ergodic sense for the proposed *** numerical experiments on LASSO regression problems and total variation-based image denoising problem illustrate that our new algorithms outperform the corresponding exact algorithms.
We propose a weighted minimum variance allocation model, denoted by WMVA, which distributes an amount of a divisible resource as fairly as possible while satisfying all demand intervals. We show that the problem WMVA ...
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We propose a weighted minimum variance allocation model, denoted by WMVA, which distributes an amount of a divisible resource as fairly as possible while satisfying all demand intervals. We show that the problem WMVA has a unique optimal solution and it can be characterized by the uniform distribution property (UDP in short). Based on the UDP property, we develop an efficient algorithm. Theoretically, our algorithm has a worst-case O(n2) complexity, but we prove that, subject to slight conditions, the worst case cannot happen on a 64-bit computer when the problem dimension is greater than 129. We provide extensive simulation results to support the argument and it explains why, in practice, our algorithm runs significantly faster than most existing algorithms, including many O(n) algorithms.
In this paper, we propose a modified positive-indefinite proximal linearized ADMM (PIPL-ADMM) with a larger Glowinski's relaxation factor for solving two-block linearly constrained separable convex programming by ...
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In this paper, we propose a modified positive-indefinite proximal linearized ADMM (PIPL-ADMM) with a larger Glowinski's relaxation factor for solving two-block linearly constrained separable convex programming by variational inequality technique. We investigate the internal relationships between the step size coefficient and the penalty coefficient to identify the convergence of PIPL-ADMM. The convergence of PIPL-ADMM and its convergence rate measured by the iteration complexity are established in the ergodic case. Numerical experiments are reported to illustrate the efficiency of the proposed methods.
We consider a dual method for solving non-strictly convex programs possessing a certain separable structure. This method may be viewed as a dual version of a block coordinate ascent method studied by Auslender [1, Sec...
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We consider a dual method for solving non-strictly convex programs possessing a certain separable structure. This method may be viewed as a dual version of a block coordinate ascent method studied by Auslender [1, Section 6]. We show that the decomposition methods of Han [6,7] and the method of multipliers may be viewed as special cases of this method. We also prove a convergence result for this method which can be applied to sharpen the available convergence results for Han's methods.
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