In this paper,we propose a new stopping criterion for Eckstein and Bertsekas’s generalized alternating direction method of *** stopping criterion is easy to verify,and the computational cost is much less than the cla...
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In this paper,we propose a new stopping criterion for Eckstein and Bertsekas’s generalized alternating direction method of *** stopping criterion is easy to verify,and the computational cost is much less than the classical stopping criterion in the highly influential paper by Boyd et al.(Found Trends Mach Learn 3(1):1–122,2011).
In this paper anew class of proximal-like algorithms for solving monotone inclusions of the form T(x) There Exists 0 is derived. It is obtained by applying linear multi-step methods (LMM) of numerical integration in o...
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In this paper anew class of proximal-like algorithms for solving monotone inclusions of the form T(x) There Exists 0 is derived. It is obtained by applying linear multi-step methods (LMM) of numerical integration in order to solve the differential inclusion (x) over dot (t) is an element of -T(x(t)), which can be viewed as a generalization of the steepest decent method for a convex function. It is proved that under suitable conditions on the parameters of the LMM, the generated sequence converges weakly to a point in the solution set T-1 (0). The LMM is very similar to the classical proximal point algorithm in that both are based on approximately evaluating the resolvants of T. Consequently, LMM can be used to derive multi-step versions of many of the optimization methods based on the classical proximal point algorithm. The convergence analysis allows errors in the computation of the iterates, and two different error criteria are analyzed, namely, the classical scheme with summable errors, and a recently proposed more constructive criterion.
The problems studied in this paper are a class of monotone constrained variational inequalities VI (S, f) in which S is a convex set with some linear constraints. By introducing Lagrangian multipliers to the linear co...
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The problems studied in this paper are a class of monotone constrained variational inequalities VI (S, f) in which S is a convex set with some linear constraints. By introducing Lagrangian multipliers to the linear constraints, such problems can be solved by some projection type prediction-correction methods. We focus on the mapping f that does not have an explicit form. Therefore, only its function values can be employed in the numerical methods. The number of iterations is significantly dependent on a parameter that balances the primal and dual variables. To overcome potential difficulties, we present a self-adaptive prediction-correction method that adjusts the scalar parameter automatically. Convergence of the proposed method is proved under mild conditions. Preliminary numerical experiments including some traffic equilibrium problems indicate the effectiveness of the proposed methods.
Keywords proximal point algorithm - variational inequality - prediction-correction
MSC 65K10 - 90C25 - 90C30 - 90C90
We study the noncommutative rank (nc-rank) computation of a symbolic matrix whose entries are linear forms in noncommutative variables. For this problem, polynomial time algorithms were given by Garg, Gurvits, Oliveir...
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We study the noncommutative rank (nc-rank) computation of a symbolic matrix whose entries are linear forms in noncommutative variables. For this problem, polynomial time algorithms were given by Garg, Gurvits, Oliveira, and Wigderson over the rational numbers, and by Ivanyos, Qiao, and Subrahmanyam over arbitrary fields. We present a significantly different polynomial time algorithm that works for any field. Our algorithm is based on a combination of submodular optimization on modular lattices and convex optimization on CAT(0) spaces.
We study a class of vector optimization problems with a C-convex objective function under linear constraints. We extend the proximal point algorithm used in scalar optimization to vector optimization. We analyze both ...
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We study a class of vector optimization problems with a C-convex objective function under linear constraints. We extend the proximal point algorithm used in scalar optimization to vector optimization. We analyze both the global and local convergence results for the new algorithm. We then apply the proximal point algorithm to a supply chain network risk management problem under bi-criteria considerations. (C) 2015 Elsevier B.V. and Association of European Operational Research Societies (EURO) within the International Federation of Operational Research Societies (IFORS). All rights reserved.
The purpose of this paper is to introduce a new iterative method that is the combination of the proximal point algorithm, viscosity approximation method and alternating resolvent method for finding the common zeros of...
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The purpose of this paper is to introduce a new iterative method that is the combination of the proximal point algorithm, viscosity approximation method and alternating resolvent method for finding the common zeros of two accretive operators in Banach spaces. And we will prove the strong convergence theorems for the iterative algorithms and give the example of the main theorems. The results of this paper are improvements and extensions of the corresponding ones announced by many others. (c) 2016 Elsevier Inc. All rights reserved.
We give a new algorithm for solving the Fermat-Weber location problem involving mixed gauges. This algorithm, which is derived from the partial inverse method developed by J.E. Spingarn, simultaneously generates two s...
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We give a new algorithm for solving the Fermat-Weber location problem involving mixed gauges. This algorithm, which is derived from the partial inverse method developed by J.E. Spingarn, simultaneously generates two sequences globally converging to a primal and a dual solution respectively. In addition, the updating formulae are very simple; a stopping rule can be defined though the method is not dual feasible and the entire set of optimal locations can be obtained from the dual solution by making use of optimality conditions.
A discussion on weak and strong approximating fixed points was presented. The nonlinear ergodic theorems of Baillon's type were analyzed. Weak and strong convergence theorems of Mann's type and Halpern's t...
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A discussion on weak and strong approximating fixed points was presented. The nonlinear ergodic theorems of Baillon's type were analyzed. Weak and strong convergence theorems of Mann's type and Halpern's type in Banach spaces were studied. Iterative methods for approximation of common fixed points for families of nonexpansive mappings were established. The feasibility problem by convex combinations of nonexpansive retractions and convex minimization problem of finding a minimizer of a convex function were considered using the results.
In the matrix completion problem, most methods to solve the nuclear norm model are relaxing it to the nuclear norm regularized least squares problem. In this paper, we propose a new unconstraint model for matrix compl...
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In the matrix completion problem, most methods to solve the nuclear norm model are relaxing it to the nuclear norm regularized least squares problem. In this paper, we propose a new unconstraint model for matrix completion problem based on nuclear norm and indicator function and design a proximal point algorithm (PPA-IF) to solve it. Then the convergence of our algorithm is established strictly. Finally, we report numerical results for solving noiseless and noisy matrix completion problems and image reconstruction.
The paper concerns with an inertial-like algorithm for approximating solutions of equilibrium problems in Hilbert spaces. The algorithm is a combination around the relaxed proximalpoint method, inertial effect and th...
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The paper concerns with an inertial-like algorithm for approximating solutions of equilibrium problems in Hilbert spaces. The algorithm is a combination around the relaxed proximalpoint method, inertial effect and the Krasnoselski-Mann iteration. The using of the proximalpoint method with relaxations has allowed us a more flexibility in practical computations. The inertial extrapolation term incorporated in the resulting algorithm is intended to speed up convergence properties. The main convergence result is established under mild conditions imposed on bifunctions and control parameters. Several numerical examples are implemented to support the established convergence result and also to show the computational advantage of our proposed algorithm over other well known algorithms.
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