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Adaptive Parallel primal-dual method for Saddle Point Problems

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Numerical Mathematics(Theory,methods and Applications) 2018年第1期11卷 187-210页

作者： Xiayang Zhang Department of Mathematics Nanjing UniversityNanjing210093China

The primal-dual hybrid gradient method is a classic way to tackle saddle-point ***,its convergence is not guaranteed in *** restric-tions on the step size parameters,e.g.,τσ≤1/||A^(T)A||,are imposed to guarantee the *** this paper,a new convergent method with no restriction on parame-ters is *** the expensive calculation of ||A^(T)A|| is *** method produces a predictor like other primal-dual methods but in a parallel fashion,which has the potential to speed up the *** new iterate is then updated by a sim-ple correction to guarantee the ***,the parameters are adjusted dynamically to enhance the efficiency as well as the robustness of the *** generated sequence monotonically converges to the solution set.A worst-case O(1/t)convergence rate in ergodic sense is also established under mild *** nu-merical efficiency of the proposed method is verified by applications in LASSO problem and Steiner tree problem.

关键词： Adaptive Parallel primal-dual method Saddle-point problem LASSO

Weighted Variational Minimization Model for Wavelet Domain Inpainting with primal-dual method

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Journal of Donghua University(English Edition) 2014年第4期31卷 458-462页

作者：许建楼郝岩郝彬彬张凤云 School of Mathematics and Statistics Henan University of Science and Technology School of Science China University of Petroleum Qingdao School of Information Engineering Shandong Youth University of Political Science

To preserve the edges and details of the image,a new variational model for wavelet domain inpainting was proposed which contained a non-convex regularizer. The non-convex regularizer can utilize the local information of image and perform better than those usual convex ones. In addition, to solve the non-convex minimization problem,an iterative reweighted method and a primaldual method were designed. The numerical experiments show that the new model not only gets better visual effects but also obtains higher signal to noise ratio than the recent method.

关键词： total variation wavelet inpainting primal-dual method

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Split method of Multipliers and Its Application to Parallel and Distributed Logistic Regression

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Chinese Journal of Electronics 2014年第2期23卷 305-310页

作者： FENG Hui TANG Xianding YANG Tao HU Bo Department of Electronic Engineering Fudan University

We consider the scenario where two variables need to be optimized simultaneously. The minimization over one variable has an analytical solution, while it is intractable for the other. Under the Lagrangian dual framework, we propose two iterative optimization algorithms, which make partial minimization and gradient descent alternatingly over two variables. The first algorithm asserts that the iteration result converges to a KKT point under proper stepsize rules, which only needs the augmented Lagrangian function to be convex over partial variable. The second algorithm provides the local attraction property around the KKT point. Our algorithms provide a general solution to parallel and distributed optimization with summable objective functions. Simulation results on parallel and distributed logistic regression classification are present, which show faster convergence rate with less computational complexity compared with other methods.

关键词： method of multiplier primal-dual method Parallel optimization Distributed optimization Logistic regression

A Modified and Simplified Full Nesterov–Todd Step O(N)Infeasible Interior-Point method for Second-Order Cone Optimization

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Journal of the Operations Research Society of China 2018年第2期6卷 301-315页

作者： Behrouz Kheirfam Department of Applied Mathematics Azarbaijan Shahid Madani UniversityTabrizIran

We present a modified and simplified version of an infeasible interior-point method for second-order cone optimization published in 2013(Zangiabadi et *** J Optim Theory Appl,2013).In the earlier version,each iteration consisted of one socalled feasibility step and a few centering ***,each iteration consists of only a feasibility ***,the new algorithm improves the number of iterations and the improvement is due to a lemma which gives an upper bound for the proximity after the feasibility *** complexity result coincides with the best-known iteration bound for infeasible interior-point methods.

关键词： Second-order cone optimization Infeasible interior-point method primal-dual method Polynomial complexity

Distributed solver for linear matrix inequalities: an optimization perspective

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Control Theory and Technology 2021年第4期19卷 507-515页

作者： Weijian Li Wen Deng Xianlin Zeng Yiguang Hong Department of Automation University of Science and Technology of ChinaHefei 230027AnhuiChina Key Lab of Systems and Control Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100190China Key Laboratory of Intelligent Control and Decision of Complex Systems School of AutomationBeijing Institute of TechnologyBeijing 100081China Department of Control Science and Engineering Tongji UniversityShanghai 201804China

In this paper,we develop a distributed solver for a group of strict(non-strict)linear matrix inequalities over a multi-agent network,where each agent only knows one inequality,and all agents co-operate to reach a consensus solution in the intersection of all the feasible *** formulation is transformed into a distributed optimization problem by introducing slack variables and consensus ***,by the primal–dual methods,a distributed algorithm is proposed with the help of projection operators and derivative ***,the convergence of the algorithm is analyzed,followed by illustrative simulations.

关键词： Distributed computation Distributed optimization Linear matrix inequalities primal-dual method

A New Kernel Function Yielding the Best Known Iteration Bounds for primal-dual Interior-Point Algorithms

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Acta Mathematica Sinica,English Series 2009年第12期25卷 2169-2178页

作者： Yan Qin BAI Jin LiGUO Cornelis ROOS Department of Mathematics Shanghai University Shanghai 200436 P. R. China Business School University of Shanghai for Science and Technology Shanghai 200093 P. R. China Faculty of Electrical Engineering Mathematics and Computer Science Delft University of Technology P.O. Box 5031 2600 GA Delft The Netherlands

Kernel functions play an important role in defining new search directions for primal-dual interior-point algorithm for solving linear optimization problems. In this paper we present a new kernel function which yields ... 详细信息

关键词： linear optimization interior-point method primal-dual method large-update method polynomial complexity

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Randomized primal–dual Proximal Block Coordinate Updates

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Journal of the Operations Research Society of China 2019年第2期7卷 205-250页

作者： Xiang Gao Yang-Yang Xu Shu-Zhong Zhang Department of Industrial and Systems Engineering University of MinnesotaMinneapolisUSA Department of Mathematical Sciences Rensselaer Polytechnic InstituteTroyUSA

In this paper,we propose a randomized primal–dual proximal block coordinate updating framework for a general multi-block convex optimization model with coupled objective function and linear *** mere convexity,we establish its O(1/t)convergence rate in terms of the objective value and feasibility *** framework includes several existing algorithms as special cases such as a primal–dual method for bilinear saddle-point problems(PD-S),the proximal Jacobian alternating direction method of multipliers(Prox-JADMM)and a randomized variant of the ADMM for multi-block convex *** analysis recovers and/or strengthens the convergence properties of several existing *** example,for PD-S our result leads to the same order of convergence rate without the previously assumed boundedness condition on the constraint sets,and for Prox-JADMM the new result provides convergence rate in terms of the objective value and the feasibility *** is well known that the original ADMM may fail to converge when the number of blocks exceeds *** result shows that if an appropriate randomization procedure is invoked to select the updating blocks,then a sublinear rate of convergence in expectation can be guaranteed for multi-block ADMM,without assuming any strong *** new approach is also extended to solve problems where only a stochastic approximation of the subgradient of the objective is available,and we establish an O(1/√t)convergence rate of the extended approach for solving stochastic programming.

关键词： primal-dual method Alternating direction method of multipliers(ADMM) Randomized algorithm Iteration complexity·First-order stochastic approximation

Complexity analysis of interior-point algorithm based on a new kernel function for semidefinite optimization

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Journal of Shanghai University(English Edition) 2008年第5期12卷 388-394页

作者：钱忠根白延琴王国强 Department of Basic Courses Jiangsu Teachers University of Technology Changzhou 213001 P. R. China Department of Mathematics College of Sciences Shanghai University Shanghai 200444 P. R. China College of Vocational Technology Shanghai University of Engineering Science Shanghai 200437 P. R. China

Interior-point methods （IPMs） for linear optimization （LO） and semidefinite optimization （SDO） have become a hot area in mathematical programming in the last decades. In this paper, a new kernel function with simple algebraic expression is proposed. Based on this kernel function, a primal-dual interior-point methods （IPMs） for semidefinite optimization （SDO） is designed. And the iteration complexity of the algorithm as O（n^3/4 log n/ε） with large-updates is established. The resulting bound is better than the classical kernel function, with its iteration complexity O（n log n/ε） in large-updates case.

关键词： interior-point algorithm primal-dual method semidefinite optimization (SDO) polynomial complexity

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primal-dual algorithm for distributed optimization with local domains on signed networks

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Primal-dual algorithm for distributed optimization with loca...