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Distributed algorithms for Robust Convex Optimization via the Scenario Approach

IEEE TRANSACTIONS ON AUTOMATIC CONTROL 2019年第3期64卷 880-895页

作者： You, Keyou Tempo, Roberto Xie, Pei Tsinghua Univ Dept Automat Beijing 100084 Peoples R China Tsinghua Univ BNRist Beijing 100084 Peoples R China Politecn Torino CNR IEIIT Inst Elect Comp & Telecommun Engn I-10129 Turin Italy

This paper proposes distributed algorithms to solve robust convex optimization (RCO) when the constraints are affected by nonlinear uncertainty. We adopt a scenario approach by randomly sampling the uncertainty set. To facilitate the computational task, instead of using a single centralized processor to obtain a "global solution" of the scenario problem (SP), we resort to multiple interconnected processors that are distributed among different nodes of a network to simultaneously solve the SP. Then, we propose a primal-dual subgradient algorithm and a random projection algorithm to distributedly solve the SP over undirected and directed graphs, respectively. Both algorithms are given in an explicit recursive form with simple iterations, which are especially suited for processors with limited computational capability. We show that, if the underlying graph is strongly connected, each node asymptotically computes a common optimal solution to the SP with a convergence rate O(1/(Sigma(k)(t=1) zeta(t))), where {zeta(t)} is a sequence of appropriately decreasing stepsizes. That is, the RCO is effectively solved in a distributed way. The relations with the existing literature on robust convex programs are thoroughly discussed and an example of robust system identification is included to validate the effectiveness of our distributed algorithms.

关键词： primal-dual algorithm random projection algorithm robust convex optimization (RCO) scenario approach uncertainty

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An approximation algorithm for the submodular multicut problem in trees with linear penalties

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OPTIMIZATION LETTERS 2021年第4期15卷 1105-1112页

作者： Hou, Chenfei Gao, Suogang Liu, Wen Wu, Weili Du, Ding-Zhu Hou, Bo Hebei Normal Univ Sch Math Sci Shijiazhuang 050024 Hebei Peoples R China Univ Texas Dallas Dept Comp Sci Richardson TX 75080 USA

In this paper, we consider the submodular multicut problem in trees with linear penalties (SMCLP( T) problem). In the SMCLP(T) problem, we are given a tree T = (V, E) with a submodular function c(.) : 2(E) -> R->= 0, a set of k distinct pairs of vertices P = {(s(1), t(1)), (s(2), t(2)), ... , (s(k), t(k))} with non-negative penalty costs pi(j) for the pairs (s(j), t(j)) is an element of P. The goal is to find a partial multicut M subset of E such that the total cost, consisting of the submodular cost of M and the penalty cost of the pairs not cut by M, is minimized. Let P-j be the unique path from s(j) to t(j) in the tree, where 1 <= j <= k and let m be the maximal length of all P-j. Our main work is to present an m-approximation algorithm for the SMCLP(T) problem via the primal-dual method.

关键词： Multicut Submodular function primal-dual algorithm

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Distributed online primal-dual subgradient method on unbalanced directed networks

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ADVANCED ROBOTICS 2024年第9-10期38卷 591-602页

作者： Tada, Keishin Hayashi, Naoki Takai, Shigemasa Osaka Univ Grad Sch Engn Suita Osaka Japan Osaka Univ Grad Sch Engn Sci Toyonaka Osaka Japan

In this paper, we investigate a distributed online optimization method on multiagent communication networks. We consider a distributed prime-dual subgradient algorithm for an online convex optimization problem with time-varying coupled constraints. Each agent updates the estimations for the primal and dual optimizers by a consensus-based online algorithm. In the proposed algorithm, the gradient direction is scaled by estimating the left eigenvector of a weight matrix associated with the directed communication network. The scaling procedure enables agents in the network to estimate the time-varying optimal solutions by counterbalancing the unbalanced communication flows. The performance of the proposed algorithm is examined by a dynamic regret and a fit, which evaluate the cumulative error against the time-varying optimal cost function and the constraint function, respectively. We provide a sufficient condition under which both the dynamic regret and the fit are sublinear. A numerical example of an online economic dispatch problem confirms the validity of the proposed method.

关键词： Distributed online optimization multiagent system primal-dual algorithm

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An Approximation algorithm for the Generalized Prize-Collecting Steiner Forest Problem with Submodular Penalties

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Journal of the Operations Research Society of China 2022年第1期10卷 183-192页

作者： Xiao-Dan Jia Bo Hou Wen Liu School of Mathematical Sciences Hebei Normal UniversityShijiazhuang 050024HebeiChina

In this paper,we consider the generalized prize-collecting Steiner forest problem with submodular penalties(GPCSF-SP problem).In this problem,we are given an undirected connected graph G=(V,E)and a collection of disjoint vertex subsets V={V_(1),V_(2),…,V_(l)}.Assume c:E→R_(+)is an edge cost function andπ:2^(V)→R_(+)is a submodular penalty *** objective of the GPCSF-SP problem is to find an edge subset F such that the total cost including the edge cost in F and the penalty cost of the subcollection S containing these Vi not connected by F is *** using the primal-dual technique,we give a 3-approximation algorithm for this problem.

关键词： Generalized prize-collecting Steiner forest problem Submodular function primal-dual algorithm

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Projective Splitting as a Warped Proximal algorithm

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APPLIED MATHEMATICS AND OPTIMIZATION 2022年第2期85卷 1-11页

作者： Bui, Minh N. North Carolina State Univ Dept Math Raleigh NC 27695 USA

We show that the asynchronous block-iterative primal-dual projective splitting framework introduced by P. L. Combettes and J. Eckstein in their 2018 Math. Program. paper can be viewed as an instantiation of the recent... 详细信息

关键词： Warped proximal algorithm Projective splitting primal-dual algorithm Splitting algorithm Monotone inclusion Monotone operator

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Rician noise and intensity nonuniformity correction (NNC) model for MRI data

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BIOMEDICAL SIGNAL PROCESSING AND CONTROL 2019年第0期49卷 506-519页

作者： Liu, Lu Yang, Huan Fan, Jiyun Liu, Ryan Wen Duan, Yuping Tianjin Univ Ctr Appl Math Tianjin Peoples R China Henan Univ Sch Math & Stat Kaifeng Peoples R China Wuhan Univ Technol Sch Nav Wuhan Hubei Peoples R China

Rician noise and intensity nonuniformity are two common artifacts and usually coexist in magnetic resonance imaging (MRI) data. Many methods have been proposed in the literature dealing with either Rician noise or intensity nonuniformity individually. We numerically verify that the existence of intensity nonuniformity may lead to the underestimation of noise, which means intensity nonuniformity influences the performance of denoising and vice versa. Thus, we propose a novel restoration model via a maximum a posteriori (MAP) estimator by regarding MRI data as a combination of two multiplicative components, namely, the true intensity and the bias field, and a noise followed a Rician distribution. We also guarantee that the proposed model has at least one positive nontrivial solution theoretically. An efficient algorithm based on alternating minimization method is developed, all subproblems of which can be solved effectively by either Newton's method or closed-form solutions. Intensive numerical results on synthetic and real MRI data confirm the robustness of the method and its better performance for MRI data restoration. (C) 2018 Elsevier Ltd. All rights reserved.

关键词： Rician noise Intensity nonuniformity Total variation Alternating minimization method primal-dual algorithm

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Inducing strong convergence into the asymptotic behaviour of proximal splitting algorithms in Hilbert spaces

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OPTIMIZATION METHODS & SOFTWARE 2019年第3期34卷 489-514页

作者： Bot, Radu Ioan Csetnek, Ernoe Robert Meier, Dennis Univ Vienna Fac Math Oskar Morgenstern Pl 1 A-1090 Vienna Austria

Proximal splitting algorithms for monotone inclusions (and convex optimization problems) in Hilbert spaces share the common feature to guarantee for the generated sequences in general weak convergence to a solution. In order to achieve strong convergence, one usually needs to impose more restrictive properties for the involved operators, like strong monotonicity (respectively, strong convexity for optimization problems). In this paper, we propose a modified Krasnosel'skii-Mann algorithm in connection with the determination of a fixed point of a nonexpansive mapping and show strong convergence of the iteratively generated sequence to the minimal norm solution of the problem. Relying on this, we derive a forward-backward and a Douglas-Rachford algorithm, both endowed with Tikhonov regularization terms, which generate iterates that strongly converge to the minimal norm solution of the set of zeros of the sum of two maximally monotone operators. Furthermore, we formulate strong convergent primal-dual algorithms of forward-backward and Douglas-Rachford-type for highly structured monotone inclusion problems involving parallel-sums and compositions with linear operators. The resulting iterative schemes are particularized to the solving of convex minimization problems. The theoretical results are illustrated by numerical experiments on the split feasibility problem in infinite dimensional spaces.

关键词： fixed points of nonexpansive mappings Tikhonov regularization splitting methods forward-backward algorithm Douglas-Rachford algorithm primal-dual algorithm

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Covering problems in edge- and node-weighted graphs

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DISCRETE OPTIMIZATION 2016年 20卷 40-61页

作者： Fukunaga, Takuro Natl Inst Informat Chiyoda Ku 2-1-2 Hitotsubashi Tokyo Japan ERATO JST Kawarabayashi Large Graph Project Tokyo Japan

This paper discusses the graph covering problem in which a set of edges in an edge- and node-weighted graph is chosen to satisfy some covering constraints while minimizing the sum of the weights. In this problem, because of the large integrality gap of a naive linear programming (LP) relaxation, LP rounding algorithms based on the relaxation yield poor performance. Here we propose a stronger LP relaxation for the graph covering problem. The proposed relaxation is applied to designing primal dual algorithms for two fundamental graph covering problems: the prize-collecting edge dominating set problem and the multicut problem in trees. Our algorithms are an exact polynomial-time algorithm for the former problem, and a 2-approximation algorithm for the latter problem. These results match the currently known best results for purely edge-weighted graphs. (C) 2016 Elsevier B.V. All rights reserved.

关键词： Edge dominating set Multicut primal-dual algorithm

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Accelerated Convergence algorithm for Distributed Constrained Optimization under Time-Varying General Directed Graphs

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IEEE TRANSACTIONS ON SYSTEMS MAN CYBERNETICS-SYSTEMS 2020年第7期50卷 2612-2622页

作者： Li, Huaqing Lu, Qingguo Liao, Xiaofeng Huang, Tingwen Southwest Univ Coll Elect & Informat Engn Chongqing Key Lab Nonlinear Circuits & Intelligen Chongqing 400715 Peoples R China Texas A&M Univ Qatar Sci Program Doha Qatar

This paper studies a class of distributed convex optimization problems by a set of agents in which each agent only has access to its own local convex objective function and the estimate of each agent is restricted to both coupling linear constraint and individual box constraints. Our focus is to devise a distributed primal-dual gradient algorithm for working out the problem over a sequence of time-varying general directed graphs. The communications among agents are assumed to be uniformly strongly connected. A column-stochastic mixing matrix and a fixed step-size are applied in the algorithm which exactly steers all the agents to asymptotically converge to a global optimal solution. Based on the standard strong convexity and the smoothness assumptions of the objective functions, we show that the distributed algorithm is capable of driving the whole network to geometrically converge to an optimal solution of the convex optimization problem only if the step-size does not exceed some upper bound. We also give an explicit analysis for the convergence rate of the proposed optimization algorithm. Simulations on economic dispatch problems and demand response problems in power systems are performed to illustrate the effectiveness of the proposed optimization algorithm.

关键词： Optimization Convergence Linear programming Convex functions Multi-agent systems Generators Couplings Distributed convex optimization geometric convergence multiagent systems primal-dual algorithm time-varying directed graphs

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A primal-dual gradient method for image decomposition based on (BV, H ^-1)

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MULTIDIMENSIONAL SYSTEMS AND SIGNAL PROCESSING 2011年第4期22卷 335-348页

作者： Yin, Haiqing Liu, Hongwei China Univ Petr Coll Math & Computat Sci Dongying 257061 Shandong Peoples R China Xidian Univ Sch Sci Xian 710071 Shaanxi Peoples R China

The main aim of this paper is to accelerate the image decomposition model based on (BV, H (-1)). It is solved with a particularly effective primal-dual gradient descent algorithm. The algorithm works on the primal-dual formulation and exploits the information of the primal and dual variables simultaneously. It converges significantly faster than some popular existing methods in numerical experiments. This approach is to some extent related to projection type methods for solving variational inequalities.

关键词： Image decomposition primal-dual algorithm Structure Texture

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