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On the Convergence Properties of a Distributed projected subgradient algorithm

IEEE TRANSACTIONS ON AUTOMATIC CONTROL 2023年第12期68卷 7546-7559页

作者： Li, Weijian Chen, Zihan Lou, Youcheng Hong, Yiguang South Univ Sci & Technol China Sch Syst Design & Intelligent Mfg Shenzhen 518055 Peoples R China Huawei Technol Co Ltd Beijing 100031 Peoples R China Chinese Acad Sci Acad Math & Syst Sci MDIS Beijing 100045 Peoples R China Tongji Univ Dept Control Sci & Engn Shanghai 200070 Peoples R China Shanghai Res Inst Intelligent Syst Shanghai 200082 Peoples R China

A weight-balanced network plays an important role in the exact convergence of a distributed optimization algorithm, but is not always satisfied in practice. Different from most of existing works focusing on designing distributed algorithms, in this article, we analyze the convergence of a well-known distributed projected subgradient algorithm over time-varying general graph sequences, i.e., the weight matrices of the network are only required to be row stochastic instead of doubly stochastic. We first show that there may exist a graph sequence such that the algorithm is not convergent when the network switches freely within finitely many graphs. Then, to guarantee its convergence under any uniformly jointly strongly connected graph sequence, we provide a necessary and sufficient condition on the cost functions, i.e., the intersection of optimal solution sets to all local optimization problems is not empty. Furthermore, we surprisingly find that the algorithm is convergent for any periodically switching graph sequence, but the converged solution minimizes a weighted sum of local cost functions, where the weights depend on the Perron vectors of some product matrices of the underlying switching graphs. Finally, we consider a slightly broader class of quasi-periodically switching graph sequences, and show that the algorithm is convergent for any quasi-periodic graph sequence if and only if the network switches between only two graphs. This work helps us understand impacts of communication networks on the convergence of distributed algorithms, and complements existing results from a different viewpoint.

关键词： Communication network constrained distributed optimization convergence analysis projected subgradient algorithm

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Distributed Optimization for Convex Mixed-Integer Programs based on projected subgradient algorithm 57

Distributed Optimization for Convex Mixed-Integer Programs b...

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57th IEEE Conference on Decision and Control (CDC)

作者： Sun, Chuangchuang Dai, Ran Ohio State Univ Mech & Aerosp Engn Dept Columbus OH 43210 USA

ISBN: (纸本)9781538613955

Convex Mixed-Integer Program (MIP) has received extensive attention due to its wide applications. This paper proposes a distributed optimization algorithm based on projected subgradient algorithm (PSA) to solve general convex MIPs. We first decouple the feasible region as the intersection of multiple local convex constraints and the relaxed integrality constraints. Then an auxiliary variable vector is assigned to each decoupled constraint with a consensus constraint to keep equivalence. A Distributed projected subgradient algorithm (DPSA) is proposed to solve the reformulated problem. The updates of the auxiliary variables associated with the convex local constraints as well as the integrality constraint are implemented in a parallel way, followed by the gossip step to drive all auxiliary variable sets into consensus. While this algorithm applies to general convex MIPs, special attention is paid to quadratic and linear constraints in order to obtain a closed-form solution for each subproblem. While convergence of such DPSA for convex problems has been studied, the presence of the integrality constraint makes it inapplicable. Consequently, convergence of the proposed DPSA and properties at the converging point under certain assumptions is presented. Numerical results on maximum clique problem are provided to sustain the effectiveness of the proposed algorithm.

关键词： distributed optimization projected subgradient algorithm convex mixed-integer program

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Robust Sensor Selection By Deep Unfolding

Robust Sensor Selection By Deep Unfolding

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2025 IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP 2025

作者： Singh, Yuvraj Rohela, Jahnvi Majumder, Kaushani Mulleti, Satish Dept. of Electrical Engineering Indian Institute of Technology Bombay India

ISBN: (纸本)9798350368741

The challenge of sensor selection involves choosing a subset of the available measurements to estimate data while aiming to minimize estimation errors. Traditionally, convex relaxation methods with solvers like the projected subgradient (PSG) algorithm have been employed to address this problem. However, they are computationally intensive due to the large number of iterations involved and necessitate accurate apriori knowledge of the noise statistics, which may be unavailable. Moreover, in the case of linear measurements, the cost functions involved are data-independent, which makes them non-adaptive. In this paper, we tackle these challenges by introducing two deep neural networks that unfold the PSG algorithm. These networks are trained on observations where knowledge of noise statistics and the parameters to estimate are implicit. This makes the approach data-adaptive without needing any priors. Moreover, in comparison to the conventional PSG algorithm, the unfoldings require fewer iterations. Experimental results demonstrate that these networks achieve a lower mean-squared error by 4 ~ 10 dB compared to the standard PSG algorithm or random selection. © 2025 IEEE.

关键词： interpretable machine learning iterative unfolding Neural network projected subgradient algorithm sensor selection

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A Privacy-Preserving Distributed Control of Optimal Power Flow

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IEEE TRANSACTIONS ON POWER SYSTEMS 2022年第3期37卷 2042-2051页

作者： Ryu, Minseok Kim, Kibaek Argonne Natl Lab Math & Comp Sci Div Lemont IL 60439 USA

We consider a distributed optimal power flow formulated as an optimization problem that maximizes a nondifferentiable concave function. Solving such a problem by the existing distributed algorithms can lead to data privacy issues because the solution information exchanged within the algorithms can be utilized by an adversary to infer the data. To preserve data privacy, in this paper we propose a differentially private projected subgradient (DP-PS) algorithm that includes a solution encryption step. We show that a sequence generated by DP-PS converges in expectation, in probability, and with probability 1. Moreover, we show that the rate of convergence in expectation is affected by a target privacy level of DP-PS chosen by the user. We conduct numerical experiments that demonstrate the convergence and data privacy preservation of DP-PS.

关键词： Data privacy Convergence Voltage Distributed databases Reactive power Generators Privacy Differential privacy projected subgradient algorithm optimal power flow dual decompositions

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Sparsity-Promoting Sensor Selection for Non-Linear Measurement Models

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IEEE TRANSACTIONS ON SIGNAL PROCESSING 2015年第3期63卷 684-698页

作者： Chepuri, Sundeep Prabhakar Leus, Geert Delft Univ Technol Fac Elect Engn Math & Comp Sci NL-2628 CD Delft Netherlands

The problem of choosing the best subset of sensors that guarantees a certain estimation performance is referred to as sensor selection. In this paper, we focus on observations that are related to a general non-linear model. The proposed framework is valid as long as the observations are independent, and its likelihood satisfies the regularity conditions. We use several functions of the Cramer-Rao bound (CRB) as a performance measure. We formulate the sensor selection problem as the design of a sparse vector, which in its original form is a nonconvex l(0)-(quasi) norm optimization problem. We present relaxed sensor selection solvers that can be efficiently solved in polynomial time. The proposed solvers result in sparse sensing techniques. We also propose a projected subgradient algorithm that is attractive for large-scale problems. The developed theory is applied to sensor placement for localization.

关键词： Convex optimization Cramer-Rao bound nonlinear models projected subgradient algorithm sensor networks sensor placement sensor selection sparse sensing sparsity

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An Event-Based Delayed Projection Row-Stochastic Method for Distributed Constrained Optimization Over Time-Varying Graphs

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IEEE TRANSACTIONS ON SYSTEMS MAN CYBERNETICS-SYSTEMS 2024年第12期54卷 7508-7520页

作者： Xing, Mingqi Ma, Dazhong Zhang, Huaguang Xie, Xiangpeng Northeastern Univ Coll Informat Sci & Engn Shenyang 110819 Peoples R China Nanjing Univ Posts & Telecommun Sch Internet Things Nanjing 210003 Peoples R China

This article investigates the distributed constrained optimization problem with event-triggered communication over time-varying weight-unbalanced directed graphs. A more generalized network model is considered where the communication topology may be variable and unbalanced over time, the information flows across agents are subject to time-varying communication delays, and agents are not required to know their out-degree information accurately. To address the above challenges, we propose a novel discrete-time distributed event-triggered delay subgradient algorithm. To facilitate convergence analysis, a consensus-only "virtual" agent technique is employed, dynamically adjusting its state (active or asleep) to ensure a delay-free information flow among agents. Additionally, an augmentation approach is proposed to ensure that the augmented time-varying weight matrix is row-stochastic. It is shown that the agents' local decision variables converge to the same optimal solution, in the case of reasonable communication delays and event-triggering thresholds. Numerical examples show the efficiency of the proposed algorithm.

关键词： Optimization Delays Linear programming Topology Heuristic algorithms Directed graphs Real-time systems Communication delays constrained optimization event-triggered communication projected subgradient algorithm time-varying unbalanced directed graph

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A STRONGLY CONVERGENT METHOD FOR NONSMOOTH CONVEX MINIMIZATION IN HILBERT SPACES

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NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION 2011年第10期32卷 1009-1018页

作者： Bello Cruz, J. Y. Iusem, A. N. Univ Fed Goias Inst Matemat & Estat BR-74001970 Goiania Go Brazil Jardim Bot Inst Matemat Pura & Aplicada Rio De Janeiro Brazil

In this article, we propose a strongly convergent variant on the projected subgradient method for constrained convex minimization problems in Hilbert spaces. The advantage of the proposed method is that it converges strongly when the problem has solutions, without additional assumptions. The method also has the following desirable property: the sequence converges to the solution of the problem which lies closest to the initial iterate.

关键词： Convex minimization Nonsmooth optimization projected subgradient algorithm Projection method Strong convergence

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Enabling Privacy-Preservation in Decentralized Optimization

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IEEE TRANSACTIONS ON CONTROL OF NETWORK SYSTEMS 2019年第2期6卷 679-689页

作者： Zhang, Chunlei Wang, Yongqiang Clemson Univ Dept Elect & Comp Engn Clemson SC 29634 USA

Decentralized optimization is crucial for the design, deployment, and functionality of many distributed systems. In this paper, we address the problem of privacy-preservation in decentralized optimization. In most existing decentralized optimization approaches, participating agents exchange and disclose intermediate estimate values to neighboring agents explicitly, which may not be desirable when the estimates contain sensitive information of individual agents. The problem is more acute when adversaries exist which try to steal information from other participating agents. To address this issue, we propose a novel approach that enables privacy-preservation in decentralized optimization by combining partially homomorphic cryptography with the unique properties of the decentralized optimization. We show that cryptographic techniques can be incorporated in a fully decentralized manner to enable privacy-preservation in decentralized optimization in the absence of any third party or aggregator. This is significant in that to our knowledge, all existing cryptographic-based optimization approaches rely on the assistance of a third-party or aggregator in order to protect the privacy of all parties. The approach is also applicable to the average consensus problem, which is finding broad applications in fields as diverse as robotic networks, distributed computing, and power systems. Numerical simulations confirm the effectiveness and low computational complexity of the proposed approach.

关键词： Decentralized optimization partially homomorphic encryption privacy preservation projected subgradient algorithm

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Extended forward-backward algorithm

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JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 2013年第1期403卷 167-172页

作者： Lassonde, Marc Nagesseur, Ludovic Univ Antilles Guyane LAMIA F-97159 Pointe A Pitre Guadeloupe France

We propose an extended forward backward algorithm for approximating a zero of a maximal monotone operator which can be split as the extended sum of two maximal monotone operators. We establish the weak convergence in average of the sequence generated by the algorithm under assumptions similar to those used in classical forward backward algorithms. This provides as a special case an algorithm for solving convex constrained minimization problems without qualification condition. (C) 2013 Elsevier Inc. All rights reserved.

关键词： Maximal monotone operator epsilon-enlargement Extended sum Forward-backward algorithm projected subgradient algorithm Splitting algorithm Convergence in average

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projected subgradient based distributed convex optimization with transmission noises

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APPLIED MATHEMATICS AND COMPUTATION 2022年 418卷 126794-126794页

作者： Zhang, Li Liu, Shuai Shandong Univ Sch Control Sci & Engn Jinan Peoples R China

This paper discusses a kind of convex optimization problem considering noises from in-formation transmission in multi-agent systems. Different from previous works, we focus on the objective function which is a summation of strictly L-0(F)-convex functions under random inner space. Our system is described by Ito formula, which leads to that it is hard to calculate second-order derivative when designing the projected subgradient algorithm. It is shown that all states in stochastic system will converge to the unique optimal state in the polyhedric set constraint by adopting projected subgradient algorithm and the con-vergence rate is also investigated. Numerical examples are provided to demonstrate the results.& nbsp;(c) 2021 Elsevier Inc. All rights reserved.

关键词： Distributed convex optimization projected subgradient algorithm Additive noise Polyhedric set constraint Random inner space

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