Decentralized Optimization Problem Research Articles

Coordinate-update algorithms can efficiently detect infeasible optimization problems

Coordinate update/descent algorithms are widely used in large-scale optimization due to their low per-iteration cost and scalability, but their behavior on infeasible or misspecified problems has not been much studied compared to the algorithms that use full updates. For coordinate-update methods to be as widely adopted to the extent so that they can be used as engines of general-purpose solvers, it is necessary to also understand their behavior under pathological problem instances. In this work, we show that the normalized iterates of randomized coordinate-update fixed-point iterations (RC-FPI) converge to the infimal displacement vector and use this result to design an efficient infeasibility detection method. We then extend the analysis to the setup where the coordinates are defined by non-orthonormal basis using the Friedrichs angle and then apply the machinery to decentralized optimization problems.

Decentralized gradient tracking with local steps

Gradient tracking (GT) is an algorithm designed for solving decentralized optimization problems over a network (such as training a machine learning model). A key feature of GT is a tracking mechanism that allows us to overcome data heterogeneity between nodes. We develop a novel decentralized tracking mechanism, K-GT, which enables communication-efficient local updates in GT while inheriting the data-independence property of GT. We prove a convergence rate for K-GT on smooth non-convex functions and prove that it reduces the communication overhead asymptotically by a linear factor K, where K denotes the number of local steps. We illustrate the robustness and effectiveness of this heterogeneity correction on convex and non-convex benchmark problems and a non-convex neural network training task with the MNIST dataset.

Push–Pull With Device Sampling

We consider decentralized optimization problems in which a number of agents collaborate to minimize the average of their local functions by exchanging over an underlying communication graph. Specifically, we place ourselves in an asynchronous model where only a random portion of nodes perform computation at each iteration, while the information exchange can be conducted between all the nodes and in an asymmetric fashion. For this setting, we propose an algorithm that combines gradient tracking with a network-level variance reduction (in contrast to variance reduction within each node). This enables the nodes to track the average of the gradients of the objective functions. Our theoretical analysis shows that the algorithm converges linearly, when the local objective functions are strongly convex, under mild connectivity conditions on the expected mixing matrices. In particular, our result does not require the mixing matrices to be doubly stochastic. In the experiments, we investigate a broadcast mechanism that transmits information from computing nodes to their neighbors, and confirm the linear convergence of our method on both synthetic and real-world datasets.

Learning Regionally Decentralized AC Optimal Power Flows With ADMM

One potential future for the next generation of smart grids is the use of decentralized optimization algorithms and secured communications for coordinating renewable generation (e.g., wind/solar), dispatchable devices (e.g., coal/gas/nuclear generations), demand response, battery & storage facilities, and topology optimization. The Alternating Direction Method of Multipliers (ADMM) has been widely used in the community to address such decentralized optimization problems and, in particular, the AC Optimal Power Flow (AC-OPF). This paper studies how machine learning may help in speeding up the convergence of ADMM for solving AC-OPF. It proposes a novel decentralized machine-learning approach, namely ML-ADMM, where each agent uses deep learning to learn the consensus parameters on the coupling branches. The paper also explores the idea of learning only from ADMM runs that exhibit high-quality convergence properties, and proposes filtering mechanisms to select these runs. Experimental results on test cases based on the French system demonstrate the potential of the approach in speeding up the convergence of ADMM significantly.

Graph neural networks-based scheduler for production planning problems using reinforcement learning

Reinforcement learning (RL) is increasingly adopted in job shop scheduling problems (JSSP). But RL for JSSP is usually done using a vectorized representation of machine features as the state space. It has three major problems: (1) the relationship between the machine units and the job sequence is not fully captured, (2) exponential increase in the size of the state space with increasing machines/jobs, and (3) the generalization of the agent to unseen scenarios. This paper presents a novel framework named GraSP-RL, GRAph neural network-based Scheduler for Production planning problems using Reinforcement Learning. It represents JSSP as a graph and trains the RL agent using features extracted using a graph neural network (GNN). While the graph is itself in the non-Euclidean space, the features extracted using the GNNs provide a rich encoding of the current production state in the Euclidean space. At its core is a custom message-passing algorithm applied to the GNN. The node features encoded by the GNN are then used by the RL agent to select the next job. Further, we cast the scheduling problem as a decentralized optimization problem in which the learning agent is assigned to all the production units individually and the agent learns asynchronously from the experience collected on all the other production units. The GraSP-RL is then applied to a complex injection molding production environment with 30 jobs and 4 machines. The task is to minimize the makespan of the production plan. The schedule planned by GraSP-RL is then compared and analyzed with a priority dispatch rule algorithm like first-in-first-out (FIFO) and metaheuristics like tabu search (TS) and genetic algorithm (GA). The proposed GraSP-RL outperforms the FIFO, TS, and GA for the trained task of planning 30 jobs in JSSP. We further test the generalization capability of the trained agent on two different problem classes: Open shop system (OSS) and Reactive JSSP (RJSSP). In these modified problem classes our method produces results better than FIFO and comparable results to TS and GA, without any further training while also providing schedules instantly.

Decentralized ADMM with compressed and event-triggered communication

This paper considers the decentralized optimization problem, where agents in a network cooperate to minimize the sum of their local objective functions by communication and local computation. We propose a decentralized second-order communication-efficient algorithm called communication-censored and communication-compressed quadratically approximated alternating direction method of multipliers (ADMM), termed as CC-DQM, by combining event-triggered communication with compressed communication. In CC-DQM, agents are allowed to transmit the compressed message only when the current primal variables have changed greatly compared to its last estimate. Moreover, to relieve the computation cost, the update of Hessian is also scheduled by the trigger condition. Theoretical analysis shows that the proposed algorithm can still maintain an exact linear convergence, despite the existence of compression error and intermittent communication, if the local objective functions are strongly convex and smooth. Finally, numerical experiments demonstrate its satisfactory communication efficiency.

Gradient-Push Algorithm for Distributed Optimization With Event-Triggered Communications

Decentralized optimization problems consist of multiple agents connected by a network. The agents have each local cost function, and the goal is to minimize the sum of the functions cooperatively. It requires the agents to communicate with each other, and reducing the cost for communication is desired for a communication-limited environment. Recently, the decentralized gradient descent algorithms involving event-triggered communication have been proposed when the network of the agents is an undirected graph. On the other hand, the network of agents is often directed graph for realistic scenarios whose communication resources are limited. In this work, we first propose a gradient-push algorithm involving event-triggered communication on a directed network. Each agent sends its current states to its neighbors only when the differences between the latest sent states and the current states are larger than thresholds. The convergence of the algorithm is established under suitable decays and summability conditions on a stepsize and triggering thresholds. Numerical experiments are presented to support the effectiveness and the convergence results of the algorithm. More precisely, the numerical results reveals that the proposed algorithm may reduce the communication cost significantly compared to the gradient-push algorithm not involving the event-triggered communication.

A Compressed Gradient Tracking Method for Decentralized Optimization With Linear Convergence

Communication compression techniques are of growing interests for solving the decentralized optimization problem under limited communication, where the global objective is to minimize the average of local cost functions over a multiagent network using only local computation and peer-to-peer communication. In this article, we propose a novel compressed gradient tracking algorithm (C-GT) that combines gradient tracking technique with communication compression. In particular, C-GT is compatible with a general class of compression operators that unifies both unbiased and biased compressors. We show that C-GT inherits the advantages of gradient tracking-based algorithms and achieves linear convergence rate for strongly convex and smooth objective functions. Numerical examples complement the theoretical findings and demonstrate the efficiency and flexibility of the proposed algorithm.

Review of bio-inspired optimization applications in renewable-powered smart grids: Emerging population-based metaheuristics

The management of renewable-powered smart grids deals with nonlinear optimization problems featuring a variety of linear or nonlinear constraints, discrete or continuous optimization variables, involving high dimensionality of the solution space, and strict time requirements to identify the optimal or near-optimal solution. One promising approach for addressing such optimization problems is to apply bio-inspired population-based optimization algorithms, many such metaheuristics emerging lately. In this paper, we have identified the metaheuristics with the highest impact published recently and reviewed their applications in the management of renewable-powered smart energy grids using the Preferred Reporting Items for Systematic reviews and Meta-Analyses (PRISMA) methodology and the Web of Science Core Collection as the reference database. Four main smart grid application domains we been analyzed: (i) energy prediction models’ optimization to reduce uncertainty (ii) energy resources coordination to handle the stochastic nature of renewables, (iii) demand response using controllable loads and flexibility while considering the consumers’ needs and constraints and (iv) optimization of grid energy efficiency and costs. The results showed the advantages of such metaheuristics for decentralized optimization problems with low computational time and resource overhead. At the same time, several issues need to be addressed to increase their adoption in the smart grid management scenarios: the lack of standard testing methodologies and benchmarks, efficient management of exploration and exploitation of the optimization search space, guidelines for metaheuristics application with clear links to the type of optimization problems, etc.

Decentralized Optimization Over the Stiefel Manifold by an Approximate Augmented Lagrangian Function

In this paper, we focus on the decentralized optimization problem over the Stiefel manifold, which is defined on a connected network of <inline-formula><tex-math notation="LaTeX">$d$</tex-math></inline-formula> agents. The objective is an average of <inline-formula><tex-math notation="LaTeX">$d$</tex-math></inline-formula> local functions, and each function is privately held by an agent and encodes its data. The agents can only communicate with their neighbors in a collaborative effort to solve this problem. In existing methods, multiple rounds of communications are required to guarantee the convergence, giving rise to high communication costs. In contrast, this paper proposes a decentralized algorithm, called DESTINY, which only invokes a single round of communications per iteration. DESTINY combines gradient tracking techniques with a novel approximate augmented Lagrangian function. The global convergence to stationary points is rigorously established. Comprehensive numerical experiments demonstrate that DESTINY has a strong potential to deliver a cutting-edge performance in solving a variety of testing problems.

A Unified and Refined Convergence Analysis for Non-Convex Decentralized Learning

We study the consensus decentralized optimization problem where the objective function is the average of <inline-formula><tex-math notation="LaTeX">$n$</tex-math></inline-formula> agents private non-convex cost functions; moreover, the agents can only communicate to their neighbors on a given network topology. The stochastic learning setting is considered in this paper where each agent can only access a noisy estimate of its gradient. Many decentralized methods can solve such problem including EXTRA, Exact-Diffusion/D<inline-formula><tex-math notation="LaTeX">$^{2}$</tex-math></inline-formula>, and gradient-tracking. Unlike the famed <small>Dsgd</small> algorithm, these methods have been shown to be robust to the heterogeneity across the local cost functions. However, the established convergence rates for these methods indicate that their sensitivity to the network topology is worse than <small>Dsgd</small>. Such theoretical results imply that these methods can perform much worse than <small>Dsgd</small> over sparse networks, which, however, contradicts empirical experiments where <small>Dsgd</small> is observed to be more sensitive to the network topology. In this work, we study a general <u>s</u>tochastic <u>u</u>nified <u>d</u>ecentralized <u>a</u>lgorithm (<b>SUDA</b>) that includes the above methods as special cases. We establish the convergence of <b>SUDA</b>&#x00A0;under both non-convex and the Polyak-&#x0141;ojasiewicz condition settings. Our results provide improved network topology dependent bounds for these methods (such as Exact-Diffusion/D<inline-formula><tex-math notation="LaTeX">$^{2}$</tex-math></inline-formula> and gradient-tracking) compared with existing literature. Moreover, our results show that these methods are often less sensitive to the network topology compared to <small>Dsgd</small>, which agrees with numerical experiments.

Introducing a Bi-Level Linear Programming Model to Reduce Patient Payment and Increase Hospital Income Simultaneously

Background: The cost of health care is a large part of every household’s budget. On the other hand, as an economic entity, the hospital is constantly faced with different aspects of cost and revenue. So, we are dealing with conflicting objectives. Objectives: The main purpose of the research is to help financial management in a specialty hospital. This article provides part of operational research under bi-level optimization for hospital managers to provide targeted financial planning. The method is based on the fact that the objective is to maximize the hospital income on one level, and on the other level, the objective is to reduce the patient’s payment. Methods: The hierarchical and decentralized optimization problem is written as a bi-level model that minimizes patient costs and maximizes hospital revenues, which is an NP-Hard problem. The optimal solution to this problem is obtained using a genetic algorithm. Then, the hospital’s performance is evaluated by the Pabon Lasso diagram. It is shown that the use of this model has a significant effect on the hospital’s performance. Results: Implementation of this model in the studied hospital shows that patient payment costs decreased and hospital income increased (reaching equilibrium point). Conclusion: Hospital performance after model implementation was evaluated by the Pabon Lasso diagram and showed that it has an effective role in hospital performance.

Detection of Insider Attacks in Distributed Projected Subgradient Algorithms

The gossip-based distributed projected subgradient algorithms (DPS) are widely used to solve decentralized optimization problems in various multi-agent applications, while they are generally vulnerable to data injection attacks by internal malicious agents as each agent locally estimates its descent direction without an authorized supervision. In this work, we explore the application of artificial intelligence (AI) technologies to detect internal attacks. We show that a general neural network is particularly suitable for detecting and localizing the malicious agents, as they can effectively explore nonlinear relationship underlying the collected data. Moreover, we propose to adopt one of the state-of-the-art approaches in decentralized federated learning, i.e., a gossip-based collaborative learning protocol, to facilitate training the neural network models by gossip exchanges. This advanced approach is expected to make our model more robust to challenges with insufficient training data, or mismatched test data. In our simulations, a least-squared problem is considered to verify the feasibility and effectiveness of AI-based methods. Simulation results demonstrate that the proposed AI-based methods are beneficial to improve performance of detecting and localizing malicious agents over score-based methods, and the peer-to-peer neural network model is indeed robust to target issues.

A Decentralized Market Framework for Procurement of Operating Reserves From District Energy Systems

District energy system (DES) has become a popular form of supplying comprehensive energy demands in local buildings. For DESs, the operational flexibility can be maintained by energy conversion and storage facilities. Hence, DESs are capable of providing operating reserves to local power systems. This paper addresses the implementation challenges of allowing DES to be deployed alongside conventional generators for providing operating reserves. A two-level hierarchical market framework is proposed which involves a wholesale energy-reserve market for conventional power plants and a subordinate DES operating reserve market. The subordinate operating reserve market determines the quantity and the marginal price of operating reserves provided by DESs. The two markets are linked by the price elastic-quantity provided in the operating reserve demand curve (ORDC) to achieve the stated market coordination. A decentralized decision-making structure is proposed to clear the subordinate operating reserve market considering DES privacy and security requirements. A nested dual-loop solution process is proposed to solve the decentralized optimization problem. A test system is used to illustrate the benefits of the proposed technique in supplying operating reserves through DESs.

Resource-Aware Exact Decentralized Optimization Using Event-Triggered Broadcasting

This article addresses the decentralized optimization problem where a group of agents with coupled private objective functions work together to exactly optimize the summation of local interests. Upon modeling the decentralized problem as an equality-constrained centralized one, we leverage the linearized augmented Lagrangian method to design an event-triggered decentralized algorithm that only requires light local computation at generic time instants and peer-to-peer communication at sporadic triggering time instants. The triggering time instants for each agent are locally determined by comparing the deviation between true and broadcast primal variables with certain triggering thresholds. Provided that the threshold is summable over time, we establish a new upper bound for the effect of triggering behavior on the primal-dual residual. Based on this, the same convergence rate O(1/k) with periodic algorithms is secured for nonsmooth convex problems. Stronger convergence results are obtained for strongly convex and smooth problems, that is, the iterates linearly converge with exponentially decaying triggering thresholds. Finally, the developed strategy is examined with two common optimization problems; comparison results illustrate its performance and superiority in exploiting communication resources.

Event-Triggering Interaction Scheme for Discrete-Time Decentralized Optimization With Nonuniform Step Sizes.

In this article, we study the discrete-time decentralized optimization problems of multiagent systems by an event-triggering interaction scheme, in which each agent privately knows its local convex cost function, and collectively minimizes the total cost functions. The underlying interaction and the corresponding weight matrix are required to be undirected connected and doubly stochastic, respectively. To resolve this optimization problem collaboratively, we propose a decentralized event-triggering algorithm (DETA) that is based on the consensus theory and inexact gradient tracking technique. DETA involves each agent interacting with its neighboring agents only at some independent event-triggering sampling time instants. Under the assumptions that the global convex cost function is coercive and has Lipschitz continuous gradient, we prove that DETA steers all agents' states to an optimal solution even with nonuniform constant step sizes. Moreover, our analysis also shows that DETA converges at a rate of O(1/√t) if the step sizes are uniform and do not exceed some upper bounds. We illustrate the effectiveness of DETA on a canonical simple decentralized parameter estimation problem.

Nash optimality based distributed model predictive control for vehicle platoon

In this paper, a distributed model predictive control algorithm (DMPC) based on Nash optimality is proposed for automated vehicle platoon control. The optimization decision of vehicle platoon is decomposed into the decentralized optimization of single vehicles, in which the Nash optimality algorithm is adopted to solve the decentralized optimization problem. Thus, each vehicle can reach the local optimal target and the whole team can reach its Nash equilibrium. The methodology employs neighborhood information of the entire platoon through on-board sensors and V2V communication to achieve coordination of the entire platoon. The ability of the methods in terms of robustness to disturbances and cyber-physical interaction is demonstrated with simulation case studies.

A Decentralized Proximal-Gradient Method With Network Independent Step-Sizes and Separated Convergence Rates

This paper proposes a novel proximal-gradient algorithm for a decentralized optimization problem with a composite objective containing smooth and non-smooth terms. Specifically, the smooth and nonsmooth terms are dealt with by gradient and proximal updates, respectively. The proposed algorithm is closely related to a previous algorithm, PG-EXTRA \cite{shi2015proximal}, but has a few advantages. First of all, agents use uncoordinated step-sizes, and the stable upper bounds on step-sizes are independent of network topologies. The step-sizes depend on local objective functions, and they can be as large as those of the gradient descent. Secondly, for the special case without non-smooth terms, linear convergence can be achieved under the strong convexity assumption. The dependence of the convergence rate on the objective functions and the network are separated, and the convergence rate of the new algorithm is as good as one of the two convergence rates that match the typical rates for the general gradient descent and the consensus averaging. We provide numerical experiments to demonstrate the efficacy of the introduced algorithm and validate our theoretical discoveries.

A Partial Augmented Lagrangian Method for Decentralized Electric Vehicle Charging in Capacity-Constrained Distribution Networks

This work deals with decentralized optimization problems where a collection of network agents operate selfishly and individually under a set of coupling constraints among them. This collection of problems are motivated by coordinating electric vehicle (EV) charging in a feeder capacity-constrained distribution network where each EV makes decisions locally to achieve the system’s global benefit. First, a centralized scheme is established and admits an optimal coordination behaviour that minimizes the aggregate generation cost and all the EVs’ local costs. However, the coupling of the charging behaviours among the EV population brings challenges to the design of the decentralized algorithm, which is expected to converge to the global optimum. A partial augmented Lagrangian method is presented to disperse the coupling constraint by introducing a penalty term. Combined with the gradient projection method, an iterative procedure is designed such that each EV and feeder line update in parallel until satisfying the terminal condition. The convergence can be guaranteed under some certain mild conditions, and the convergence rate to the global optimum is analysed as well. We include some simulation results to demonstrate the performance of the developed results.

A Primal-Dual Quasi-Newton Method for Exact Consensus Optimization

We introduce the primal-dual quasi-Newton (PD-QN) method as an approximated second order method for solving decentralized optimization problems. The PD-QN method performs quasi-Newton updates on both the primal and dual variables of the consensus optimization problem to find the optimal point of the augmented Lagrangian. By optimizing the augmented Lagrangian, the PD-QN method is able to find the exact solution to the consensus problem with a linear rate of convergence. We derive fully decentralized quasi-Newton updates that approximate second order information to reduce the computational burden relative to dual methods and to make the method more robust in ill-conditioned problems relative to first order methods. The linear convergence rate of PD-QN is established formally and strong performance advantages relative to existing dual and primal-dual methods are shown numerically.