Multi-agent Optimization Problem Research Articles

Distributed Subgradient Algorithm for Multi-Agent Optimization With Dynamic Stepsize

In this paper, we consider distributed convex optimization problems on multi-agent networks. We develop and analyze the distributed gradient method which allows each agent to compute its dynamic stepsize by utilizing the time-varying estimate of the local function value at the global optimal solution. Our approach can be applied to both synchronous and asynchronous communication protocols. Specifically, we propose the distributed subgradient with uncoordinated dynamic stepsizes (DS-UD) algorithm for synchronous protocol and the AsynDGD algorithm for asynchronous protocol. Theoretical analysis shows that the proposed algorithms guarantee that all agents reach a consensus on the solution to the multi-agent optimization problem. Moreover, the proposed approach with dynamic stepsizes eliminates the requirement of diminishing stepsize in existing works. Numerical examples of distributed estimation in sensor networks are provided to illustrate the effectiveness of the proposed approach.

A Stochastic Multiagent Optimization Framework for Interdependent Transportation and Power System Analyses

We study the interdependence between transportation and power systems considering decentralized renewable generators and electric vehicles (EVs). We formulate the problem in a stochastic multiagent optimization framework considering the complex interactions between EV/conventional vehicle drivers, renewable/conventional generators, and independent system operators, with locational electricity and charging prices endogenously determined by markets. We show that the multiagent optimization problems can be reformulated as a single convex optimization problem and prove the existence and uniqueness of the equilibrium. To cope with the curse of dimensionality, we propose the alternating direction method of multipliers (ADMM)-based decomposition algorithm to facilitate parallel computing. Numerical insights are generated using standard test systems in the transportation and power system literature.

Distributed Convex Optimization With State-Dependent (Social) Interactions and Time-Varying Topologies

In this paper, an unconstrained collaborative optimization of a sum of convex functions is considered where agents make decisions using local information from their neighbors. The communication between nodes are described by a time-varying sequence of possibly state-dependent weighted networks. A new framework for modeling multi-agent optimization problems over networks with state-dependent interactions and time-varying topologies is proposed. A gradient-based discrete-time algorithm using diminishing step size is proposed for converging to the optimal solution under suitable assumptions. The algorithm is totally asynchronous without requiring B-connectivity assumption for convergence. The algorithm still works even if the weighted matrix of the graph is periodic and irreducible in synchronous protocol. Finally, a case study on a network of robots in an automated warehouse is provided in order to demonstrate the results.

Distributed fixed-time optimization for multi-agent systems over a directed network

This paper proposes some distributed algorithms to solve the multi-agent optimization problem with equality constraints, in which the team objective is a sum of local convex objective functions. Firstly, a directed network related to equality constraints is constructed before converting the constrained optimization problem into an unconstrained one. Secondly, a continuous algorithm is designed by using local information of agents, and the objective function converges to the global optimum in a fixed-time interval. Moreover, in order to reduce the communication cost, an event-triggered algorithm with sign function is devised. It is found that the optimal value can be achieved in a fixed-time interval, but the sign function can cause high-frequency chattering when the sate variables converge to the optimal value. Therefore, an event-triggered algorithm with saturation function is proposed, which can effectively overcome this disadvantage. Finally, the proposed algorithms are verified by some numerical simulations.

Distributed Nonconvex Optimization of Multiagent Systems Using Boosting Functions to Escape Local Optima

In this article, we address the problem of multiple local optima arising due to nonconvex objective functions in cooperative multiagent optimization problems. To escape such local optima, we propose a systematic approach based on the concept of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">boosting functions</i> . The underlying idea is to temporarily transform the gradient at a local optimum into a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">boosted</i> <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">gradient</i> with a nonzero magnitude. We develop a distributed boosting scheme based on a gradient-based optimization algorithm using a novel optimal variable step size mechanism so as to guarantee convergence. Even though our motivation is based on the coverage control problem setting, our analysis applies to a broad class of multiagent problems. Simulation results are provided to compare the performance of different boosting functions families and to demonstrate the effectiveness of the boosting function approach in attaining improved (still generally local) optima.

Byzantine-Resilient Multiagent Optimization

We consider the problem of multiagent optimization wherein an unknown subset of agents suffer Byzantine faults and thus behave adversarially. We assume that each agent i has a local cost function f <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> , and the overarching goal of the good agents is to collaboratively minimize a global objective that properly aggregates these local cost functions. To the best of our knowledge, we are among the first to study Byzantine-resilient optimization where no central coordinating agent exists, and we are the first to characterize the structures of the convex coefficients of the achievable global objectives. Dealing with Byzantine faults is very challenging. For example, in contrast to fault-free networks, reaching Byzantine-resilient agreement even in the simplest setting is far from trivial. We take a step toward solving the proposed Byzantine-resilient multiagent optimization problem by focusing on scalar local cost functions. Our results might provide useful insights for the general local cost functions.

Distributed initialization-free algorithms for multi-agent optimization problems with coupled inequality constraints

This paper studies a resource optimization problem for a multi-agent network where all agents have local objective functions and local constraint sets. Meanwhile, the decision variables of all the agents need to satisfy a set of globally coupled inequality constraints. Then, a distributed continuous-time algorithm is designed with the help of the nonsmooth analysis theory and projection operator method. The proposed algorithm does not require each agent to send the gradient information of the cost function to its neighbors, which prevents the gradient information of agents leaking out. Moreover, the convergence analysis shows that the proposed algorithm can converge to the optimal solution starting from any initial allocation. Finally, a case study is presented for a resource allocation problem of a hybrid water-power network.

Distributed optimization for structured programs and its application to energy management in a building district

This paper deals with structured multi-agent optimization problems that involve coupled local and global decision variables. We propose an iterative distributed algorithm that explicitly accounts for this structure, and requires the agents to communicate only their tentative solutions for the global variables throughout iterations. Our approach extends to structured multi-agent optimization a proximal-based distributed methodology that has recently appeared in the literature. Privacy of local information is preserved and communication effort is reduced with respect to alternative distributed solutions where local and global optimization variables are grouped together and treated as a single decision vector. Multi-agent optimization problems with the considered structural properties appear in various contexts. In this paper, we apply our approach to energy management in a district where multiple buildings can communicate over a possibly time-varying network and aim at optimizing the use of shared and local resources. We illustrate the efficacy of the resulting distributed energy management algorithm by means of a detailed simulation study on a cooling problem.

A scenario-based approach to multi-agent optimization with distributed information

In this paper, we consider optimization problems involving multiple agents. Each agent introduces its own constraints on the optimization vector, and the constraints of all agents depend on a common source of uncertainty. We suppose that uncertainty is known locally to each agent through a private set of data (multi-agent scenarios), and that each agent enforces its scenario-based constraints to the solution of the multi-agent optimization problem. Our goal is to assess the feasibility properties of the corresponding multi-agent scenario solution. In particular, we are able to provide a priori certificates that the solution is feasible for a new occurrence of the global uncertainty with a probability that depends on the size of the datasets and the desired confidence level. The recently introduced wait-and-judge approach to scenario optimization and the notion of support rank are used for this purpose. Notably, decision-coupled and constraint-coupled uncertain optimization programs for multi-agent systems fit our framework and, hence, any distributed optimization scheme to solve the associated multi-agent scenario problem can be accompanied with our a priori probabilistic feasibility certificates.

A Proximal Diffusion Strategy for Multiagent Optimization With Sparse Affine Constraints

This article develops a proximal primal–dual decentralized strategy for multiagent optimization problems that involve multiple coupled affine constraints, where each constraint may involve only a subset of the agents. The constraints are generally sparse, meaning that only a small subset of the agents are involved in them. This scenario arises in many applications, including decentralized control formulations, resource allocation problems, and smart grids. Traditional decentralized solutions tend to ignore the structure of the constraints and lead to degraded performance. We instead develop a decentralized solution that exploits the sparsity structure. Under constant step-size learning, the asymptotic convergence of the proposed algorithm is established in the presence of nonsmooth terms, and it occurs at a linear rate in the smooth case. We also examine how the performance of the algorithm is influenced by the sparsity of the constraints. Simulations illustrate the superior performance of the proposed strategy.

Fenchel Dual Gradient Methods for Distributed Convex Optimization Over Time-Varying Networks

To date, a large collection of distributed algorithms for convex multi-agent optimization have been reported, yet only few of them converge to an optimal solution at guaranteed rates when the topologies of the agent networks are time-varying. Motivated by this, we develop a family of distributed Fenchel dual gradient methods for solving strongly convex yet non-smooth multi-agent optimization problems with nonidentical local constraints over time-varying networks. The proposed algorithms are constructed based on the application of weighted gradient methods to the Fenchel dual of the multiagent optimization problem. They are able to drive all the agents to dual optimality at an O(1/k) rate and to primal optimality at an O(1/√k) rate under a standard network connectivity condition. The competent convergence performance of the Fenchel dual gradient methods is demonstrated via numerical examples.

A New Randomized Block-Coordinate Primal-Dual Proximal Algorithm for Distributed Optimization

This paper proposes Triangularly Preconditioned Primal- Dual algorithm, a new primal-dual algorithm for minimizing the sum of a Lipschitz-differentiable convex function and two possibly nonsmooth convex functions, one of which is composed with a linear mapping. We devise a randomized block-coordinate (BC) version of the algorithm which converges under the same stepsize conditions as the full algorithm. It is shown that both the original as well as the BC scheme feature linear convergence rate when the functions involved are either piecewise linear-quadratic, or when they satisfy a certain quadratic growth condition (which is weaker than strong convexity). Moreover, we apply the developed algorithms to the problem of multiagent optimization on a graph, thus obtaining novel synchronous and asynchronous distributed methods. The proposed algorithms are fully distributed in the sense that the updates and the stepsizes of each agent only depend on local information. In fact, no prior global coordination is required. Finally, we showcase an application of our algorithm in distributed formation control.

Randomized Gradient-Free Distributed Optimization Methods for a Multiagent System With Unknown Cost Function

This paper proposes a randomized gradient-free distributed optimization algorithm to solve a multiagent optimization problem with set constraints. Random gradient-free oracle instead of the true gradient information is built locally such that the estimated gradient information is utilized in guiding the update of decision variables. Thus, the algorithm requires no explicit expressions but only local measurements of the cost functions. The row-stochastic and column-stochastic matrices are used as the weighting matrices during the communication with neighbors, making the algorithm convenient to implement in directed graphs as compared with the doubly stochastic weighting matrix. Without the true gradient information, we establish asymptotic convergence to the approximated optimal solution, where the optimality gap can be set arbitrarily small. Moreover, it is shown that the proposed algorithm achieves the same rate of convergence $O(\ln t/\sqrt{t})$ as the state-of-the-art gradient-based methods with similar settings, but having the advantages of less required information and more practical communication topologies.

Distributed Subgradient-Based Multiagent Optimization With More General Step Sizes

A wider selection of step sizes is explored for the distributed subgradient algorithm for multi-agent optimization problems, for both time-invariant and time-varying communication topologies. The square summable requirement of the step sizes commonly adopted in the literature is removed. The step sizes are only required to be positive, vanishing and non-summable. It is proved that in both unconstrained and constrained optimization problems, the agents' estimates reach consensus and converge to the optimal solution with the more general choice of step sizes. The idea is to show that a weighted average of the agents' estimates approaches the optimal solution, but with different approaches. In the unconstrained case, the optimal convergence of the weighted average of the agents' estimates is proved by analyzing the distance change from the weighted average to the optimal solution and showing that the weighted average is arbitrarily close to the optimal solution. In the constrained case, this is achieved by analyzing the distance change from the agents' estimates to the optimal solution and utilizing the boundedness of the constraints. Then the optimal convergence of the agents' estimates follows because consensus is reached in both cases. These results are valid for both a strongly connected time-invariant graph and time-varying balanced graphs that are jointly strongly connected.

A fixed-time convergent algorithm for distributed convex optimization in multi-agent systems

This technical paper presents a distributed continuous-time algorithm to solve multi-agent optimization problem with the team objective being the sum of all local convex objective functions while subject to an equality constraint. The optimal solutions are achieved within fixed time which is independent of the initial conditions of agents. This advantage makes it possible to off-line preassign the settling time according to task requirements. The fixed-time convergence for the proposed algorithm is rigorously proved with the aid of convex optimization and fixed-time Lyapunov theory. Finally, the algorithm is valuated via an example.

On the distributed optimization over directed networks

In this paper, we propose a distributed algorithm, called Directed-Distributed Subgradient Descent (D-DSD), to solve multi-agent optimization problems over directed graphs. Existing algorithms mostly deal with similar problems under the assumption of undirected networks, i.e., requiring the weight matrices to be doubly-stochastic. The row-stochasticity of the weight matrix guarantees that all agents reach consensus, while the column-stochasticity ensures that each agent’s local (sub)gradient contributes equally to the global objective. In a directed graph, however, it may not be possible to construct a doubly-stochastic weight matrix in a distributed manner. We overcome this difficulty by augmenting an additional variable for each agent to record the change in the state evolution. In each iteration, the algorithm simultaneously constructs a row-stochastic matrix and a column-stochastic matrix instead of only a doubly-stochastic matrix. The convergence of the new weight matrix, depending on the row-stochastic and column-stochastic matrices, ensures agents to reach both consensus and optimality. The analysis shows that the proposed algorithm converges at a rate of O(lnkk), where k is the number of iterations.

Об одной задаче многоагентной оптимизации в позиционных играх

Об одной задаче многоагентной оптимизации в позиционных играх

A Stochastic Multi-agent Optimization Model for Energy Infrastructure Planning under Uncertainty in An Oligopolistic Market

This paper presents a mathematical model for analyzing long-term infrastructure investment decisions in a deregulated electricity market, such as the case in the United States. The interdependence between different decision entities in the system is captured in a network-based stochastic multi-agent optimization model, where new entrants of investors compete among themselves and with existing generators for natural resources, transmission capacities, and demand markets. To overcome computational challenges involved in stochastic multi-agent optimization problems, we have developed a solution method by combining stochastic decomposition and variational inequalities, which converts the original problem to many smaller problems that can be solved more easily.

Randomized gradient-free method for multiagent optimization over time-varying networks.

In this brief, we consider the multiagent optimization over a network where multiple agents try to minimize a sum of nonsmooth but Lipschitz continuous functions, subject to a convex state constraint set. The underlying network topology is modeled as time varying. We propose a randomized derivative-free method, where in each update, the random gradient-free oracles are utilized instead of the subgradients (SGs). In contrast to the existing work, we do not require that agents are able to compute the SGs of their objective functions. We establish the convergence of the method to an approximate solution of the multiagent optimization problem within the error level depending on the smoothing parameter and the Lipschitz constant of each agent's objective function. Finally, a numerical example is provided to demonstrate the effectiveness of the method.

An improved RBF neural network method for information security evaluation

Itis well-known that information security means the protection of information,and ensuring the availability, confidentiality and integrity of information. The purposeof this paper is to present an improved RBF neural network method forinformation evaluation. Ant colony optimization is a multi-agent approach fordifficult combinatorial optimization problems, which has been applied tovarious NP hard problems. Here, ant colony optimization algorithm is applied tooptimize the parameters of RBF neural network. In this paper, we employ “unauthorized access”, “unauthorized accessto a system resource”, “data leakage”, “denial of service”, “unauthorizedmodification data and software”, “system crash” as the features of informationsecurity evaluation. It is indicated that the information security evaluation error of the improvedRBF neural network is smaller than that of the RBF neural network. Thus, theimproved RBF neural network is very suitable for information security evaluation. DOI : http://dx.doi.org/10.11591/telkomnika.v12i4.4806