Distributed Convex Optimization Problem Research Articles

Distributed Continuous-Time Nonsmooth Convex Optimization With Coupled Inequality Constraints

This paper studies distributed convex optimization problems over continuous-time multiagent networks subject to two types of constraints, i.e., local feasible set constraints and coupled inequality constraints, where all involved functions are not necessarily differentiable, only assumed to be convex. In order to solve this problem, a modified primal-dual continuous-time algorithm is proposed by projections on local feasible sets. With the aid of constructing a proper Lyapunov function candidate, the existence of solutions of the algorithm in the Caratheodory sense and the convergence of the algorithm to an optimal solution for the distributed optimization problem are established. Additionally, a sufficient condition is provided for making the algorithm fully distributed. Finally, the theoretical result is corroborated by a simulation example.

Continuous-Time Distributed Subgradient Algorithm for Convex Optimization With General Constraints

The distributed convex optimization problem is studied in this paper for any fixed and connected network with general constraints. To solve such an optimization problem, a new type of continuous-time distributed subgradient optimization algorithm is proposed based on the Karuch–Kuhn–Tucker condition. By using tools from nonsmooth analysis and set-valued function theory, it is proved that the distributed convex optimization problem is solved on a network of agents equipped with the designed algorithm. For the case that the objective function is convex but not strictly convex, it is proved that the states of the agents associated with optimal variables could converge to an optimal solution of the optimization problem. For the case that the objective function is strictly convex, it is further shown that the states of agents associated with optimal variables could converge to the unique optimal solution. Finally, some simulations are performed to illustrate the theoretical analysis.

Event-triggered asynchronous distributed optimization algorithm with heterogeneous time-varying step-sizes

This paper concerns distributed convex optimization problems over time-varying undirected graphs, in which the global objective function is expressed as the sum of individual objective functions of the agents. Each agent only knows its local objective functions. To figure out such problems, an event-triggered asynchronous distributed optimization algorithm (termed as EV-ADOA) with time-varying heterogeneous step-sizes is proposed, which is suitable for undirected graphs changing over time. Under two standard assumptions on strongly convex and smoothness of local objective functions, the EV-ADOA can achieve linear convergence with a proper upper bound of the heterogeneous time-varying step-sizes. EV-ADOA with event-triggered scheme can decrease network communication, and the Zeno-like behavior strictly is excluded. The efficiency of EV-ADOA is demonstrated by experiments.

A Geometrically Converging Dual Method for Distributed Optimization Over Time-Varying Graphs

In this article, we consider a distributed convex optimization problem over time-varying undirected networks. We propose a dual method, primarily averaged network dual ascent (PANDA), that is proven to converge R-linearly to the optimal point given that the agents' objective functions are strongly convex and have Lipschitz continuous gradients. Like dual decomposition, PANDA requires half the amount of variable exchanges per iterate of methods based on DIGing, and can provide with practical improved performance as empirically demonstrated.

Primal-Dual Subgradient Algorithm for Distributed Constraint Optimization Over Unbalanced Digraphs

This paper investigates the distributed convex optimization problem with coupled inequality constraints over unbalanced digraphs depicted as row-stochastic matrices, where each agent in the network only has access to its local information, while the local constraint functions of all the agents are coupled in inequality constraints. To solve this kind of problems, a novel distributed iterative algorithm is proposed via consensus scheme and the projected primal-dual subgradient method. Different from the previous related results, our algorithm can not only deal with coupling inequality constraints but also conquer the unbalanced topology caused by directed graphs. Moreover, it is proved that under certain assumptions, the optimal solution of the problem can be asymptotically obtained by performing the designed algorithm. Finally, the numerical examples are presented to further illustrate the efficacy of the proposed approach.

Distributed Parallel Sparse Multinomial Logistic Regression

Sparse Multinomial Logistic Regression (SMLR) is widely used in the field of image classification, multi-class object recognition, and so on, because it has the function of embedding feature selection during classification. However, it cannot meet the time and memory requirements for processing large-scale data. We have reinvestigated the classification accuracy and running efficiency of the algorithm for solving SMLR problems using the Alternating Direction Method of Multipliers (ADMM), which is called fast SMLR (FSMLR) algorithm in this paper. By reformulating the optimization problem of FSMLR, we transform the serial convex optimization problem to the distributed convex optimization problem, i.e., global consensus problem and sharing problem. Based on the distributed optimization problem, we propose two distribute parallel SMLR algorithms, sample partitioning-based distributed SMLR (SP-SMLR), and feature partitioning-based distributed SMLR (FP-SMLR), for a large-scale sample and large-scale feature datasets in big data scenario, respectively. The experimental results show that the FSMLR algorithm has higher accuracy than the original SMLR algorithm. The big data experiments show that our distributed parallel SMLR algorithms can scale for massive samples and large-scale features, with high precision. In a word, our proposed serial and distribute SMLR algorithms outperform the state-of-the-art algorithms.

Distributed Convex Optimization with Inequality Constraints over Time-Varying Unbalanced Digraphs

This paper considers a distributed convex optimization problem with inequality constraints over time-varying unbalanced digraphs, where the cost function is a sum of local objective functions, and each node of the graph only knows its local objective and inequality constraints. Although there is a vast body of literature on distributed optimization, most of them require the graph to be balanced, which is quite restrictive and not necessary. To solve it, this work proposes a novel idea of using the epigraph form of the constrained optimization, which can be easily used to study time-varying unbalanced digraphs. Under local communications, a simple iterative algorithm is then designed for each node. We prove that if the graph is uniformly jointly strongly connected, each node asymptotically converges to some common optimal solution.

A distributed Newton–Raphson-based coordination algorithm for multi-agent optimization with discrete-time communication

This paper proposes a novel distributed continuous-time Newton–Raphson algorithm for distributed convex optimization problem, where the components of the goal are obtainable at different agents. To accelerate convergence speed, we focus on introducing Newton descent idea in our algorithm and extending it in a distributed setting. It is proved that the proposed algorithm can converge to the global optimal point with exponential convergence rate under weight-balanced directed graphs. Motivated by practical considerations, an event-triggered broadcasting strategy is further developed for each agent. Therein, the implementation of communication is driven by the designed triggered condition monitored by agents. Consequently, the proposed continuous-time algorithm can be executed with discrete-time communication, thus being able to greatly save communication expenditure. Moreover, the strategy is proved to be free of Zeno behavior. Eventually, the simulation results illustrate the advantages of the proposed algorithm.

Approximate Projection Methods for Decentralized Optimization With Functional Constraints

We consider distributed convex optimization problems that involve a separable objective function and nontrivial functional constraints, such as linear matrix inequalities. We propose a decentralized and computationally inexpensive algorithm, which is based on the concept of approximate projections. Our algorithm is one of the consensus-based methods in that, at every iteration, each agent performs a consensus update of its decision variables followed by an optimization step of its local objective function and local constraints. Unlike other methods, the last step of our method is not a Euclidean projection onto the feasible set, but instead a subgradient step in the direction that minimizes the local constraint violation. We propose two different averaging schemes to mitigate the disagreements among the agents’ local estimates. We show that the algorithms converge almost surely, i.e., every agent agrees on the same optimal solution, under the assumption that the objective functions and constraint functions are nondifferentiable and their subgradients are bounded. We provide simulation results on a decentralized optimal gossip averaging problem, which involves semidefinite programming constraints, to complement our theoretical results.

Finite‐Time Average Consensus Based Approach for Distributed Convex Optimization

AbstractIn this paper, we consider a distributed convex optimization problem where the objective function is an average combination of individual objective function in multi‐agent systems. We propose a novel Newton Consensus method as a distributed algorithm to address the problem. This method utilises the efficient finite‐time average consensus method as an information fusion tool to construct the exact Newtonian global gradient direction. Under suitable assumptions, this strategy can be regarded as a distributed implementation of the classical standard Newton method and eventually has a quadratic convergence rate. The numerical simulation and comparison experiment show the superiority of the algorithm in convergence speed and performance.

Distributed zero‐gradient‐sum algorithm for convex optimization with time‐varying communication delays and switching networks

SummaryThe distributed convex optimization problem subject to time‐varying communication delays and switching network topologies is addressed in this paper. Based on continuous‐time Zero‐Gradient‐Sum scheme, the novel distributed algorithms are proposed to minimize the global objective function which is composed of a sum of strictly convex local cost functions. In the fixed network topology case, by constructing a new Lyapunov‐Krasovskii function, two explicit sufficient conditions for the maximum admissible time delay are derived to guarantee that all agents' states converge to the optimal solution. In the switching network topology case, the stability condition is derived by the common Lyapunov function theory. In addition, two sufficient conditions about the maximum admissible time delays are also derived for the fixed and switching weight‐balanced network topologies, respectively. Several simulation tests are used to illustrate the effectiveness of our obtained theoretical results.

Distributed Nonsmooth Optimization With Coupled Inequality Constraints via Modified Lagrangian Function

This technical note considers a distributed convex optimization problem with nonsmooth cost functions and coupled nonlinear inequality constraints. To solve the problem, we first propose a modified Lagrangian function containing local multipliers and a nonsmooth penalty function. Then we construct a distributed continuous-time algorithm by virtue of a projected primal-dual subgradient dynamics. Based on the nonsmooth analysis and Lyapunov function, we obtain the existence of the solution to the nonsmooth algorithm and its convergence.

Event-triggered zero-gradient-sum distributed convex optimisation over networks with time-varying topologies

ABSTRACTThis paper focuses on a distributed convex optimisation problem over networks with time-varying topologies. First, an event-triggered strategy is employed to reduce computation and communication load in the networked systems. Second, the exponential convergence of event-triggered zero-gradient-sum algorithm is guaranteed if the corresponding time-varying network topology satisfies a new connected condition, called cooperatively connected condition. This condition does not require topologies constantly connected or jointly connected but only requires the integral of the Laplacian matrix of the network topology over a period of time is connected. Hence, it is suitable for more general time-varying topologies. Third, a convergence analysis technique is developed which is based on the difference of the Lyapunov function rather than its differentiation. Finally, a simulation example is provided to verify the results obtained in this paper.

Distributed finite-time optimization for second order continuous-time multiple agents systems with time-varying cost function

In this paper, a finite time distributed convex optimization problem is studied for continuous time multiple agents systems. The distributed optimization objective is not only to drive the agents to reach a consensus but also to cooperatively minimize the sum of the objective functions of each agent in finite time. Different from the traditional optimization problems, in this problem, the objective function can be time varying and each agent can only acquire the information of its neighbours local objective function through the network. In order to solve this problem, firstly, we proposed a centralized finite-time optimization control algorithm. Then, the distributed finite-time optimization controllers for both the first-order and second-order multiple agents systems are designed based on the centralized finite-time optimization algorithm. Theoretical studies indicate that the proposed algorithms can minimize the team objective function in finite time and all agents will reach a consensus in finite time. Finally, a simulation example is presented to show the validity of the theoretical results.

Distributed Continuous-Time Convex Optimization With Time-Varying Cost Functions

In this paper, a time-varying distributed convex optimization problem is studied for continuous-time multi-agent systems. Control algorithms are designed for the cases of single-integrator and double-integrator dynamics. Two discontinuous algorithms based on the signum function are proposed to solve the problem in each case. Then in the case of double-integrator dynamics, two continuous algorithms based on, respectively, a time-varying and a fixed boundary layer are proposed as continuous approximations of the signum function. Also, to account for inter-agent collision for physical agents, a distributed convex optimization problem with swarm tracking behavior is introduced for both single-integrator and double-integrator dynamics.

Sampled-data based distributed convex optimization with event-triggered communication

This paper studies the distributed convex optimization problem for multi-agent systems over undirected and connected networks. Motivated by practical considerations, we propose a new distributed optimization algorithm with event-triggered communication. The proposed event detection is decentralized, sampled-data and not requires periodic communications among agents to calculate the threshold. Based on Lyapunov approaches, we show that the proposed algorithm is asymptotically converge to the unknown optimizer if the design parameters are chosen properly. We also give an upper bound on the convergence rate. Finally, we illustrate the effectiveness of the proposed algorithm by a numerical simulation.

Distributed convex optimisation with event-triggered communication in networked systems

ABSTRACTThis paper studies the distributed convex optimisation problem over directed networks. Motivated by practical considerations, we propose a novel distributed zero-gradient-sum optimisation algorithm with event-triggered communication. Therefore, communication and control updates just occur at discrete instants when some predefined condition satisfies. Thus, compared with the time-driven distributed optimisation algorithms, the proposed algorithm has the advantages of less energy consumption and less communication cost. Based on Lyapunov approaches, we show that the proposed algorithm makes the system states asymptotically converge to the solution of the problem exponentially fast and the Zeno behaviour is excluded. Finally, simulation example is given to illustrate the effectiveness of the proposed algorithm.

Event-triggered zero-gradient-sum distributed consensus optimization over directed networks

This paper focuses on the event-triggered zero-gradient-sum algorithms for a distributed convex optimization problem over directed networks. The communication process is driven by trigger conditions monitored by nodes. The proposed trigger conditions are decentralized and just depend on each node’s own state. In the continuous-time case, we propose an algorithm based on a sample-based monitoring scheme. In the discrete-time case, we propose a new event-triggered zero-gradient-sum algorithm which is suitable for more general network models. It is proved that two proposed event-triggered algorithms are exponentially convergent if the design parameters are chosen properly and the network topology is strongly connected and weight-balanced. Finally, we illustrate the advantages of the proposed algorithms by numerical simulation.

Stochastic sub-gradient algorithm for distributed optimization with random sleep scheme

In this paper, we consider a distributed convex optimization problem of a multi-agent system with the global objective function as the sum of agents’ individual objective functions. To solve such an optimization problem, we propose a distributed stochastic sub-gradient algorithm with random sleep scheme. In the random sleep scheme, each agent independently and randomly decides whether to inquire the sub-gradient information of its local objective function at each iteration. The algorithm not only generalizes distributed algorithms with variable working nodes and multi-step consensus-based algorithms, but also extends some existing randomized convex set intersection results. We investigate the algorithm convergence properties under two types of stepsizes: the randomized diminishing stepsize that is heterogeneous and calculated by individual agent, and the fixed stepsize that is homogeneous. Then we prove that the estimates of the agents reach consensus almost surely and in mean, and the consensus point is the optimal solution with probability 1, both under randomized stepsize. Moreover, we analyze the algorithm error bound under fixed homogeneous stepsize, and also show how the errors depend on the fixed stepsize and update rates.

Primal Recovery from Consensus-Based Dual Decomposition for Distributed Convex Optimization

Dual decomposition has been successfully employed in a variety of distributed convex optimization problems solved by a network of computing and communicating nodes. Often, when the cost function is separable but the constraints are coupled, the dual decomposition scheme involves local parallel subgradient calculations and a global subgradient update performed by a master node. In this paper, we propose a consensus-based dual decomposition to remove the need for such a master node and still enable the computing nodes to generate an approximate dual solution for the underlying convex optimization problem. In addition, we provide a primal recovery mechanism to allow the nodes to have access to approximate near-optimal primal solutions. Our scheme is based on a constant stepsize choice, and the dual and primal objective convergence are achieved up to a bounded error floor dependent on the stepsize and on the number of consensus steps among the nodes.