Convex Constraint Set Research Articles

Distributed Online Learning Algorithm for Noncooperative Games Over Unbalanced Digraphs.

This article investigates constrained online noncooperative games (NGs) of multiagent systems over unbalanced digraphs, where the cost functions of players are time-varying and are gradually revealed to corresponding players only after decisions are made. Moreover, in the problem, the players are subject to local convex set constraints and time-varying coupling nonlinear inequality constraints. To the best of our knowledge, no result about online games with unbalanced digraphs has been reported, let alone constrained online games. To seek the variational generalized Nash equilibrium (GNE) of the game online, a distributed learning algorithm is proposed based on gradient descent, projection, and primal-dual methods. Under the algorithm, sublinear dynamic regrets and constraint violations are established. Finally, online electricity market games illustrate the algorithm.

Exponential stabilization of linear systems with feedback optimization and its application to microgrids control

This paper proposes a feedback-optimization-based control method for linear time-invariant systems, which is aimed to exponentially stabilize the system and, meanwhile, drive the system output to an approximate solution of an optimization problem with convex set constraints and affine inequality constraints. To ensure the exponential stability of the closed-loop system, the original optimization problem is first reformulated into a counterpart that has only convex set constraints. It is shown that the optimal solution of the new optimization problem is an approximate optimal solution of the original problem. Then, based on this new optimization problem, the projected primal–dual gradient dynamics algorithm is used to design the controller. By using the singular perturbation method, sufficient conditions are provided to ensure the exponential stability of the closed-loop system. The proposed method is also applied to microgrid control.

A Recurrent Neural Network Approach for Constrained Distributed Fuzzy Convex Optimization.

This article investigates a class of constrained distributed fuzzy convex optimization problems, where the objective function is the sum of a set of local fuzzy convex objective functions, and the constraints include partial order relation and closed convex set constraints. In undirected connected node communication network, each node only knows its own objective function and constraints, and the local objective function and partial order relation functions may be nonsmooth. To solve this problem, a recurrent neural network approach based on differential inclusion framework is proposed. The network model is constructed with the help of the idea of penalty function, and the estimation of penalty parameters in advance is eliminated. Through theoretical analysis, it is proven that the state solution of the network enters the feasible region in finite time and does not escape again, and finally reaches consensus at an optimal solution of the distributed fuzzy optimization problem. Furthermore, the stability and global convergence of the network do not depend on the selection of the initial state. A numerical example and an intelligent ship output power optimization problem are given to illustrate the feasibility and effectiveness of the proposed approach.

Distributed Discrete-Time Convex Optimization With Closed Convex Set Constraints: Linearly Convergent Algorithm Design.

The convergence rate and applicability to directed graphs with interaction topologies are two important features for practical applications of distributed optimization algorithms. In this article, a new kind of fast distributed discrete-time algorithms is developed for solving convex optimization problems with closed convex set constraints over directed interaction networks. Under the gradient tracking framework, two distributed algorithms are, respectively, designed over balanced and unbalanced graphs, where momentum terms and two time-scales are involved. Furthermore, it is demonstrated that the designed distributed algorithms attain linear speedup convergence rates provided that the momentum coefficients and the step size are appropriately selected. Finally, numerical simulations verify the effectiveness and the global accelerated effect of the designed algorithms.

Normalized penalty gradient flow: A continuous-time approach for solving constrained nonconvex nonsmooth optimization problems

To avoid calculating penalty parameters, this paper introduces a continuous-time approach that combines the normalized gradient flow with the penalty method to solve the nonconvex nonsmooth optimization problem with a convex constraint set. Subsequently, this approach is extended to solve nonconvex nonsmooth optimization with a box constraint set and affine equality constraints. Compared with the current methods, the proposed approach does not require calculating the penalty parameters, allows the objective function to be nonconvex or nonsmooth, and allows the initial point to be arbitrarily selected. It is proved that the solution trajectory with any initial point converges to the critical point set of the optimization problem. Finally, the effectiveness of the proposed approach is authenticated through several numerical experiments.

Distributed strategies for mixed equilibrium problems: Continuous-time theoretical approaches

In this paper, the mixed equilibrium problem is studied by employing a continuous-time multi-agent system, where the objective of agents is to cooperatively find a point from the feasible set such that the sum of bifunctions with a free-variable is nonnegative. Different from existing works on mixed equilibrium problems with only convex set constraints, we consider a more general case where the feasible set is constrained by a set of convex inequalities. Moreover, each agent only has access to the information of its own bifunction and inequality constraint function, as well as the information of a local convex set. To handle this problem, the Karush–Kuhn–Tucker condition that is suitable for continuous-time theoretical approaches is provided, and a continuous-time distributed primal–dual algorithm is proposed. Under mild assumptions on the graph and bifunctions, we prove that the states of all agents asymptotically converge to the common solution of the mixed equilibrium problem. Finally, a simulation example is worked out to demonstrate the effectiveness of our theoretical results.

Distributed algorithm for nonsmooth multi-coalition games and its application in electricity markets

In this paper, we study distributed nonsmooth multi-coalition games (MCGs), where the players are subject to local convex set constraints and coupled inequality constraints. Different from existing distributed MCGs, our problem involves nonsmooth payoff functions and constraints, relaxes the requirement for communication networks, and only relies on the strict monotonicity of pseudo-gradients. Due to the nonsmoothness of payoff functions and constraints, the weaker communication networks as well as the coexistence of cooperation and competition among players, existing generalized Nash equilibrium (GNE) seeking algorithms cannot solve the problem. Also, they pose obstacles to the algorithm design and analysis, mainly because of the non-global Lipschitz continuity of subgradients and the unconnectedness of subnetworks. To seek the variational GNE (vGNE) of the nonsmooth MCGs, we design a distributed subgradient-based algorithm. We prove that the algorithm converges to the exact vGNE of the nonsmooth MCGs from any initial states. Finally, our result is applied to the electricity market games (EMGs) of smart grids.

Linear Convergence of distributed estimation with constraints and communication delays

Motivated by the multi-sensor estimation problem, we consider a distributed constraint parameter estimation problem on a directed graph. Our goal is to design a fast distributed algorithm to handle constraint parameter estimation problems, especially considering the communication delays when sensors interact on a directed graph. Based on the gradient-tracking algorithm and indirect projection method, the Projected Distributed Parameter Estimation Algorithm (PDPEA) is proposed, which can deal with the closed convex set constraint and use the historical information stored by the sensors to achieve fast and accurate mean gradient estimation. The analysis shows that the PDPEA converges to a global and consensual minimizer even under the influence of measurement noise, arbitrarily bounded communication delays, and directed information topologies. Under the assumption that the local cost function is strongly convex and smooth, we show that the algorithm converges at a linear rate for an appropriate choice of stepsize. Lastly, we demonstrate the effectiveness of the proposed algorithm by setting up a multi-sensor parameter estimation network.

Distributed constrained optimization algorithms with linear convergence rate over time-varying unbalanced graphs

In this work, the constrained optimization problem is studied, where the global objective function is the sum of N convex functions and a closed convex set constraint is involved. The aim of this work is to design distributed optimization algorithms with linear convergence rate over time-varying unbalanced graphs for solving the studied problem. Towards this end, the Push-DIGing based accelerated distributed constrained optimization algorithm (PDADCO) and the Push-Pull based accelerated distributed constrained optimization algorithm (PPADCO) are developed. Specially, during the algorithm developments, the method of feasible direction, which can be successfully integrated into the Push-DIGing framework and the Push-Pull framework, is newly introduced to deal with the involved closed convex set constraint. Furthermore, based on the small gain theorem, the linear convergence properties of both the PDADCO and the PPADCO are rigorously established, with the feasible regions of the step-sizes being given. Finally, a simulation example is provided to verify the effectiveness of the established theoretical results.

Event-Triggered Distributed Stochastic Mirror Descent for Convex Optimization.

This article is concerned with the distributed convex constrained optimization over a time-varying multiagent network in the non-Euclidean sense, where the bandwidth limitation of the network is considered. To save the network resources so as to reduce the communication costs, we apply an event-triggered strategy (ETS) in the information interaction of all the agents over the network. Then, an event-triggered distributed stochastic mirror descent (ET-DSMD) algorithm, which utilizes the Bregman divergence as the distance-measuring function, is presented to investigate the multiagent optimization problem subject to a convex constraint set. Moreover, we also analyze the convergence of the developed ET-DSMD algorithm. An upper bound for the convergence result of each agent is established, which is dependent on the trigger threshold. It shows that a sublinear upper bound can be guaranteed if the trigger threshold converges to zero as time goes to infinity. Finally, a distributed logistic regression example is provided to prove the feasibility of the developed ET-DSMD algorithm.

Well-posedness of fractional Moreau’s sweeping processes of Caputo type

In this paper, we study the well-posedness (in the sense of existence and uniqueness of a solution) of a sweeping process involving fractional order derivative of Caputo type in Hilbert spaces. The convex constraint set C(t) is assumed to satisfy a Hölder continuous variation. The notion of a solution of this fractional sweeping process is proposed, the existence and uniqueness of such solution are proved and a stability result is provided, as well. More precisely, we present new arguments for the proof of existence of the fractional sweeping process. It is still based on the ideas of the catching-up algorithm of J.J. Moreau (see Kunze and Monteiro Marques (2000) and Moreau (1977)). We establish a catching-up algorithm for fractional sweeping process. This new algorithm gives us a sequence of approximate solutions. The technical problems are to prove that we can extract a weakly convergent subsequence, and to check that this limit function is a solution of the fractional sweeping process. Here we propose a new approach to overcome these problems. The existence of solutions follows, to some extent, from an important differential inequality on the Caputo fractional derivative of the squared norm CD0α‖u(t)‖2≤2u(t),CD0αu(t), which was studied in Aguila-Camacho et al. (2014), Gomoyunov (2018) and Kamenskii et al. (2021), that applied to two approximate solutions, by the monotonicity of normal cone NC(t)(⋅) and the minus sign on the right hand side, yields that their distance is nonincreasing. This reasoning allows us to prove that the sequence of approximate solutions is of the fractional sweeping process, converging to a solution. Our approach is constructive and the algorithm is implemented numerically. The results of the paper are applied in the study of an application to non-regular electrical circuit containing nonsmooth electronic device like diode and fractional inductor, which represent an additional novelty of our paper. Finally, two numerical simulations to valid the theoretical results are given.

Seeking strategy design for distributed nonsmooth games and its application

This paper investigates the nonsmooth noncooperative games (NGs) over unbalanced digraphs, where each player has a nondifferentiable payoff function. Meanwhile, these players are subject to local nonsmooth inequality constraints, local convex set constraints and coupled equality constraints. Due to the coexistence of nonsmooth payoff functions, unbalanced digraphs and constraints, existing generalized Nash equilibrium (GNE) seeking strategies are fail to address our problem. Based on subgradient descents, differential inclusions, projection and primal-dual methods, we develop a distributed seeking strategy. Besides, the convergence of the strategy is proved. Finally, the proposed method is applied to electricity market games (EMGs) of smart grids.

Distributed Algorithm for Nonsmooth Resource Allocation Problems With Nonlinear Constraints and Its Application to Smart Grids

In this article, we study nondifferentiable resource allocation problems (RAPs). In our problem, the decisions of agents are subject to coupling inequality constraints, local inequality constraints, and local convex set constraints. In contrast to existing RAPs, the cost functions are nonsmooth and the inequality constraints are nonsmooth and nonlinear in our problem. Based on differential inclusions and projection methods, we exploit a fully distributed subgradient-based resource allocation algorithm to optimally allocate the network resources. With the help of the set-valued LaSalle invariance principle, we prove the global convergence of the algorithm to the optimal resource allocation of our problem. Finally, our method is applied to the economic dispatch problems of smart grids. With our method, the generations of generating units converge to the optimal power generation.

Consensus Tracking and Containment in Multiagent Networks With State Constraints

The ability of tracking is an important prerequisite for multiagent networks to perform collective activities. This article investigates the problem of containment for a weighted multiagent network with continuous-time agents under state constraints. The network is composed of uninformed and informed agents, where the latter receive external inputs. A new general class of distributed nonlinear controllers is designed for accomplishing both containment and consensus tracking, where the state of each agent is required to stay in its desired convex constraint set. We show that, by using matrix analysis, convex analysis, and Lyapunov theory, all agents eventually converge to the convex hull formed by the external inputs while they obey their constraints during the transience. No relationship is assumed between the convex hull and the intersection of all constraint sets. The consensus tracking problem with a single external input is also solved under this framework. As a generalization, we tackle the multiscaled constrained containment problem, where agents can specify their desired buffer zones by either zooming in or zooming out the convex hull. Numerical examples are provided to illustrate the theoretical results.

Adaptive Exact Penalty Design for Optimal Resource Allocation.

In this article, a distributed adaptive continuous-time optimization algorithm based on the Laplacian-gradient method and adaptive control is designed for resource allocation problem with the resource constraint and the local convex set constraints. In order to deal with local convex sets, a distance-based exact penalty function method is adopted to reformulate the resource allocation problem instead of the widely used projection operator method. By using the nonsmooth analysis and set-valued LaSalle invariance principle, it is proven that the proposed algorithm is capable of solving the nonsmooth resource allocation problem. Finally, two simulation examples are presented to substantiate the theoretical results.

Convergence analysis of an improved Bregman-type Peaceman–Rachford splitting algorithm for nonconvex nonseparable linearly constrained optimization problems

This study refers to a class of nonconvex nonseparable optimization problems with linear constraints and closed convex set constraints. Firstly, the equivalent transformation of the discussed problem is carried out by introducing the indicator function and the auxiliary variable. Secondly, to solve the equivalent problem of transformation, an improved Bregman-type Peaceman–Rachford splitting method is proposed by improving the Lagrange multiplier updating technique in the traditional splitting algorithm. Under the general assumptions, the global convergence of the proposed algorithm is proven. Furthermore, if the augmented Lagrange function satisfies the Kurdyka–Łojasiewicz property, the strong convergence of the proposed algorithm is established. Finally, some preliminary numerical results are shown to indicate the effectiveness of the proposed algorithm.

Majorization-minimization-based Levenberg–Marquardt method for constrained nonlinear least squares

A new Levenberg–Marquardt (LM) method for solving nonlinear least squares problems with convex constraints is described. Various versions of the LM method have been proposed, their main differences being in the choice of a damping parameter. In this paper, we propose a new rule for updating the parameter so as to achieve both global and local convergence even under the presence of a convex constraint set. The key to our results is a new perspective of the LM method from majorization-minimization methods. Specifically, we show that if the damping parameter is set in a specific way, the objective function of the standard subproblem in LM methods becomes an upper bound on the original objective function under certain standard assumptions. Our method solves a sequence of the subproblems approximately using an (accelerated) projected gradient method. It finds an varepsilon-stationary point after O(varepsilon ^{-2}) computation and achieves local quadratic convergence for zero-residual problems under a local error bound condition. Numerical results on compressed sensing and matrix factorization show that our method converges faster in many cases than existing methods.

A consensus algorithm based on multi-agent system with state noise and gradient disturbance for distributed convex optimization

Almost all systems are inevitably subject to various uncertainties or disturbances from the external environment in practical applications. Taking these factors into consideration, in this paper a distributed algorithm with state noise and gradient disturbance is proposed for solving distributed optimization problem with closed convex set constraint based on multi-agent system under weight-balanced graph. Moreover, based on the gradient tracking and projection methods, the proposed distributed algorithm with gradient tracking can improve the convergence rate by introducing a projection error term and an auxiliary parameter. In contrast to some existing constrained distributed gradient algorithms, the proposed one can make the convergence faster and enhance the performance of convergence. The proposed algorithm is illustrated with two simulation examples to show its effectiveness and robustness.

Generalized Nash equilibrium seeking algorithm design for distributed multi-cluster games

In this article, we investigate multi-cluster games, where the cost functions of every player depends on its own decision as well as the decisions of players in other clusters. The decisions of players is constrained by not only local convex set constraints but also coupling constraints. For the purpose of seeking the variational generalized Nash equilibrium (GNE) of the multi-cluster game, we design a distributed algorithm by virtue of gradient descent and projections. Moreover, the asymptotical convergence of the algorithm to the variational GNE is analyzed with the help of Lyapunov stability theory and variational analysis. Finally, a simulation example is proposed to verify the validity of the algorithm.

A Distributed Resource Allocation Algorithm for the General Linear Multiagent Systems

We study the distributed resource allocation problem for heterogeneous multiagent systems over an undirected graph, an essential issue in multiagent system coordinated control and complex network system control. The decision variable is subject to global equality and local convex set constraints, and the objective is smooth and convex. It aims to minimize the global objective function by exchanging neighboring information between agents. An adaptive distributed algorithm is designed using the distance function-based exact penalty function method. A state- and time-based triggering condition is designed to avoid continuous communication and reduce the communication burden. From a random initial state, the proposed algorithm asymptotically converges to the optimal value by means of LaSalle’s invariance principle. Finally, examples are provided to demonstrate the effectiveness of our algorithm.