A continuous-time consensus algorithm using neurodynamic system for distributed time-varying optimization with inequality constraints
Introduction
Compared with centralized optimization, distributed optimization has the ability to quickly deal with large-scale optimization problems by requiring only local information of agents rather than global information. Hence, it has attracted great attention of researchers in recent years due to its broad applications such as, in smart grids, sensor networks, machine learning, etc.(e.g., [1], [2], [3]). The aim is to minimize the sum of the agents’ local objective functions under certain conditions, where each agent only needs to share local information with their neighbors. With the development of research on distributed problems, the discrete-time method and the continuous-time method are generated. In [4], a discrete-time distributed primal-dual subgradient method was proposed for multiagent optimization. In [5], a continuous-time auxiliary system was designed for a class of nonlinear multi-agent systems.
In real-time optimization, neural networks as parallel computational models are wildly used(see [6], [7], [8]), in which the gradient method has been well developed for neurodynamic optimization. For example, in [9], a one-layer recurrent neural network was proposed based on gradient method for constrained pseudoconvex optimization. In this latter, a generalized neural network was proposed in [10] based on a novel auxiliary function and gradient method for distributed nonsmooth optimization. In [11], both continuous-time and discrete-time distributed optimization algorithms were proposed based on collective neurodynamic system for distributed constrained optimization.
Generally, a common assumption for literature is that objective functions and constraints are time-invariant (see [12], [13], [14]). However, a number of interesting applications are time-varying. Resource allocation in time-varying environment [15], traffic engineering [16], robot navigation [17] and online optimization [18] are all typically time-varying problems. Since the solution of the time-varying problem is changing over time, it’s impossible to find the optimal solution precisely. Therefore, the primary target is to design algorithms to enable the decision variable to track the time-varying solution in real-time. The basic idea is to sample the time-varying problem at particular times and solve the corresponding sequence of optimization problems. As a result, the original problem can be seen as a time-invariant problem in each time interval (, ). There are already existing majority of literature by using this idea to solve time-varying problems. In [19], a discrete gradient method was proposed to solve an nonstationary unconstrained optimization problem. In [20], discrete time-sampling algorithms were proposed to track the optimal solution of the time-varying problem.
However, many existing works on time-varying optimization do not take time-varying constraints into consideration. In [21], gradient-based searching methods were proposed for distributed unconstrained time-varying quadratic optimization problems. In [22], a distributed finite-time algorithm was designed to solve unconstrained time-varying optimization for continuous-time multi-agent system. To the best of my knowledge, very few continuous-time distributed algorithms for time-varying constrained optimization problem have been discussed. Therefore, this paper discusses a distributed time-varying convex optimization problem with inequality constraints based on neurodynamic system. The main contributions of the paper are as follows:
(1) Many existing works on distributed optimization are build on the time-invariant objective function and time-invariant inequality constraints. However, a number of interesting applications are time-varying, such as Resource allocation in time-varying environment, traffic engineering, robot navigation, etc. The paper discusses a distributed time-varying convex optimization problem with time-varying inequality constraints based on neurodynamic system.
(2) A distributed continuous-time consensus algorithm, which can track the optimal solution of the time-varying convex problem, is proposed by using only local information and local interaction. Log-barrier penalty functions are used to include the inequality constraints into the objective function and signum functions are used for agents to reach consensus.
(3) The proposed algorithm is continuous instead of discontinuous. The consistency and convergence are both analyzed by using Lyapunov theory. Theoretical studies show that all agents are able to achieve consensus and track the time-varying optimal solution by utilizing the proposed methods.
The rest of this paper is organized as follows. In Section 2, the distributed time-varying constrained optimization problem and related preliminaries are presented. In Section 3, a distributed continuous-time consensus algorithm is proposed to solve the distributed time-varying convex optimization problem using neurodynamic system. In Section 4, the tracking properties of the proposed algorithm is analyzed. In Section 5, two numerical examples are presented to show the effectiveness of the proposed algorithm. Finally, Section 6 gives the concluding remarks.
Section snippets
Preliminaries
Notations: The first and the second partial derivatives of the objective function with respect to can be denoted as and , respectively. Let be the derivative of with respect to . denotes an ()-dimensional identity matrix. () denotes an n-dimensional vector (an ()-dimensional matrix) with all elements being 1 (0). denotes the Kronecker product. Denote the set of real and non-negative numbers by and respectively. Let and be the
Distributed continuous‐time consensus algorithm
In this section, a distributed continuous-time consensus algorithm for problem (1) will be proposed. Similar to the distributed time-invariant optimization problem considered in [12] and the distributed time-varying optimization problem considered in [21], let be the ith agent’s estimation of the optimal solution , and . Based on the Assumption 1, problem (1) can be written as follows:
Trajectory tracking analysis
Based on the previous analysis that the optimal solution is changing over time, the distributed time-varying optimization problems in the paper can be understood as a tracking problem. Theorem 2 Consider the time-varying optimization problem in Eq. (5) and the proposed algorithm in Eq. (8). Define as the solution of Eq. (8) with initial states , and is a decreasing exponential function, satisfying . When conditions: and are met, there
Simulations
Example 1 A distributed time-varying convex optimization problem with three agents is considered, i.e., .
Adding time-varying inequality constraints to all agents, agent 1 has local time-varying constraint as . Agent 2 has local time-varying constraint as . Agent 3 has local time-varying constraint as .
It can be clearly seen that each agent has different objective functions and
Conclusion
This paper addressed the distributed optimization problem with time-varying strongly convex objective functions and convex inequality constraints using neurodynamic system. Based on some reasonable assumptions, by employing the signum function and log-barrier penalty functions, a distributed continuous-time consensus algorithm is proposed to achieve the consensus of agents and track the optimal solution of the time-varying convex problem. Taking both time-varying equation constraints and
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper
Acknowledgments
This work is supported by Natural Science Foundation of China (Grant nos: 61773320), Fundamental Research Funds for the Central Universities (Grant no. XDJK2020TY003), and also supported by the Natural Science Foundation Project of Chongqing CSTC (Grant no. cstc2018jcyjAX0583).
References (37)
- et al.
Distributed wireless power transfer in sensor networks with multiple mobile chargers
Comput. Netw.
(2015) - et al.
A one-layer recurrent neural network for constrained pseudoconvex optimization and its application for dynamic portfolio optimization
Neural Netw.
(2012) - et al.
A generalized neural network for distributed nonsmooth optimization with inequality constraint
Neural Netw.
(2019) - et al.
A consensus algorithm based on collective neurodynamic system for distributed optimization with linear and bound constraints
Neural Netw.
(2020) - et al.
Distributed gradient algorithm for constrained optimization with application to load sharing in power systems
Syst. Control Lett.
(2015) - et al.
A penalty-like neurodynamic approach to constrained nonsmooth distributed convex optimization
Neurocomputing
(2020) Gradient methods for nonstationary unconstrained optimization problems
Autom. Remote Control
(2005)- et al.
Nonlinear Programming: Theory and Algorithms
New York, NY, USA: Wiley
(2005) - et al.
Distributed intrusion detection system in a multi-layer network architecture of smart grids
IEEE Trans. Smart Grid
(2011) - et al.
Communication efficient distributed machine learning with the parameter server
Adv. Neural Inf. Process. Syst.
(2014)
Distributed primal?dual subgradient method for multiagent optimization via consensus algorithms
IEEE Trans. Syst. Man Cybern. Part B (Cybern.)
Distributed time-varying convex optimization for a class of nonlinear multiagent systems
IEEE Trans. Autom. Control
Neural networks and physical systems with emergent collective computational abilities
Proc. Natl. Acad. Sci. U.S.A.
Gradient methods for the optimization of dynamical systems containing neural networks
IEEE Trans. Neural Netw.
Neural Networks: A Comprehensive Foundation (3rd Edition)
Continuous-time algorithm for approximate distributed optimization with affine equality and convex inequality constraints
IEEE Trans. Syst. Man Cybern. Syst.
Distributed time-varying resource allocation optimization based on finite-time consensus approach
IEEE Control Syst. Lett.
Cited by (10)
Distributed dual consensus algorithm for time-varying optimization with coupled equality constraint
2024, Applied Mathematics and ComputationET-PDA: An event-triggered parameter distributed accelerated algorithm for economic dispatch problems
2024, Journal of the Franklin InstituteDistributed online convex optimization with multiple coupled constraints: A double accelerated push–pull algorithm
2023, Journal of the Franklin InstituteTime-varying distributed optimization problem with inequality constraints
2023, Journal of the Franklin Institute