A continuous-time consensus algorithm using neurodynamic system for distributed time-varying optimization with inequality constraints

doi:10.1016/j.jfranklin.2021.07.007

Journal of the Franklin Institute

Volume 358, Issue 13, September 2021, Pages 6741-6758

https://doi.org/10.1016/j.jfranklin.2021.07.007 Get rights and content

Abstract

In this paper, a distributed time-varying convex optimization problem with inequality constraints is discussed based on neurodynamic system. The goal is to minimize the sum of agents’ local time-varying objective functions subject to some time-varying inequality constraints, each of which is known only to an individual agent. Here, the optimal solution is time-varying instead of constant. Under an undirected and connected graph, a distributed continuous-time consensus algorithm is designed by using neurodynamic system, signum functions and log-barrier penalty functions. The proposed algorithm can be understood through two parts: one part is used to reach consensus and the other is used to achieve gradient descent to track the optimal solution. Theoretical studies indicate that all agents will achieve consensus and the proposed algorithm can track the optimal solution of the time-varying convex problem. Two numerical examples are provided to validate the theoretical results.

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 ( $t \in [t_{k}, t_{k + 1})$ , $k \in {0, 1, 2, \dots}$ ). 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 $f (x, t)$ with respect to $x$ can be denoted as $\nabla_{x} f (x, t)$ and $\nabla_{x x} f (x, t)$ , respectively. Let $\dot{x}$ be the derivative of $x$ with respect to $t$ . $I_{m}$ denotes an ( $m \times m$ )-dimensional identity matrix. $1_{n}$ ( $0_{m n}$ ) denotes an n-dimensional vector (an ( $m \times n$ )-dimensional matrix) with all elements being 1 (0). $\otimes$ denotes the Kronecker product. Denote the set of real and non-negative numbers by $R$ and $R^{+}$ respectively. Let ${∥ x ∥}_{1}$ and $∥ x ∥$ 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 $x_{i} (t)$ be the ith agent’s estimation of the optimal solution $x^{*} (t)$ , and $\bar{x} (t) = {[x_{1}^{T} (t), x_{2}^{T} (t), \dots, x_{n}^{T} (t)]}^{T}$ . Based on the Assumption 1, problem (1) can be written as follows: $\bar{x} (t) = {\begin{matrix} \arg & \min \bar{f} (\bar{x}, t) = \sum_{i = 1}^{n} f_{i} (x_{i}, t) + \frac{1}{2} β {\bar{x}}^{T} L \otimes I_{m} \bar{x}, \\ s.t. & L \otimes I_{m} \bar{x} = 0_{m n}, \\ g_{j} (x_{i}, \end{matrix}$

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 $x (t)$ as the solution of Eq. (8) with initial states $x (0) \in R^{n}$ , $c (t) > 0$ and $s (t)$ is a decreasing exponential function, satisfying $s (0) > max_{1 \leq j \leq p} g_{j} (x_{i}, t)$ . When conditions: $lim_{t ⟶ \infty} c (t) = \infty$ and $lim_{t ⟶ \infty} s (t) = 0$ are met, there

Simulations

Example 1

A distributed time-varying convex optimization problem with three agents is considered, i.e., $f (x, t) = \sum_{i = 1}^{3} f_{i} (x, t)$ .

$f_{1} (t) = \frac{1}{2} x^{2} + 2 x \sin t + 1$

$f_{2} (t) = \frac{1}{2} {(x - 2 \sin t)}^{2} + 3 x$

$f_{3} (t) = \frac{1}{2} x^{2} + 3 x + 7 \sin t$

Adding time-varying inequality constraints to all agents, agent 1 has local time-varying constraint as $g_{1} (t) = x \sin t + 1 \leq 0$ . Agent 2 has local time-varying constraint as $g_{2} (t) = x - \sin t \leq 0$ . Agent 3 has local time-varying constraint as $g_{3} (t) = x \sin t - 3 \leq 0$ .

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).

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A continuous-time consensus algorithm using neurodynamic system for distributed time-varying optimization with inequality constraints

Abstract

Introduction

Section snippets

Preliminaries

Distributed continuous‐time consensus algorithm

Trajectory tracking analysis

Simulations

Conclusion

Declaration of Competing Interest

Acknowledgments

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