# A Distributed Resource Allocation Algorithm for the General Linear Multiagent Systems

Authors: M. Li; B. Shao; X. Shi
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

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.


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## Body

## 1. Introduction

With the increasing scale and complexity of modern networked control systems owing to their potential applications in many fields, including, but not limited to large-scale machine learning, resource scheduling in energy networks and source location in wireless network systems [1, 2], distributed constrained convex optimization problems based on multiagent systems have attracted extensive attention in recent decades. The primary purpose of the distributed constrained convex optimization problem is to minimize the sum of all the local objective functions, whereas the decision variable is subject to some constraints. The conventional centralized method of gradient flow algorithm and intelligent optimization algorithm cannot be used effectively in distributed multiagent systems, since a central control agent is required for collecting and calculating the global objective functions and constraints over the whole network. Recently, an increasing number of scholars have designed a series of distributed optimization algorithms, which significantly reduce the computational burden and thus become a critical branch of optimization [3].This study mainly focuses on a special class of distributed convex optimization problems, namely, resource allocation problems, in which each agent has its own known local objective function and local constraints. All the agents are coupled with each other by a global equality constraint. Many distributed optimization algorithms have been developed to solve this problem. For example, the consensus-based method in [4], surplus-based algorithm in [5], push-sum-based economic dispatch algorithm in [6], Nesterov accelerated economic dispatch algorithm in [7], and references therein.In the discrete-time algorithms proposed above, the dynamics of each agent are not considered. With the development of big data and cyber-physical systems, an increasing number of scholars have tried to design continuous-time algorithms and analyze their convergence properties employing the well-defined Lyapunov stability [8]. For the unconstrained optimization problem, an event-triggered distributed optimization algorithm based on a primal-dual structure is proposed to reduce the communication burden [9]. Besides, the historical information of the dual variable is applied to balance the local gradient value. It thus ensures the constant control parameter instead of the diminishing step size in [6]. For the resource allocation problem, with only a global equality constraint, a fixed-time distributed algorithm is designed in [10], where the individual gradient information of neighboring agents is required to obtain the optimal allocation result within a fixed time. In addition, the initial state must satisfy the global equality constraint, which reduces the universality of the algorithm and cannot deal with the dynamic resource allocation problem efficiently. An initialization-free distributed algorithm by virtue of the sliding mode control scheme is established in [11] to eliminate this initialization constraint. Unlike the fixed-time method, an accelerated distributed algorithm is designed on the basis of the continuous-time Nesterov acceleration method [12]. For the resource allocation problem involving both global equality and local set constraints, an initialization-free distributed algorithm over an undirected network is proposed based on the projection operator [13]. However, the projection operator is applied to the state variable. Instead, an output-based distributed algorithm for the undirected and directed networks is designed in [14]. Different from the previous projection-based algorithms, a novel projection-based distributed optimization algorithm is designed in [15], where the dual projection is applied to the dual variable and then directly obtains the corresponding state value instead of the differential of state. Thus, we obtain a relatively high convergence rate. With the aid of the primal-dual gradient flow approach, an initialization-free distributed optimization algorithm is provided to solve the nonsmooth constrained optimization problems over the undirected network in [16], in which the dual variable of local convex function inequality constraint (penalty factor) is adaptively changed until the local inequality constraint is satisfied. Similarly, for the dual variable of the global equality constraint, the corresponding control parameter adaptively increases until a consensus is achieved in [17]. Sometimes, the disturbance is unavoidable. To suppress these disturbances, an internal model observer was designed to track the disturbance and then develop an optimization algorithm for the nonsmooth resource allocation problem [18].It is noted that the above-mentioned distributed algorithms focus on the first-order multiagent systems. However, second-order and higher-order multiagent systems are more common in practical applications. Therefore, several studies try to design distributed algorithms for the linear and nonlinear second-order multiagent systems [19–23]. Based on the tracking control idea, the resource allocation with equality constraint is researched in [24] for the general linear heterogeneous multiagent systems. For the same problem, a predefined time-convergent distributed optimization algorithm is designed in [25], in which a time-varying control parameter is utilized for obtaining higher feedback gain. Unlike [25], the exact solution instead of the suboptimal solution of the resource allocation problem is obtained in [26] within a fixed time. Notably, the local constraints are not considered in the above results. Practically, agents’ decisions are subject to some local constraints, such as the security considered in robots and the production capacity of the generator in smart grids. Although a novel adaptive distributed algorithm is proposed in [27] based on the Karush–Kuhn–Tucker (KKT) condition, the proposed algorithm tends to diverge as time goes to infinity. In addition, with the continuous expansion of multiagent systems’ scale, each agent’s communication burden becomes heavier and heavier. Thus, the event-triggered communication mechanism is introduced in the distributed coordination control problem of multiagent systems [28–31]. For the resource allocation problem, many discrete-time communication algorithms have been proposed by means of the event-triggered communication mechanism in recent years, such as the state-based triggering condition for the quadratic economic dispatch problem in [32] and general convex energy management in [33], the differential projection-based algorithm in [34], and the resource allocation problem for second-order multiagent systems [35].This paper aims to design an adaptive distributed optimization algorithm for the resource allocation problem with heterogeneous multiagent systems over an undirected network. The contributions of this study are as follows: (1) Compared with [24–26], the local convex set constraint is considered in this paper. Furthermore, an adaptive distributed optimization is proposed utilizing the distance-function-based exact penalty function method. (2) To reduce the communication burden, an event-triggered algorithm is designed.We organize the remainder of this study into four sections. In Section2, the resource problem formulation over a general linear heterogeneous multiagent system is provided, and the corresponding communication network is given after that. Section 3 provides a state-based distributed algorithm and its convergence analysis. The effectiveness of our proposed algorithms is given in Section 4, and the conclusion of this study is rendered in Section 5.Notations:T denotes the transpose symbol. ℝ represents the real number set. 1n,0n, and In denote all ones, all zeros, and identity matrix with n dimensions. Occasionally, the subscript for the all-zero vector is omitted. ⊗ denotes the Kronecker product. intΩ and bdΩ denote the interior point set and boundary point set, respectively. The distance function is defined by dx,Ω=infx−y,∀y∈Ω, where the nearest point of x is denoted by PΩx=argmin∀y∈Ωx−y. The normal cone of the set Ω is defined by NΩx=z|zTy−x≤0,∀y∈Ω. ln⋅ is the natural logarithmic function.


## 2. Communication Network and Problem Formulation

### 2.1. Communication Network

The communication network of the multiagent system consists ofn agents and is represented by G=V,A, in which the agent set is denoted by V=1,2,…,n and the adjacency-weighted matrix is denoted by A. In this paper, the graph is assumed to be undirected. If there exists a communication link between agents i and j, then aij=aji>0; aij=aji=0, otherwise. Besides, we assume aii=0,∀i∈V. For agents i,j, if there exists a series of agents such that agent i can communicate with agent j, then agents i and j are reachable from each other. If this property holds for any two agents in G, then G is a connected network. Finally, the Laplacian matrix L is defined as Lij=−aij,∀i≠j,Lii=∑j=1,j≠inaij. Besides, the corresponding eigenvalues for a connected undirected network are orderly defined as 0=ρ1<ρ2≤,…,≤ρn.Lemma 1.
(see [15]). If G is undirected and connected, then we have(1)RΛ1−1RTL=LRΛ1−1RT=Kn,where Kn=In−1/n1nT1n,Λ1=diagρ2,…,ρn, and 1/n1n,R∈ℝn×n is an orthogonal matrix with R∈ℝn×n−1.


### 2.2. Problem Formulation

In the distributed resource allocation problem, all agents collaboratively seek the optimal solution through neighboring information exchange. Specifically, the individual’s objective function is denoted byfiyi:ℝq⟶ℝ with decision variable yi. The total network resource is denoted as ∑i=1ndi with local demand resource di. In addition, the decision variable yi is subject to a local convex set Ωi. Eventually, the mathematical form can be summarized as follows:(2)minfy=∑i=1nfiyi,∑i=1nyi=∑i=1ndi,yi∈Ωi,∀i∈V,where fy=∑i=1nfiyi is the global objective function. From (2), it is noted that the useful information di,fiyi is only known to themselves. However, we need to minimize the global objective function under the coupled global equality constraint. This increases the difficulty of algorithm design. The above mathematical form can be found in many practical applications. For example, the economic dispatch problem in smart grid, where it aims to give the optimal power generation plan for the distributed generator systems.The dynamic of each agent is described as follows:(3)xi=Aixi+Biui,yi=Cixi,where xi∈ℝni,ui∈ℝpi and yi∈ℝq are the state variable, input variable, and output variable for every agent i, respectively. Besides, yi is the decision variable of problem (2). Ai∈ℝni×ni,Bi∈ℝni×pi, and Ci∈ℝq×ni are the state, input, and output matrices, respectively. From the dynamics of (3), it is noted that the distributed resource allocation in this paper is an optimal output control problem, which aims to design the suitable controller ui such that the output yi achieves the optimal solution of problem (2). For this purpose, the following assumptions are required in similar works [5, 24–26].Assumption 1.
For system (3), Ai,Bi is controllable, and the following rank condition is satisfied:(4)rankCiBi0−AiBiBi=ni+q.Lemma 2.
(see [36]). Considering Assumption 1, there exist matrixes Γi,Φi,Ψi such that the following equations hold:(5)CiΨi=Iq,BiΦi=AiΨi,BiΓi=Ψi.Assumption 2.
The global functionfy is μ− strongly convex with a constant number μ>0. For problem (2), there exists at least one point y˜, such that ∑i=1ny˜i=∑i=1ndi and y˜i∈Ωi,∀i∈V (Slater’s condition).
With the help of the distance function, problem (2) can be rewritten as follows:(6)minf^y=∑i=1nfiyi+cidyi,Ωi,∑i=1nyi=∑i=1ndi,where ci is the penalty factor. According to the gradient flow method, we need to calculate the gradient/subgradient of cidyi,Ωi. Therefore, the following lemma is provided.Lemma 3.
Sincedyi,Ωi is convex, then the gradient of dyi,Ωi is defined as follows:(7)∂dyi,Ωi=0,yi∈intΩi,NΩiyi∩B0,1,yi∈bdΩi,yi−PΩiyidyi,Ωi,yi∉Ωi.
In addition,B0,1 denotes a set B0,1=zi|zi≤1. Obviously, if yi∈Ωi, we have ∂dyi,Ωi⊂NΩiyi.Lemma 4.
(see [18]). y∗ is the optimal solution of problem (2); if there exists a vector ϑ∗∈ℝq such that(8)ζ∗+NΩy∗−1n⊗ϑ∗=0,∑i=1nyi∗=∑i=1di,where ζ∗∈∂fy∗⊂ℝnq.
The LaSalle’s invariant principle is widely used in the algorithm design of the distributed optimization problem, such as the distributed optimal coordinated problem [9] and distributed optimal resource allocation problem [10, 13, 14, 16–18, 27]. Therefore, we provide the LaSalle’s invariant principle in the following lemma.Lemma 5.
(see [37, 38]). For an autonomous system with the following given nonlinear dynamics: p˙t=Φpt, where the map Φ:ℝm⟶ℝm. Let V:ℝm⟶ℝ be a continuous Lyapunov-like function and Q be a compact invariance set for the given system. If we have V˙p≤0,∀p∈Q, then all the solutions starting from set Q converge to element of set E, where E is largest invariance set of the set p|V˙p=0.


## 2.1. Communication Network

The communication network of the multiagent system consists ofn agents and is represented by G=V,A, in which the agent set is denoted by V=1,2,…,n and the adjacency-weighted matrix is denoted by A. In this paper, the graph is assumed to be undirected. If there exists a communication link between agents i and j, then aij=aji>0; aij=aji=0, otherwise. Besides, we assume aii=0,∀i∈V. For agents i,j, if there exists a series of agents such that agent i can communicate with agent j, then agents i and j are reachable from each other. If this property holds for any two agents in G, then G is a connected network. Finally, the Laplacian matrix L is defined as Lij=−aij,∀i≠j,Lii=∑j=1,j≠inaij. Besides, the corresponding eigenvalues for a connected undirected network are orderly defined as 0=ρ1<ρ2≤,…,≤ρn.Lemma 1.
(see [15]). If G is undirected and connected, then we have(1)RΛ1−1RTL=LRΛ1−1RT=Kn,where Kn=In−1/n1nT1n,Λ1=diagρ2,…,ρn, and 1/n1n,R∈ℝn×n is an orthogonal matrix with R∈ℝn×n−1.


## 2.2. Problem Formulation

In the distributed resource allocation problem, all agents collaboratively seek the optimal solution through neighboring information exchange. Specifically, the individual’s objective function is denoted byfiyi:ℝq⟶ℝ with decision variable yi. The total network resource is denoted as ∑i=1ndi with local demand resource di. In addition, the decision variable yi is subject to a local convex set Ωi. Eventually, the mathematical form can be summarized as follows:(2)minfy=∑i=1nfiyi,∑i=1nyi=∑i=1ndi,yi∈Ωi,∀i∈V,where fy=∑i=1nfiyi is the global objective function. From (2), it is noted that the useful information di,fiyi is only known to themselves. However, we need to minimize the global objective function under the coupled global equality constraint. This increases the difficulty of algorithm design. The above mathematical form can be found in many practical applications. For example, the economic dispatch problem in smart grid, where it aims to give the optimal power generation plan for the distributed generator systems.The dynamic of each agent is described as follows:(3)xi=Aixi+Biui,yi=Cixi,where xi∈ℝni,ui∈ℝpi and yi∈ℝq are the state variable, input variable, and output variable for every agent i, respectively. Besides, yi is the decision variable of problem (2). Ai∈ℝni×ni,Bi∈ℝni×pi, and Ci∈ℝq×ni are the state, input, and output matrices, respectively. From the dynamics of (3), it is noted that the distributed resource allocation in this paper is an optimal output control problem, which aims to design the suitable controller ui such that the output yi achieves the optimal solution of problem (2). For this purpose, the following assumptions are required in similar works [5, 24–26].Assumption 1.
For system (3), Ai,Bi is controllable, and the following rank condition is satisfied:(4)rankCiBi0−AiBiBi=ni+q.Lemma 2.
(see [36]). Considering Assumption 1, there exist matrixes Γi,Φi,Ψi such that the following equations hold:(5)CiΨi=Iq,BiΦi=AiΨi,BiΓi=Ψi.Assumption 2.
The global functionfy is μ− strongly convex with a constant number μ>0. For problem (2), there exists at least one point y˜, such that ∑i=1ny˜i=∑i=1ndi and y˜i∈Ωi,∀i∈V (Slater’s condition).
With the help of the distance function, problem (2) can be rewritten as follows:(6)minf^y=∑i=1nfiyi+cidyi,Ωi,∑i=1nyi=∑i=1ndi,where ci is the penalty factor. According to the gradient flow method, we need to calculate the gradient/subgradient of cidyi,Ωi. Therefore, the following lemma is provided.Lemma 3.
Sincedyi,Ωi is convex, then the gradient of dyi,Ωi is defined as follows:(7)∂dyi,Ωi=0,yi∈intΩi,NΩiyi∩B0,1,yi∈bdΩi,yi−PΩiyidyi,Ωi,yi∉Ωi.
In addition,B0,1 denotes a set B0,1=zi|zi≤1. Obviously, if yi∈Ωi, we have ∂dyi,Ωi⊂NΩiyi.Lemma 4.
(see [18]). y∗ is the optimal solution of problem (2); if there exists a vector ϑ∗∈ℝq such that(8)ζ∗+NΩy∗−1n⊗ϑ∗=0,∑i=1nyi∗=∑i=1di,where ζ∗∈∂fy∗⊂ℝnq.
The LaSalle’s invariant principle is widely used in the algorithm design of the distributed optimization problem, such as the distributed optimal coordinated problem [9] and distributed optimal resource allocation problem [10, 13, 14, 16–18, 27]. Therefore, we provide the LaSalle’s invariant principle in the following lemma.Lemma 5.
(see [37, 38]). For an autonomous system with the following given nonlinear dynamics: p˙t=Φpt, where the map Φ:ℝm⟶ℝm. Let V:ℝm⟶ℝ be a continuous Lyapunov-like function and Q be a compact invariance set for the given system. If we have V˙p≤0,∀p∈Q, then all the solutions starting from set Q converge to element of set E, where E is largest invariance set of the set p|V˙p=0.


## 3. Algorithm Design and Convergence Analysis

### 3.1. Continuous-Time Communication Algorithm

From Lemma4, it can be noted that the optimal solution y∗ must satisfy the equality constraint and the key is to seek the optimal dual Lagrange multiplier ϑ∗. Therefore, the following designed algorithm has the following three aspects: (1) the historical information of the local Lagrange multipliers λi is used for trading off the difference between real output yi and local demand resource di; (2) the multiagent consensus method is used to guarantee that all local Lagrange multipliers λi converge to the same optimal dual Lagrange multiplier ϑ∗; (3) since we cannot directly control the output value yi, we provide an auxiliary variable ηi, which is driven by the gradient information of the local objective function and local distance function. Subsequently, based on the deviation tracking control scheme, a specific algorithm is designed as follows:(9a)ui=−Kixi++Γiη˙i−Φi−KiΨiηi,(9b)η˙i=−∇fiyi+ci∂dyi,Ωi+λi,(9c)λ.i=−yi+di−α∑j=1naijλi−λj−βzi,(9d)z˙i=α∑j=1naijλi−λj,(9e)c˙i=dyi,Ωi,where zi is used to store the historical information of ∑j=1naijλi−λj and a proportional-integral control scheme is developed combining the terms ∑j=1naijλi−λj and zi. The control parameters α,β are greater than zero. Since A,B is controllable, then there exists feedback gain K such that A−BK is Hurwitz-stable, which means that all the eigenvalues of matrix A−BK have the negative real part. To guarantee the optimality, we need to set ∑i=1nzi0=0,ci0=0 and other variables xi,yi,λi,ηi initialization-free.Remark 1.
Many practical systems can be formulated by the general linear multiagent systems with dynamic (3) or can be linearized to (3). Typically, the high-order system is a special linear system. For example, the economic dispatch problem over the turbine-generator system, where the dynamics of each generator is described by a linear system [22]. However, the rank assumption of the turbine generator is not held. Besides, the dynamics of aircraft systems can also be linearized to a linear system [24, 25], and our proposed algorithm is effective for those aircraft systems.Remark 2.
To solve the distributed constrained optimization problem over a heterogeneous multiagent system, several distributed optimization algorithms have been proposed. In detail, based on the tracking control idea, the resource allocation problem incorporating with a global equality constraint is solved in [24]. Besides, the disturbance is suppressed by the designed internal-model observer. For the same problem, the solution is obtained within predefined time using the time-based generator technology in [25]. Similarly, the solution can be achieved within fixed time in [26]. However, the local constraints for each agent are not considered in the above three papers. More importantly, the above results cannot be directly extended to problem (2) by using the well-known projection operator. Moreover, only the suboptimal solution is obtained at the predefined time in [25], and the rank condition in [26] is stronger than ours. The global equality constraint is not considered in [39]. In addition, the dynamics of each agent are more complex than those in (3), in which a projection operator is applied to the output variable, that is, PΩiyi∈Ωi, instead of yi∈Ωi. The output variable yi is actually unconstrained in [39]. Although the same problem is solved in [27] under a milder rank condition, the designed algorithm depends on an equality CiBiKi=CiAi, and thus, the designed algorithm can be reduced to a first-order multiagent system. Also because of this equality, the designed algorithm will diverge as time goes to infinity. This equality is difficult to hold strictly, and then the small deviations may lead to system divergence. In addition, the event-triggered communication is also not considered in [27].Remark 3.
Among algorithm (9a)–(9e), the state information xi is used in (9a). Thus, the state-based feedback control is constructed. However, the state information xi is not available. For example, some state quantities do not have practical significance. A state observer or state estimator can be designed to estimate the unaccessible internal state. For instance, the designed linear Luenberger observer converges to the real state very quickly when the original system is observable [40]. Besides, the sliding mode observer is an alternative method [41].
Based on Lemma2, together with dynamic equation (3), we have the following compact form:(10)x˙=A−BKx−Ψη+Ψη˙,η˙=−∇fy+DT∂dy,Ω+λ,λ.=−y−d−αL⊗Iqλ−βz,z˙=αL⊗Iqλ,c˙=dy,Ω,where x,y,η,λ,z,c,∇fy are the aggregate vector of themselves. D,K,Ψ are the aggregate diagonal matrix of vectors ci,Ki,Ψi. Besides, we can see that system (10) is an autonomous system since the time t is not explicitly displayed in right side of (10) [42]. Next, let x∗,y∗,λ∗,z∗,c∗ be the equilibrium point of system (10); then, we prove that y∗ is the optimal solution to problem (2) in the following lemma.Lemma 6.
Ifx∗,η∗,λ∗,z∗,c∗ is the equilibrium of system (10), then y∗=Cx∗ is the optimal allocation scheme of problem (2).Proof.
Since the considered network is undirected and connected,λ=1n⊗λ0,λ0∈ℝ follows L⊗Iqλ=0. In addition, from the definition of ∂dy∗,Ω, the first equality holds owing to the facts ci∗∂dyi∗,Ωi⊂Ωi and ci∗≥0. Then, in terms of the zero-sum property of z0, the second equality is obtained, which follows that y∗−d+αL⊗Iqλ∗+βz∗=0. Thus, from Lemma 4, we can prove that y∗ is the optimal solution of problem (2).
The convergence proof is provided in Theorem1.Theorem 1.
Under Assumptions1 and 2, the feedback gain K is selected such that A−BK is Hurwitz stable; then, the output value yi generated from system (3) with controllers (9a)–(9e) achieves the optimal solution of problem (2). Besides, the control parameters satisfy that α>0,β>1/2μ.Proof.
Letx¯=x−x∗,y¯=y−y∗,η¯=η−η∗,z¯=z−z∗,c¯=c−c∗,λ¯=λ−λ∗; then, we choose the Lyapunov function V1 as V1=Cx¯2+1/2λ¯2+1/2λ¯+z¯2+β/αz¯TRΛ1−1RTz¯+1/2c¯2. Taking the time derivative over system (10) yields(11)V˙1=2Cx¯TCA−BKx¯−Ψη¯−2Cx¯Th+2Cx¯Tλ¯−λTy−αλ¯TL⊗Iqλ¯−2βλ¯Tz¯−βz¯2−λ¯Ty¯−y¯Tθ¯+2βλ¯Tθ¯+c¯Tdy,Ω=2Cx¯TCA−BKx¯−Ψη¯−2Cx¯Th−αλ¯TL⊗Iqλ¯−βz¯2−y¯Tz¯+c¯Tdy,Ω,in which h=∂fy−∂fy∗+DT∂dy,Ω−D∗T∂dy∗,Ω. Next, we focus on the term θ=yi−yi∗Tci∂dyi,Ωi−ci∗∂dyi∗,Ωi+ci−ci∗dyi,Ωi.
For the first case,yi∈Ωi. Then, we have dyi,Ωi=0 and ∂dyi,Ωi∈NΩiyi,∂dyi∗,Ωi∈NΩiyi. Based on the definition of the normal cone over set Ωi, we have yi−yi∗T∂dyi∗,Ωi≤0 and yi∗−yiT∂dyi,Ωi≤0. Thus, one can ensure that θi≤0.
For the second case,yi∉Ωi. We have that ∂dyi,Ωi=yi−PΩiyi/dyi,Ωi follows the definition of the gradient of the distance function. If ci≤ci∗, we have(12)θi=ci∗−ciyi−yi∗T∂dyi∗,Ωi+ci−ci∗dyi,Ωi−ciyi−yi∗T∂dyi,Ωi−∂dyi∗,Ω≤ci∗−ciyi−yi∗T∂dyi∗,Ωi+ci−ci∗dyi,Ωi≤ci∗−cidyi,Ωi+ci−ci∗dyi,Ωi=0.
In the above procedure, considering the convexity property of distance function, we have used the fact thatyi−yi∗T∂dyi,Ωi−∂dyi∗,Ω≥0 in the first inequality and its equivalent form yi−yi∗T∂dyi∗,Ωi≤dyi,Ωi−dyi∗,Ωi in the second inequality.
Ifci>ci∗, we conclude that(13)κi=−ci−ci∗yi,ΩiT∂dyi,Ωi+ci−ci∗dyi,Ωi−ci∗yi−yi∗T∂dyi,Ωi−∂dyi∗,Ω≤−ci−ci∗yi−yi∗T∂dyi,Ωi+ci−ci∗dyi,Ωi=−ci−ci∗yi−PΩiyi+PΩiyi−yi∗T⋅yi−PΩiyidyi,Ωi+ci−ci∗dyi,Ωi≤−ci−ci∗yi−PΩiyiTyi−PΩiyidyi,Ωi+ci−ci∗dyi,Ωi=0.
We have used the fact thatPΩiyi−yi∗Tyi−PΩiyi≥0 by means of the property of projection operator in the above second inequality. Therefore, we also can ensure that θi≤0. Substituting this result into (11), we have(14)V˙1≤2Cx¯TCA−BKx¯−Ψη¯−2μy¯2−αλ¯TL⊗Iqλ¯−βz¯2−y¯Tz¯≤−μy¯2+2Ac22μx¯−Ψη¯2−αλ¯L⊗Iqλ¯−β−12μθ¯2.
The following inequalities are applied in the above inequality by means of Young’s inequality:(15)2Cx¯TCA−BKx¯−Ψη¯≤12μy¯2+2CAc2μx¯−Ψη¯2,−y¯Tθ¯≤12μy¯2+12μθ¯2.
In the following, we defineV2=x¯−Ψη¯TPx¯−Ψη¯ with positive definite matrix P such that A−BKTP+PA−BK≤−νIN with N=∑i=1nni and ν>2Ac2/2μ. Then, the time derivative of V2 over system (10) gives(16)V˙2=x¯−Ψη¯TA−BKTP+PA−BKx¯−Ψη¯≤−νx¯−Ψη¯2.
Now, combining (14) and (16) gives(17)V˙1+V˙2≤−μy¯2−ν−2Ac22μx¯−Ψη¯2−αλ¯TL⊗Iqλ¯−β−12μz¯2.
From (17), we can conclude that the variable ct is bounded all the time and follows the boundedness of V1+V2. Thus, we have dyi,Ωi⟶0 owing to the dynamic of ci. There is a constant vector c˜ such as c˜i>cit,∀t≥0,i∈V. Let the candidate Lyapunov function be V=V1+V2+1nTc˜−c; then, we have(18)V˙≤−μy¯2−ν−2Ac22μx¯−Ψη¯2−αλ¯TL⊗Iqλ¯−β−12μz¯2−dy,Ω.
It is notable thatV is radially unbounded. Then, using LaSalle’s invariance principle and (18), we can conclude that the proposed algorithm converges to the largest invariance set of the set x˜,η˜,λ˜,z˜,c˜|V˙x˜,η˜,λ˜,z˜,c˜=0. From (18), we directly have y˜=y∗,z˜=z∗,λ˜=λ∗. Since c˜˙=dy˜,Ω=0, we have c˜=c∗. Based on the result x˜=Ψη˜, we have x˜˙=0,η˜˙=0. Finally, we have that x˜,η˜,λ˜,z˜,c˜ is the equilibrium of system (10). Together with Lemma 6, it is claimed that yt converges to the optimal solution of problem (2). Thus, the convergence proof is complete.


### 3.2. Event-Triggered Algorithm

In algorithm (9a)–(9e), neighboring information on the dual variable λj is assumed to be continuously accessible to agent i, which means that agent j must broadcast its Lagrange multiplier λj to its neighbors all the time. This broadcast mechanism requires agents to spend considerable communication resources, and in most cases, this communication is meaningless and redundant. To save the communication resource, motivated by the event-triggered communication mechanism, we modify (9a)–(9e) as follows:(19a)ui=−Kixi+Γiη˙i−Φi−KiΨiηi,(19b)η˙i=−∇fiyi+ci∂dyi,Ωi+λi,(19c)λ.i=−yi+di−α∑j=1naijλ^i−λ^j−βzi,(19d)z˙i=α∑j=1naijλ^i−λ^j,(19e)c˙i=dyi,Ωi,where λ^i is the lasting triggering information of the Lagrange multiplier λi and is defined as λ^i=λitik,t∈tik,tik+1 with the triggering time sequence tik. Additionally, the triggering time is determined by the following triggering condition:(20)tik+1=maxr≥tikr|αLii+2βeit2≤α2qit+δexp−σt,∀t∈tik,r,where the control parameters δ,σ are positive numbers and the estimation error is defined as ei=λ^i−λi. Compared with (9), each agent only broadcasts its information at some discrete time when the triggering condition (20) is satisfied. This forms the on-demand communication mode, which is also a kind of aperiodic communication.The convergence proof is provided in Theorem2.Theorem 2.
Under Assumptions1 and 2, the feedback gain K is selected such that A−BK is Hurwitz-stable; then, the output value yi generated from system (3) with controllers (19a)–(19e) and triggering condition (20) achieves the optimal solution of problem (2). Besides, the control parameters satisfy that α>0,β>1/μ,δ>0,σ>0 and the Zeno behavior is avoided in triggering condition (20).Proof.
Let the candidate Lyapunov function be the same as Theorem1; similarly, we have(21)V˙≤−μy¯2−ν−2Ac22μx¯−Ψη¯2−β−12μz¯2−dx,Ω−αλTL⊗Iqλ^+2βλ^−λ∗z¯≤−μy¯2−ν−2Ac22μx¯−Ψη¯2−12β−1μz¯2−dx,Ω−∑i=1nα12qit−αLii+2βei2,where qit=1/2∑j=1naijλ^i−λ^j2 and the following results are used:(22)2βeiTz¯≤β2z¯2+2βei2,and(23)−λTL⊗Iqλ^=∑i=1n∑j=1neiTLijλ^j−∑i=1n∑j=1nλ^iTLijλ^j=−∑i=1nqit∑i=1n∑j=1,j≠inaijeiTλ^i−λ^j≤−∑i=1nqit+∑i=1n∑j=1,j≠inaijei2+14λ^i−λ^j2.
Then, together with triggering condition (20), we have(24)V˙≤−μz¯2−ν−2Ac22μx¯−Ψη¯2−12β−1μz¯2−dx,Ω+nδexp−σt.
From (24), it has Vt≤V0+n∫0texp−στdτ. Thus, we can calculate that μ∫0∞y¯2dt≤V0−V∞+n/σ. Therefore, ∫0∞y¯2dt is bounded. Finally, we conclude that yi reaches to the optimal solution as time goes to infinity.
Now, we try to prove that the Zeno behavior does not exist in event-triggered condition (20), that is the intertime tik+1−tik is strictly positive. From (24), there exists a constant M such that λ.≤M. Therefore, we have(25)eit≤Mt−tik.
Then, substituting the triggering condition in the above inequality yields(26)tik+1−tik≥1MLii+1+β2/2βα2qit+δexp−σt>0.
Therefore, the Zeno behavior is excluded in the proposed event-triggered system.Remark 4.
As mentioned earlier, the advantage of event-triggered communication is to reduce the communication burden. However, its disadvantages are also obvious. (a) To design the triggering condition, the errorei is needed. Besides, the neighboring information λ^j is also needed to be stored in agent i. Thus, this will lead to much cost and high complexity with the number increasing of its neighboring agents [43]. On the other hand, with the development of science and technology, the gradual improvement of the storage and computing capacity of agents is enough to overcome the above challenges. (b) As shown in Theorem 2, the bound of the differential V˙ is greater than those in Theorem 1. Thus, the introduction of the event-triggered communication slows down the convergence rate [44]. From Figure 1, we can see that the proposed two algorithms have a similar convergence rate since the event is triggered when the estimator error ei becomes larger.Figure 1
The convergence rate comparison between algorithms (9a)–(9e) and (19a)–(19e).


## 3.1. Continuous-Time Communication Algorithm

From Lemma4, it can be noted that the optimal solution y∗ must satisfy the equality constraint and the key is to seek the optimal dual Lagrange multiplier ϑ∗. Therefore, the following designed algorithm has the following three aspects: (1) the historical information of the local Lagrange multipliers λi is used for trading off the difference between real output yi and local demand resource di; (2) the multiagent consensus method is used to guarantee that all local Lagrange multipliers λi converge to the same optimal dual Lagrange multiplier ϑ∗; (3) since we cannot directly control the output value yi, we provide an auxiliary variable ηi, which is driven by the gradient information of the local objective function and local distance function. Subsequently, based on the deviation tracking control scheme, a specific algorithm is designed as follows:(9a)ui=−Kixi++Γiη˙i−Φi−KiΨiηi,(9b)η˙i=−∇fiyi+ci∂dyi,Ωi+λi,(9c)λ.i=−yi+di−α∑j=1naijλi−λj−βzi,(9d)z˙i=α∑j=1naijλi−λj,(9e)c˙i=dyi,Ωi,where zi is used to store the historical information of ∑j=1naijλi−λj and a proportional-integral control scheme is developed combining the terms ∑j=1naijλi−λj and zi. The control parameters α,β are greater than zero. Since A,B is controllable, then there exists feedback gain K such that A−BK is Hurwitz-stable, which means that all the eigenvalues of matrix A−BK have the negative real part. To guarantee the optimality, we need to set ∑i=1nzi0=0,ci0=0 and other variables xi,yi,λi,ηi initialization-free.Remark 1.
Many practical systems can be formulated by the general linear multiagent systems with dynamic (3) or can be linearized to (3). Typically, the high-order system is a special linear system. For example, the economic dispatch problem over the turbine-generator system, where the dynamics of each generator is described by a linear system [22]. However, the rank assumption of the turbine generator is not held. Besides, the dynamics of aircraft systems can also be linearized to a linear system [24, 25], and our proposed algorithm is effective for those aircraft systems.Remark 2.
To solve the distributed constrained optimization problem over a heterogeneous multiagent system, several distributed optimization algorithms have been proposed. In detail, based on the tracking control idea, the resource allocation problem incorporating with a global equality constraint is solved in [24]. Besides, the disturbance is suppressed by the designed internal-model observer. For the same problem, the solution is obtained within predefined time using the time-based generator technology in [25]. Similarly, the solution can be achieved within fixed time in [26]. However, the local constraints for each agent are not considered in the above three papers. More importantly, the above results cannot be directly extended to problem (2) by using the well-known projection operator. Moreover, only the suboptimal solution is obtained at the predefined time in [25], and the rank condition in [26] is stronger than ours. The global equality constraint is not considered in [39]. In addition, the dynamics of each agent are more complex than those in (3), in which a projection operator is applied to the output variable, that is, PΩiyi∈Ωi, instead of yi∈Ωi. The output variable yi is actually unconstrained in [39]. Although the same problem is solved in [27] under a milder rank condition, the designed algorithm depends on an equality CiBiKi=CiAi, and thus, the designed algorithm can be reduced to a first-order multiagent system. Also because of this equality, the designed algorithm will diverge as time goes to infinity. This equality is difficult to hold strictly, and then the small deviations may lead to system divergence. In addition, the event-triggered communication is also not considered in [27].Remark 3.
Among algorithm (9a)–(9e), the state information xi is used in (9a). Thus, the state-based feedback control is constructed. However, the state information xi is not available. For example, some state quantities do not have practical significance. A state observer or state estimator can be designed to estimate the unaccessible internal state. For instance, the designed linear Luenberger observer converges to the real state very quickly when the original system is observable [40]. Besides, the sliding mode observer is an alternative method [41].
Based on Lemma2, together with dynamic equation (3), we have the following compact form:(10)x˙=A−BKx−Ψη+Ψη˙,η˙=−∇fy+DT∂dy,Ω+λ,λ.=−y−d−αL⊗Iqλ−βz,z˙=αL⊗Iqλ,c˙=dy,Ω,where x,y,η,λ,z,c,∇fy are the aggregate vector of themselves. D,K,Ψ are the aggregate diagonal matrix of vectors ci,Ki,Ψi. Besides, we can see that system (10) is an autonomous system since the time t is not explicitly displayed in right side of (10) [42]. Next, let x∗,y∗,λ∗,z∗,c∗ be the equilibrium point of system (10); then, we prove that y∗ is the optimal solution to problem (2) in the following lemma.Lemma 6.
Ifx∗,η∗,λ∗,z∗,c∗ is the equilibrium of system (10), then y∗=Cx∗ is the optimal allocation scheme of problem (2).Proof.
Since the considered network is undirected and connected,λ=1n⊗λ0,λ0∈ℝ follows L⊗Iqλ=0. In addition, from the definition of ∂dy∗,Ω, the first equality holds owing to the facts ci∗∂dyi∗,Ωi⊂Ωi and ci∗≥0. Then, in terms of the zero-sum property of z0, the second equality is obtained, which follows that y∗−d+αL⊗Iqλ∗+βz∗=0. Thus, from Lemma 4, we can prove that y∗ is the optimal solution of problem (2).
The convergence proof is provided in Theorem1.Theorem 1.
Under Assumptions1 and 2, the feedback gain K is selected such that A−BK is Hurwitz stable; then, the output value yi generated from system (3) with controllers (9a)–(9e) achieves the optimal solution of problem (2). Besides, the control parameters satisfy that α>0,β>1/2μ.Proof.
Letx¯=x−x∗,y¯=y−y∗,η¯=η−η∗,z¯=z−z∗,c¯=c−c∗,λ¯=λ−λ∗; then, we choose the Lyapunov function V1 as V1=Cx¯2+1/2λ¯2+1/2λ¯+z¯2+β/αz¯TRΛ1−1RTz¯+1/2c¯2. Taking the time derivative over system (10) yields(11)V˙1=2Cx¯TCA−BKx¯−Ψη¯−2Cx¯Th+2Cx¯Tλ¯−λTy−αλ¯TL⊗Iqλ¯−2βλ¯Tz¯−βz¯2−λ¯Ty¯−y¯Tθ¯+2βλ¯Tθ¯+c¯Tdy,Ω=2Cx¯TCA−BKx¯−Ψη¯−2Cx¯Th−αλ¯TL⊗Iqλ¯−βz¯2−y¯Tz¯+c¯Tdy,Ω,in which h=∂fy−∂fy∗+DT∂dy,Ω−D∗T∂dy∗,Ω. Next, we focus on the term θ=yi−yi∗Tci∂dyi,Ωi−ci∗∂dyi∗,Ωi+ci−ci∗dyi,Ωi.
For the first case,yi∈Ωi. Then, we have dyi,Ωi=0 and ∂dyi,Ωi∈NΩiyi,∂dyi∗,Ωi∈NΩiyi. Based on the definition of the normal cone over set Ωi, we have yi−yi∗T∂dyi∗,Ωi≤0 and yi∗−yiT∂dyi,Ωi≤0. Thus, one can ensure that θi≤0.
For the second case,yi∉Ωi. We have that ∂dyi,Ωi=yi−PΩiyi/dyi,Ωi follows the definition of the gradient of the distance function. If ci≤ci∗, we have(12)θi=ci∗−ciyi−yi∗T∂dyi∗,Ωi+ci−ci∗dyi,Ωi−ciyi−yi∗T∂dyi,Ωi−∂dyi∗,Ω≤ci∗−ciyi−yi∗T∂dyi∗,Ωi+ci−ci∗dyi,Ωi≤ci∗−cidyi,Ωi+ci−ci∗dyi,Ωi=0.
In the above procedure, considering the convexity property of distance function, we have used the fact thatyi−yi∗T∂dyi,Ωi−∂dyi∗,Ω≥0 in the first inequality and its equivalent form yi−yi∗T∂dyi∗,Ωi≤dyi,Ωi−dyi∗,Ωi in the second inequality.
Ifci>ci∗, we conclude that(13)κi=−ci−ci∗yi,ΩiT∂dyi,Ωi+ci−ci∗dyi,Ωi−ci∗yi−yi∗T∂dyi,Ωi−∂dyi∗,Ω≤−ci−ci∗yi−yi∗T∂dyi,Ωi+ci−ci∗dyi,Ωi=−ci−ci∗yi−PΩiyi+PΩiyi−yi∗T⋅yi−PΩiyidyi,Ωi+ci−ci∗dyi,Ωi≤−ci−ci∗yi−PΩiyiTyi−PΩiyidyi,Ωi+ci−ci∗dyi,Ωi=0.
We have used the fact thatPΩiyi−yi∗Tyi−PΩiyi≥0 by means of the property of projection operator in the above second inequality. Therefore, we also can ensure that θi≤0. Substituting this result into (11), we have(14)V˙1≤2Cx¯TCA−BKx¯−Ψη¯−2μy¯2−αλ¯TL⊗Iqλ¯−βz¯2−y¯Tz¯≤−μy¯2+2Ac22μx¯−Ψη¯2−αλ¯L⊗Iqλ¯−β−12μθ¯2.
The following inequalities are applied in the above inequality by means of Young’s inequality:(15)2Cx¯TCA−BKx¯−Ψη¯≤12μy¯2+2CAc2μx¯−Ψη¯2,−y¯Tθ¯≤12μy¯2+12μθ¯2.
In the following, we defineV2=x¯−Ψη¯TPx¯−Ψη¯ with positive definite matrix P such that A−BKTP+PA−BK≤−νIN with N=∑i=1nni and ν>2Ac2/2μ. Then, the time derivative of V2 over system (10) gives(16)V˙2=x¯−Ψη¯TA−BKTP+PA−BKx¯−Ψη¯≤−νx¯−Ψη¯2.
Now, combining (14) and (16) gives(17)V˙1+V˙2≤−μy¯2−ν−2Ac22μx¯−Ψη¯2−αλ¯TL⊗Iqλ¯−β−12μz¯2.
From (17), we can conclude that the variable ct is bounded all the time and follows the boundedness of V1+V2. Thus, we have dyi,Ωi⟶0 owing to the dynamic of ci. There is a constant vector c˜ such as c˜i>cit,∀t≥0,i∈V. Let the candidate Lyapunov function be V=V1+V2+1nTc˜−c; then, we have(18)V˙≤−μy¯2−ν−2Ac22μx¯−Ψη¯2−αλ¯TL⊗Iqλ¯−β−12μz¯2−dy,Ω.
It is notable thatV is radially unbounded. Then, using LaSalle’s invariance principle and (18), we can conclude that the proposed algorithm converges to the largest invariance set of the set x˜,η˜,λ˜,z˜,c˜|V˙x˜,η˜,λ˜,z˜,c˜=0. From (18), we directly have y˜=y∗,z˜=z∗,λ˜=λ∗. Since c˜˙=dy˜,Ω=0, we have c˜=c∗. Based on the result x˜=Ψη˜, we have x˜˙=0,η˜˙=0. Finally, we have that x˜,η˜,λ˜,z˜,c˜ is the equilibrium of system (10). Together with Lemma 6, it is claimed that yt converges to the optimal solution of problem (2). Thus, the convergence proof is complete.


## 3.2. Event-Triggered Algorithm

In algorithm (9a)–(9e), neighboring information on the dual variable λj is assumed to be continuously accessible to agent i, which means that agent j must broadcast its Lagrange multiplier λj to its neighbors all the time. This broadcast mechanism requires agents to spend considerable communication resources, and in most cases, this communication is meaningless and redundant. To save the communication resource, motivated by the event-triggered communication mechanism, we modify (9a)–(9e) as follows:(19a)ui=−Kixi+Γiη˙i−Φi−KiΨiηi,(19b)η˙i=−∇fiyi+ci∂dyi,Ωi+λi,(19c)λ.i=−yi+di−α∑j=1naijλ^i−λ^j−βzi,(19d)z˙i=α∑j=1naijλ^i−λ^j,(19e)c˙i=dyi,Ωi,where λ^i is the lasting triggering information of the Lagrange multiplier λi and is defined as λ^i=λitik,t∈tik,tik+1 with the triggering time sequence tik. Additionally, the triggering time is determined by the following triggering condition:(20)tik+1=maxr≥tikr|αLii+2βeit2≤α2qit+δexp−σt,∀t∈tik,r,where the control parameters δ,σ are positive numbers and the estimation error is defined as ei=λ^i−λi. Compared with (9), each agent only broadcasts its information at some discrete time when the triggering condition (20) is satisfied. This forms the on-demand communication mode, which is also a kind of aperiodic communication.The convergence proof is provided in Theorem2.Theorem 2.
Under Assumptions1 and 2, the feedback gain K is selected such that A−BK is Hurwitz-stable; then, the output value yi generated from system (3) with controllers (19a)–(19e) and triggering condition (20) achieves the optimal solution of problem (2). Besides, the control parameters satisfy that α>0,β>1/μ,δ>0,σ>0 and the Zeno behavior is avoided in triggering condition (20).Proof.
Let the candidate Lyapunov function be the same as Theorem1; similarly, we have(21)V˙≤−μy¯2−ν−2Ac22μx¯−Ψη¯2−β−12μz¯2−dx,Ω−αλTL⊗Iqλ^+2βλ^−λ∗z¯≤−μy¯2−ν−2Ac22μx¯−Ψη¯2−12β−1μz¯2−dx,Ω−∑i=1nα12qit−αLii+2βei2,where qit=1/2∑j=1naijλ^i−λ^j2 and the following results are used:(22)2βeiTz¯≤β2z¯2+2βei2,and(23)−λTL⊗Iqλ^=∑i=1n∑j=1neiTLijλ^j−∑i=1n∑j=1nλ^iTLijλ^j=−∑i=1nqit∑i=1n∑j=1,j≠inaijeiTλ^i−λ^j≤−∑i=1nqit+∑i=1n∑j=1,j≠inaijei2+14λ^i−λ^j2.
Then, together with triggering condition (20), we have(24)V˙≤−μz¯2−ν−2Ac22μx¯−Ψη¯2−12β−1μz¯2−dx,Ω+nδexp−σt.
From (24), it has Vt≤V0+n∫0texp−στdτ. Thus, we can calculate that μ∫0∞y¯2dt≤V0−V∞+n/σ. Therefore, ∫0∞y¯2dt is bounded. Finally, we conclude that yi reaches to the optimal solution as time goes to infinity.
Now, we try to prove that the Zeno behavior does not exist in event-triggered condition (20), that is the intertime tik+1−tik is strictly positive. From (24), there exists a constant M such that λ.≤M. Therefore, we have(25)eit≤Mt−tik.
Then, substituting the triggering condition in the above inequality yields(26)tik+1−tik≥1MLii+1+β2/2βα2qit+δexp−σt>0.
Therefore, the Zeno behavior is excluded in the proposed event-triggered system.Remark 4.
As mentioned earlier, the advantage of event-triggered communication is to reduce the communication burden. However, its disadvantages are also obvious. (a) To design the triggering condition, the errorei is needed. Besides, the neighboring information λ^j is also needed to be stored in agent i. Thus, this will lead to much cost and high complexity with the number increasing of its neighboring agents [43]. On the other hand, with the development of science and technology, the gradual improvement of the storage and computing capacity of agents is enough to overcome the above challenges. (b) As shown in Theorem 2, the bound of the differential V˙ is greater than those in Theorem 1. Thus, the introduction of the event-triggered communication slows down the convergence rate [44]. From Figure 1, we can see that the proposed two algorithms have a similar convergence rate since the event is triggered when the estimator error ei becomes larger.Figure 1
The convergence rate comparison between algorithms (9a)–(9e) and (19a)–(19e).


## 4. Simulations

This section provides a four-agent multiagent system with two dimensions for illustrating the effectiveness of the proposed algorithm. For all agents, the structure matrixes are defined asA1=A2=0,1;0,0,A3=0,−1;1,−2, A4=0,1,0;0,0,1;1,1,−1, B1=B2=0,1;1,−2, B3=1,0;3,−1, B4=1,0,1;0,1,0;1,0,0, C1=C2=1,1;1,−1, C3=−1,1;1,1, and C4=1,−1,1;0.5,0,0.5. Based on Lemma 2, the feedback matrixes are chosen by Φ1=Φ2=1,−1;0.5,−0.5, Φ3=−0.5,−0.5;0,−1, Φ4=−1,2;0,1;0,0, Ψ1=Ψ2=0.5,0.5;0.5,−0.5, Ψ3=−0.5,0.5;0.5,0.5, Ψ4=0,1;−1,2;0,1, Γ1=Γ2=1.5,0.5;0.5,0.5, Γ3=−0.5,0.5;−2,1, Γ4=0,1;−1,2;0,0, K1=K2=3,5;1.5,1, K3=0.75,−1;1.25,−4, and K4=2.167,1,0.333;0,3,1;1,1,1. In addition, the local payoff function and set constraint are given by(27)f1y1=y112+y122+y11−22+y12−22,f2y2=y212+y222+y21220y212+1+y22220y222+1,f3y3=y31−22+y32−32,f4y4=lne−0.05y41+e0.05y41+y412+y422+lne−0.05y42+e0.05y42,Ω1=y1∈ℝ2y1−2,2T≤2,Ω2=y2∈ℝ2|1≤y21≤2,0≤y22≤1,Ω3=y3∈ℝ2|y31≥0.5,y32≥1,y31+y32≤6,Ω4=y4∈ℝ2|y4−3,5T≤2.The local demand resource is subject tod=2;2;2;1;1;3;4;5T∈ℝ8. Let the control parameters be α=10,β=2, which satisfy the domain given in Theorem 1. From the initial x0=1;1;1;0.5;0;1;6;8T∈ℝ8, the trajectories of proposed algorithm (9a)–(9e) are shown in Figures 2 and 3. From Figure 2, we can see that the proposed algorithm converges to the theoretical optimal result y∗=1.8884,3.4729,1.8237,1,1.4396,4.5604,1.8483,3.9666, where the optimal solutions are denoted by a star symbol ∗ and the boundary is denoted by the dotted line. Besides, we know that the solutions of agents 2 and 3 are in the boundary. From Figure 3, the penalty factor ci is significantly affected by the distance function. The value of c2 remains zero all the time, which means that the corresponding variable y2 satisfies the local set constraint and this can be proved by Figure 2.Figure 2
(a) The trajectories of resource variabley1. (b) The trajectories of resource variable y2. (c) The trajectories of resource variable y3. (d) The trajectories of resource variable y4.
(a)(b)(c)(d)Figure 3
The trajectories of penalty factorci.Now, let us examine the effectiveness of triggering condition (20). The step size is set to 0.001 s. The control parameters are chosen by α=10,β=0.5,δ=0.1,σ=0.1; then, detailed triggering times are shown in Figure 4. In summary, each agent needs to broadcast 457 times for its neighboring agents, which makes up 7.6% of the continuous communication times. From triggering condition (20), a smaller β leads to fewer communications. However, it results in a larger zi∗ and thus slows down the convergence rate. In addition, the control parameter β should satisfy the convergence domain given in Theorem 2. Let eopt=ln1/2y−y∗2 The convergence rate comparison between algorithms (9a)–(9e) and (19a)–(19e) is shown in Figure 1. From Figure 1, it can be concluded that the two algorithms have similar convergence rates. Furthermore, from the partial enlarged drawing, the delayed information λ^j can accelerate the convergence speed. However, the disadvantage of event-triggered algorithm (19a)–(19e) is that its convergence accuracy is lower than continuous-time communication algorithm (9a)–(9e). Specifically, we have y−y∗=0.009. This convergence accuracy has been able to meet the requirements of practical applications. Therefore, we can claim that the proposed two algorithms are all effective in the general heterogeneous multiagent systems.Figure 4
The triggering time sequencestik.


## 5. Conclusions

This paper focuses on the distributed resource allocation problem of the general linear heterogeneous multiagent systems by designing an initialization-free distributed optimization algorithm based on the proportional-integral control approach, consensus technology of multiagent systems, and the gradient flow method. In addition, the penalty factor increases adaptively with the distance function. However, the considered network is assumed to be undirected. In the future, we will attempt to research the resource allocation problem over directed networks.



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*Source: 1434099-2022-08-22.xml*