1. Introduction
With the development of network information technology and the era of artificial intelligence, multi-agent systems (MASs) have received extensive attention in view of their applications in the machining industry [
1], synchronous generators [
2], microservice-based cloud applications [
3], USVs [
4], and other fields. It is worth noting that consensus is one of the most fundamental and important problems in MASs, and there have been many studies about it [
5,
6,
7,
8]. In essence, the distributed optimization problem is that a group of agents achieve a goal by exchanging local information with neighbors and minimizing the sum of all the local cost functions. In contrast to conventional consensus, distributed optimization problems require both achieving consensus and solving optimization problems. Up to now, distributed optimization problems have already appeared widely in power systems [
9], MPC and network flows [
10], wireless ad hoc networks [
11], etc.
Early distributed optimization problems were mainly solved by centralized optimization algorithms. The feature of a centralized optimization algorithm is that all agents have a central node that centrally stores all of the information to address the optimization problem [
12,
13]. However, centralized optimization algorithms are unsuitable for large-scale networks, because collecting information from all agents in the network requires a lot of communication and computational overhead, and there will be the single point of failure problem. Consequently, distributed optimization algorithms have emerged as the times require. In recent years, distributed optimization algorithms are divided into two main categories, i.e., discrete-time algorithms and continuous-time algorithms. More specifically, discrete-time distributed optimization algorithms have been utilized in the optimal solution of the saddle point dynamics problem [
14], epidemic control resource allocation [
15], and tactical production planning [
16]. Additionally, many researchers have made substantial explorations of continuous-time distributed optimization algorithms recently. For instance, a continuous-time optimization model was developed in [
17] for source-sink matching in carbon capture and storage systems. In [
18], the application of a continuous-time optimization algorithm was investigated in power system load distribution, and the distributed continuous-time approximate projection protocol was proposed in [
19] for solving the shortest distance optimization problem.
Many of the above optimization algorithms communicate in continuous time, which can lead to frequent algorithm updating and then cause unnecessary communication resource consumption, so it is necessary to solve the system’s resource problem. Therefore, applying event-triggered strategies to distributed optimization algorithms [
20,
21,
22,
23,
24,
25,
26] is a feasible and promising scheme that can effectively reduce the energy waste of the system. Only when the designed event-triggered condition is satisfied, is the system allowed to communicate and update the protocol, which helps to reduce the cost and burden of communication and computing as well as the collection of gradient information. Primarily, for static event-triggered (SET) mechanisms, which include the constant trigger thresholds independent of time, it is theoretically difficult to rule out Zeno behavior. Furthermore, as the working time increases, the inter-event time intervals become larger, which results in more trigger actions and wasting the system’s resources. Furthermore, the event-triggered strategy has undergone a paradigm shift from the SET strategy to the dynamic event-triggered (DET), which introduces an auxiliary parameter for each agent to dynamically adjust its threshold. Moreover, in most cases, the DET strategy can well extend the average event intervals, thus further reducing the consumption of communication resources compared to SET communication. Therefore, the DET strategy has aroused much interest and it holds great applicability value, which was considered in [
27,
28,
29,
30,
31,
32,
33]. An improved event-triggered strategy, independent of the initial conditions, was leveraged in [
34] to solve the topology separation problems caused by critical communication link failures. In [
35], the corresponding DET mechanism was presented for two cases based on nonlinear relative and absolute states coupling, and it was also proved that the continuous communication between agents can be effectively avoided. Under the DET strategy, each agent transmits information to all neighbors synchronously when its trigger condition is met, which is usually called the one-to-all DET strategy. Nevertheless, under the one-to-all DET strategy, it is unreasonable to ignore the possibility that each agent has different triggering sequences. Therefore, to overcome the limitation of the one-to-all DET strategy, it is essential to design a DET strategy that allows each agent to decide its own triggering sequences and transmit information asynchronously to its neighbors according to different event-triggered conditions designed for each of its neighbors, which is referred to as the one-to-one DET strategy. Under the one-to-one DET strategy, owing to its characteristics, an agent is not constrained by any synchronous execution of its neighbors’ transmission information, so it can adjust the information transmission more flexibly, especially in the case of cyber-attacks. In [
36], under an adaptive DET strategy, the fully distributed observer-based strategy was developed, which guarantees asymptotic consensus and eliminates Zeno behavior.
So far, note that many distributed optimization algorithms have been leveraged to solve the resource allocation problem (RAP), such as in [
37,
38,
39]. Therefore, it is necessary and significant to combine DET strategies to solve the RAP. Motivated by the above discussions, we further investigate distributed optimization algorithms with two novel synchronous and asynchronous DET strategies to address the RAP. The main contributions of this article are developed as follows.
- (1)
This work combines the consensus idea and one-to-all DET strategy to design a new distributed optimization algorithm to solve RAP, in which the algorithm can keep the equality constraint constant. In addition, unlike the SET strategies of [
40,
41], the DET in this work has a lower trigger frequency, which means that the system resources can be saved.
- (2)
In order to improve the flexibility and practicality of the algorithm, the one-to-all DET strategy is extended to a one-to-one DET strategy. Based on this strategy, a distributed optimization algorithm is developed to address the RAP.
- (3)
The two types of proposed distributed optimization algorithms only use the information of the decision variable
to avoid the communication among agents, which ingeniously reduces the resource consumption, while the algorithm in [
42] needs to exchange information about the variables
and
. In addition, the introduced internal dynamic parameters in this work are not only effective in solving RAP, but also crucial in successfully excluding Zeno behavior.
The organization of the remaining parts of this paper is as follows. Some algebraic graph theory preliminaries, a basic definition and assumptions, and the optimization problem formulation are given in
Section 2. In
Section 3 and
Section 4, distributed optimization algorithms under the proposed one-to-all and one-to-one DET strategies are presented to solve the RAP. Furthermore, the proof of the exclusion of Zeno behavior is included. In
Section 5, numerical simulation results are given to illustrate the effectiveness of the proposed algorithms. Finally, we show our conclusions and future work direction in
Section 6.
Notation 1. The symbols appearing in this article are listed in Table 1. 3. The One-to-All DET Strategy
In this section, we construct the one-to-all DET strategy, which allows each agent to transmit information synchronously. Moreover, a distributed optimization algorithm with the proposed DET is introduced and the consensus is derived, which solves the RAP (1).
For the one-to-all DET, the triggering time sequence is determined by
The measurement error of each agent is defined as
Then, we propose the one-to-all DET triggering sequence
as follows
where
and
,
, are positive constants.
Remark 1. If setting , the DET strategy reduces to the SET strategy. Then, the one-to-all SET triggering sequence is as followsConsequently, the SET strategy is a special case of the DET strategy, and the DET strategy is a more general situation. In addition, due to the internal dynamic variables of the DET function, it is easier to exclude Zeno behavior than for SET. Inspired by [
43], we design an internal dynamic variable
satisfying
where
,
and
with
, are positive constants.
Let
and
. The distributed optimization algorithm is designed as follows to solve the RAP (1):
where
,
is an auxiliary variable.
According to the distributed optimization algorithm (4), one obtains where the initial value satisfies
In addition,
in matrix form can be described as
where
,
and
.
Then, the distributed optimization problem is transformed into a multi-agent consensus, which implies when
,
, the RAP (1) is obtained for any agents. Then,
is the final value of
when it reaches consensus. The detailed procedure of the one-to-all DET strategy is given as Algorithm 1.
Algorithm 1 Distributed optimization algorithm with the one-to-all DET strategy |
- Require:
-
Initialize all parameters, such as the states and of the agent i and so on. During the initialization process, it is required that and . -
Input last triggering times and state , . - Ensure:
-
for to do -
for to n do -
Compute measurement errors with . -
Compute the trigger threshold . -
if trigger condition (2) holds then -
The event is triggered, and the event time is recorded as . -
Update the state of agent i at event time . -
Communicate information between state and its neighbor state . -
else -
Update the state of agent i at instant t which belongs to interval . -
end if -
end for -
end for
|
Remark 2. For the quadratic original optimization problem with the equality constraint, based on the Lagrange multiplier method, we construct the Lagrangian function as . Then, under Assumption 2, , is the optimal solution, where is the optimal Lagrange multipliers if and only if , . Therefore, we need to let the Lagrange multiplier of each agent update so that all reach consensus at the value , which means that the optimization problem with equality constraint is transformed to a MASs consensus problem completely. Therefore, as long as the equation holds, the algorithm can achieve consensus and the optimization problem can be addressed.
Remark 3. Algorithm (4) only uses the information of variable , which is beneficial to save communication resources in the case of limited bandwidth. Furthermore, let , from Assumption 1, i.e., , the proposed zero-initial-value distributed optimization algorithm, i.e., , satisfies the equality constraint at all times. The initial values of the algorithm are composed of the decision variable initial value and the auxiliary variable initial value . Then, we can prove that , and , because the equation holds. Therefore, when the equation is satisfied, the equality constraint holds as well at any time.
Theorem 1. Under Assumptions 1 and 2, assume that , , then the RAP (1) is solved under the distributed optimization algorithm (4) and the DET strategies (2) and (3). Moreover, Zeno behavior is excluded.
Proof. Construct the Lyapunov function
of the following form
where
□
The rest of the proof is the similar to Theorem 2.
4. The One-to-One DET Strategy
In this section, in consideration of the existence of asynchronous transmission needs, the one-to-one DET strategy is introduced, which has the unique characteristics that each agent transmits its information to all of its neighbors asynchronously, unlike the one-to-all DET strategy. Furthermore, based on the one-to-one DET strategy, a more flexible distributed optimization algorithm is similarly presented and the consensus is achieved, which also solves the RAP (1). Then, we prove that the Zeno behavior will not occur, which strongly ensures that the algorithm is implementable.
For the one-to-one DET strategy, the edge-dependent triggering time sequence is raised, i.e., , which essentially differs from the one-to-all case.
Corresponding to the one-to-one DET case, the measurement error is described as
Then, we propose the one-to-one DET triggering sequence
as follows
where
and
,
, are positive constants.
Remark 4. Similarly, if setting , the DET strategy reduces to the SET strategy. Then, the one-to-one SET triggering sequence is as follows Inspired by [
43], we design an internal dynamic variable
satisfying
where
,
and
, are positive constants. In addition,
,
and thus
.
Let
. The distributed optimization algorithm is determined as follows to solve the RAP (1):
for
. In addition, one obtains
where the initial value
,
, satisfies the equation
The detailed one-to-one DET procedure is given as Algorithm 2.
Theorem 2. Under Assumptions 1 and 2, if the parameters and in (5a,b) and (6) satisfy , , then the RAP (1) is solved under the distributed optimization algorithm (7) and the DET strategies (5a,b) and (6). Moreover, Zeno behavior is excluded.
Algorithm 2 Distributed optimization algorithm with the one-to-one DET strategy |
- Require:
-
Initialize all parameters, such as the states and of the agent i and so on. During the initialization process, it is required that and . -
Input last triggering times and state , . - Ensure:
-
for to do -
for to n do -
Compute measurement errors with . -
Compute the triggered threshold . -
if trigger condition (5a,b) holds then -
The event is triggered, and the event time is recorded as . -
Update the state of agent i to agent j at event time . -
Communicate information between state and its neighbor state . -
else -
Update the state of agent i at instant t which belongs to interval . -
end if -
end for -
end for
|
- (i)
Define the following Lyapunov function:
where
From (8), we have
Note that
From Young’s inequality, one has
Substituting (10) and (11) into (9) yields
According to Formula (12), taking the derivative of the Lyapunov function
can be derived as
Then, we can obtain from (5a,b) and (6) that
Since , , , one obtains . This implies that cannot increase and that and are bounded. In addition, , which leads to .
By LaSalle’s invariance principle in [
44], one obtains
,
Thus, the RAP (1) is solved eventually.
- (ii)
In this part, we prove that Zeno behavior does not occur by contradiction. Assume that the triggering sequence determined by (7) and (8) leads to Zeno behavior, which indicates that for any there exists a such that for any ,
For , , .
Then, for
, from (8),
where
and
.
Therefore, for any
,
By the trigger conditions (5a,b) and (6), when
,
Noting
By using the comparison principle,
Combining (14) and (15), it has
Therefore,
For
it is not difficult to see from (16) that
, which is obviously contradictory to (13). Consequently, there is no Zeno behavior. □
Remark 5. In contrast to the one-to-all DET strategy mentioned in Theorem 1, under the one-to-one DET strategy, the triggering sequences of each agent is different, which contributes to flexibly adjusting the transferred information to each of its neighbors . Furthermore, the remarkable feature of the one-to-one DET strategy is that each agent is allowed to design its own distinctive triggering instant which is immune to any synchronous executions and the requirements of or , , and so on. Therefore, in practice, one-to-one DET strategies potentially offer greater flexibility and efficiency in terms of adjusting the transmission of information, which is significant to designing a good DET strategy.
Remark 6. The proposed algorithms (4) and (7) can effectively solve RAP, but both of them need to satisfy and , which means with initialization constraints. In our future research, we will consider eliminating state initialization.
5. Numerical Example
In this section, two numerical examples are provided to illustrate the effectiveness of the theoretical results. The proposed one-to-all and one-to-one DET strategies are applied to the RAP (1) in case 1 and case 2, respectively.
Figure 1 depicts the connection topology, which satisfies Assumption 1. The chosen cost coefficients
,
, and
of the quadratic cost function
are listed in
Table 2. The load demand
D is assumed to be 145. Then, the initial values of
are selected as
,
,
,
, and
,
.
Case 1. One-to-all DET
First, consider the one-to-all DET based on Theorem 1. Given the scalars
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
Figure 2 shows that
converges to
, which essentially guarantees that all agents reach asymptotic consensus. Then, from
Figure 3,
converges to the optimal values.
Figure 4 shows the triggering instants of the one-to-all DET strategy. In addition, the equality constraint
can be obtained from
Figure 5.
Furthermore, the equality constraint
can be obtained from
Figure 6. The trajectories of
are shown in
Figure 7. Moreover,
Figure 8 shows the minimum value of
, where
is the optimal solution of the RAP (1).
Figure 9 exhibits the trajectory of the dynamic variable
.
Case 2. One-to-one DET
Consider the one-to-one DET based on Theorem 2. Different from case 1, given
,
Figure 10 shows that
converges to
, which also implies that consensus is indeed achieved. Then, as seen in
Figure 11,
converges to the minimum value.
Figure 12 shows the triggering instants of the one-to-one DET strategy. In addition, the equation constraint
is guaranteed from
Figure 13.
Besides, the equation constraint
is guaranteed from
Figure 14. The motion trajectory of
is shown in
Figure 15. Furthermore,
Figure 16 depicts the minimum value of
, where
is the optimal solution of the RAP (1).
Figure 17 shows that
converges to 0 and
always holds.
Case 3. DET vs. SET
By letting
and
in (2) and (5), one has the one-to-all SET and one-to-one SET versions (2b) and (5b). Then, the one-to-all DET and SET strategies are compared in
Figure 18 and
Figure 19. Moreover, the one-to-one DET strategy and the corresponding SET strategy are compared in
Figure 20 and
Figure 21. Since
in (2) and
in (5), the DET strategies are likely to have fewer triggering times, as compared with the SET strategies, which are also displayed in
Figure 18,
Figure 19,
Figure 20 and
Figure 21 and
Table 3 and
Table 4, which means that DET is beneficial for saving system resources with a slower update frequency.