2.1 Graph theory

and represent the set of real numbers and a vector set respectively. A graph is employed to model a communication network with n nodes. The graph consists of a vertex set and an edge set . If there exists a path between any two nodes, then is connected. Define as the weighted adjacency matrix of , ; . denotes the degree matrix, where . The Laplace matrix is equal to the degree matrix minus the adjacency matrix, i.e. . When the graph is an undirected graph, its Laplacian matrix is a symmetric matrix.

2.2 Convex function

If there exists some positive constant which makes the following inequality hold, then the cost function can be called strongly convex [11]. (2) (3) (4) where ϕ_i is a positive constant, ∇g_i and ∇²g_i are the gradient and the Hessian matrix respectively.

Lemma 1 Suppose the graph is strongly connected and balanced, for any , the following property holds. (5) where α₂ is the smallest positive eigenvalue on and β_n is the largest eigenvalue on .

3.1 The distributed DET-ZGS algorithm with time delay

In this paper, we adopt a periodic sampling method to obtain the state of the agent, where the sampling period is defined as h. This means that we sample the state of the agent at intervals of h. Specifically, we use x_i(t) to represent the state of the agent at time t, and this state changes dynamically over time. Firstly, the states of the neighbors and the agent’s own state at the previous triggering time are sent to the controller to obtain the agent’s current state. Simultaneously, an event detector is installed on each agent, which only checks if the triggering condition is met at the sampling time. Subsequently, the agent’s state from the previous sampling time is sent to a constructed trigger for evaluation. When the preset triggering condition is satisfied, the agent’s state at the triggering time is obtained. Finally, the obtained local state is used to update the control actions of both the agent itself and its neighbors. In practice, when the value at the sampling point exactly meets the DET condition, we have . Due to the existence of time delay in practical transmission, this section considers satisfying a delay of τ ∈ (0, h). In addition, for t ∈ [ςh + τ, (ς + 1)h + τ), ς = 0, 1, 2, …, we define and deviation variable . The main structure of the algorithm is designed as follows: (6) (7)

Then the DET condition is designed as follows: (8) where (9) and (10) where d_i is the degree of the agent i, γ is a positive constant. ε_i, μ_i, ω_i are self-designable parameters. The deviation variable .

Remark 1 Considering that it takes some time to make ET judgments on agent states. When designing the DET condition (8), we use the agent state q_i(ςh − h) at the previous sampling moment, which is more logical. In addition, the agent state is taken into account when designing DET conditions, which makes the frequency of triggering not increase when the agent state is close to the optimal value. In the design of DET conditions, different parameters ε_i, μ_i, ω_i can be set for each agent according to different needs to ensure that it can make the most accurate response in a specific situation. For the convenience of analysis, this paper chooses to set the same ε_i, μ_i, ω_i for each agent in order to study its behavior and response ability more clearly.

3.2 Main results and analysis

Theorem 1. Assume that the communication graph is balanced. If the following stability conditions are valid, the proposed algorithm (6) based on DET conditions (8) enables to address problem (1), i.e. lim_t→∞x_i(t) = x*. (11) (12) (13) where ϕ_min = min{ϕ₁, …, ϕ_n}.

Proof: For t ∈ [ςh + τ, (ς+ 1) h + τ), define a Lyapunov function (14)

Let and .

According to the property of convex function (2) yield (15)

According to (8) and (10) have (16) Furthermore, we obtain (17)

Based on (12), we get . Then, ξ_i(ςh) > 0. Furthermore, it follows (18)

Therefore, one obtains (19)

Due to (15) and (19), we can derive . Then we derive and yields (20)

Furthermore, one has that (21)

Due to the following properties of undirected graphs . (22)

Furthermore, due to , one has that (23) where , .

Based on (6) yield (24) where and , , , Δ(t) = [δ₁(t), δ₂(t), …, δ_n(t)]^T, i = 1, 2, …, n.

Substituting Eqs (24) into (23) yields (25)

As the inequality (4), there exists (26)

Furthermore, , , where ϕ_min = min{ϕ_1,…,ϕ_n}.

Note that (27)

Based on Lemma 1 yield (28)

By utilizing Lemma 1 and Young inequality, one has (29)

Then, we have (30)

Based on (10), we get (31)

Then, we obtain (32)

Add and subtract to right equation, one obtain (33)

Take the integration of yields (34) for t ∈ [ςh + τ, (ς + 1) h + τ).

Since the Lyapunov function is positive definite, it follows that (35)

By accumulating from ς = 1 to ς yield (36) while (37) we have (38)

Then, we can obtain, (39)

Given that q_i(ςh) ≥ 0 and , Eq (39) implies that: (40)

Utilizing the definition of q_i(ςh) and the Laplace matrix properties, it is established that the ultimate state of each agent is equivalent to a common constant value, i.e. (41)

The second order derivative of the cost function has (42) where 1 = [1 …, 1]_n.

Then, we have (43)

Combining (41) and (43) yields (44) Theorem 1 is proved.

Corollary 1 When the time delay is not considered and the following stability conditions hold, the proposed algorithm can effectively address problem (1) for any and t > 0. (45) (46) (47) where ϕ_min = min{ϕ₁, …, ϕ_n}, . Due to the exclusion of time delay, the proof process is comparatively simpler than that of Theorem 1.

Proof: We adopt the identical Lyapunov function as presented in Theorem 1, (48) where ,

According to the proof above, we can get .

Based on (6) and (7) yield (49)

Since t ∈ [ςh, ςh + h), we obtain (50) where , , i = 1, 2, …, n and .

Then, we obtain (51)

Then, one obtains (52)

Based on (34), we can have (53)

Through the iterative operation of (53), we can derive (54) Then we use the iterative method that is similar to Theorem 1 and finally get the result of Corollary 1.

Remark 2 λ_n is a global information required by each agent. The following inequality gives the bound on λ_n. Then, the bounds for h and γ can be obtained from Eq (45). (55) Thus, condition (45) can be changed as: (56)

3.3 The distributed DET-ZGS algorithm with arbitrary sampling period

According to the above analysis, the sampling period significantly affects the performance of the algorithm. When the period interval is selected to be large, it cannot be guaranteed that the algorithm will find the global optimal value. This subsection handles the sampling period on the basis of algorithm (6) and proposes a distributed DET-ZGS algorithm where the sampling period can be designed arbitrarily.

The algorithm is designed as: (57) (58)

The DET condition changes as (59) (60) (61)

Theorem 2 Suppose the undirected graph is connected. For and t > 0, when condition (62)–(64) is satisfied, the system state converges to the optimal solution x* under the effect of algorithm (57) and the DET condition (59). (62) (63) (64)

Proof: For t ∈ [ςh + τ, (ς + 1) h + τ), define a Lyapunov function (65) where and .

Then we have (66)

According to (28) and (29), we obtain (67)

From (61), it follows that (68) Combine (67) and (68) yields (69)

From Eq (34), it follows that (70)

Then iterate to obtain (71)

Then use the previous similar analysis to complete the proof of Theorem 2.

Remark 3 The stability conditions derived from Theorem 2 reveal that when the sampling period h is chosen to be small, τ and h can be eliminated due to 0 < τ < h; When the sampling period h takes a large value, the term can be ignored. From the perspective of simulation experiments, longer sampling periods require more time steps, but still lead to optimal solutions. However, in [26], excessively large sampling periods make it difficult to satisfy the stability conditions, resulting in the algorithm’s failure to converge and find the optimal solution. Therefore, the algorithm proposed in this paper allows the sampling period to be arbitrarily large, which is of great significance.