Distributed Optimization for Mobile Robots under Mobile Edge Computing Environment

Driven by the development of the Internet industry, mobile robots (MRs) technology has become increasingly mature and widely used in all walks of life. Since MRs are densely distributed in the network system, how to establish a reliable communication architecture to achieve good cooperation and resource sharing between MRs has become a research hotspot. In this respect, mobile edge computing (MEC) technology and millimeter wave (mmW) technology can provide powerful support. *is paper proposes a mmW communication network architecture for distributed MRs in MEC environment. *e mmW base station provides reliable communication services for MRs under the coverage of information cloud (IC). We design a joint resource and power allocation strategy aimed at minimizing network energy consumption. First, we use the Lyapunov optimization technique to transform the original infinite horizon Markov decision process (MDP) problem. *en, a semidistributed algorithm is introduced to solve the distributed optimization problem in the mmW network. By improving the autonomous decision-making ability of the mmW base station, the signaling overheads caused by information interaction are reduced, and information leakage is effectively avoided. Finally, the global optimal solution is obtained. Simulation results demonstrate the superiority of the proposed strategy.


Introduction
With the rapid development of the Internet industry, a variety of intelligent terminals emerge in an endless stream. Among them, intelligent MRs can set environmental perception, dynamic decision-making and planning, behavior control and execution, and other functions in one, widely used in the industry, agriculture, medical, service, and other industries, and has aroused universal attention all over the world. In order to realize good cooperation and resource sharing among multiple MRs, so that they can assist or replace human work to the greatest extent, we need to establish a communication framework among MRs. In the current research, the multirobot wireless network system can realize flexible cooperation through a central server [1]. Based on TCP and UDP in the local area network (LAN), the distributed communication method proposed in [2] allows robots to subscribe to target topics independently to achieve multirobot cross-platform communication. Each robot in [3] can establish communication paths to maintain a consistent communication with the cloud to effectively realize cooperative communication and link prediction. e authors in [4] proposed a scheme of mobile robot object recognition in Internet of ings (IOT) and used edge nodes to coordinate the data from robot vision.
In the traditional distributed communication network architecture, a centralized management scheme centered on the core network is usually adopted. e core network collects global network information and makes global decisions. With the gradual expansion and complexity of network structure, centralized management scheme will inevitably lead to huge signaling overheads, time delay, and privacy leakage. Among the existing distributed algorithms, the authors in [5,6] proposed a distributed downlink resource allocation and multidomain interference management algorithm to alleviate the co-channel interference in interference, and power. First, we use the Lyapunov optimization technique to transform the infinite horizon problem in the objective function into a single frame problem that can be processed. Second, we introduce a semidistributed algorithm for distributed optimization issues on complex mmW MECs. Compared with traditional distributed programs, this scheme improves the independent decision-making capacity of mmW APs and only needs to exchange a small amount of Lagrangian parameters through IC without sharing global information, which greatly reduces signaling overheads and protects the users' privacy. e rest of the paper is organized as follows. In Section 2, we describe the system model of the distributed MRs communication network. Section 3 formulates and transforms the optimization problem.
e semidistributed resource and power allocation are proposed in Section 4. Section 5 shows the simulation evaluation, and we draw the conclusion in Section 6.
Notations: ‖ · ‖ is the Euclidean norm. Let diag(x 1 , . . . , x K ) be the diagonal matrix. I denotes the identity matrix. A ⊗ B means the Kronecker product of matrixes A and B. E[·] represents the statistical expectation.

System Model
In the system, under the coverage of the IC that introduces MEC technology and is responsible for data calculation, storage, and interaction, there are I mmW APs and each of them provides services to J MRs within its coverage as shown in Figure 1. e set of mmW APs and MRs is denoted as I � W 1 , W 2 , . . . , W I and J � R 1 , R 2 , . . . , R J . Each mmW AP equips with mmW transceivers to overcome the significantly high path loss though highly directional transmission, and each MR introduces MEC technology and equips with a mmW beam steerable array to support transmission over IC and mmW network.
We have each MR implementing the analog beamforming, allowing one transmission link each time, and each mmW AP implementing the hybrid beamforming, allowing multiple transmission links each time. e mmW AP W i equipped with C i radio frequency (RF) chains can serve most C i MRs. Moreover, we assume that the proposed network transmits in a slotted system, which is represented as t � {0, 1, 2, . . .}. Denote x i,j (t) � 1 if the j-th MR is in the coverage of the mmW AP W i at time slot t, and x i,j (t) � 0 otherwise. We assume each MR is attached to not more than one mmW AP, i.e., I i�1 x i,j (t) ≤ 1, ∀j ∈ J. We further assume that the total number of MRs attached to the mmW AP does not exceed the maximum chain number, i.e., J j�1 x i,j (t) ≤ C i , ∀i ∈ I.

mmW Transmission Model.
e mmW network is susceptible to interference and has severe channel attenuation. By applying beamforming technology, directional transmission can be formed between mmW APs and MRs to increase data rate. According to [16], we build the model of mmW network transmission as shown in Figure 2. Let θ a i (t) 2 Complexity andθ r j (t) be the operating beamwidths of mmW AP W i and MR R j , respectively. Let ζ a ij (t) be the angle between the positive x-axis and the direction in which the mmW AP W i sees MR R j , and let ζ r ij (t) be similarly defined by changing the roles of the mmW AP W i and MR R j. . e values of ζ a ij (t) and ζ r ij (t) satisfy |ζ a ij (t) − ζ r ij (t)| � π. We further denote φ a ij (t) and φ r j (t) as the boresight angles of the mmW AP W i and MR R j relative to the positive x-axis. Let g a ij,ji (t) be the transmission gain between the beam from the mmW AP W i to MR R j. and g r ji,ij (t) be the reception gain between the beam from MR R j to the mmW AP W i , which can be written as follows.

Interference
Management. According to [16], there are two types of interference in the proposed mmW network that should be managed. e intracell interference exists among devices within a cell and can be mitigated by using proper scheduling and beamforming design. e intercell interference exists among different cells and can be  Complexity significantly reduced by resource blocks scheduling and omnidirectional communication. erefore, to ensure that the intercell interference in the mmW network is small enough to be ignored, the following constraints can be introduced: where I T is the intercell interference threshold, p a ij (t) denotes the transmit power from the mmW AP W i to MR R j , andg c ij (t) refers the channel gain between the mmW AP W i to MR R j .

Data Transmission.
Let Q j (t) be the data queue length of MR R j at time slot t. e data entering the queue of MR R j at time slot t are denoted as A j (t), whose maximum value is A j, max (t). e traffic buffer queue Q j (t) of MR R j evolves as where [x] + � max x, 0 { } is on account that the actual served packets cannot be larger than the current queue size. τ is the duration of each time slot. We further let Q j (0) � 0, ∀j ∈ J for simplicity. e achievable data rate of MR R j served by the mmW AP W i at time slot t can be written as where B a is the mmW bandwidth and N 0 is the noise power spectral density. Based on (4), the achievable data rate of MR R j served by the mmW network at time slot t is given by

Problem Formulation.
Our goal in this work is to design a resource and power allocation strategy to minimize the energy consumption of devices and guarantee power and rate limitation and traffic latency of MRs as well as ensuring performance of the mmW network. e energy aware optimization problem can be formulated as subject to (2), wherep a j (t) � i∈I p a ij (t), Q j is the predefined average queue backlog tolerance of MR R j , and R T j (t) and P T j are the minimum data rate requirement and the maximum power consumption of MR R j at time slot t. Constraint (7a) ensures the time-averaged latency of MR, which is proportional to the average queue backlog length according to Little's theorem. Constraint (7d) is the limitation of parameters.

Lyapunov Drift Penalty.
Since the optimization function P1 is a long-term average network energy consumption problem, it is an infinite horizon Markov decision process (MDP) problem [17]. To tackle the infinite horizon problem P1 with lower computational complexity and less memory consumption, we introduce the Lyapunov drift penalty technique to transform P1 into a tractable single frame problem [18]. We define a virtual queue Z n for constraint (7a) as For each slot t, the Lyapunov function can be formulated as a sum of virtual queue length squares, given by where Moreover, the Lyapunov drift function is expressed as en, the Lyapunov drift penalty function is defined to minimize the drift-plus-penalty expression boundary, which can be formulated as where V is the nonnegative control parameter, which represents the weight of the objective function.

Lemma 1.
e upper bound for the Lyapunov drift penalty (11) can be written as 4 Complexity Here, , and Q j, max and Z j, max are upper bounds for Q j (t) and Z j (t).

Proof. See Appendix A.
From Lemma 1, P1 can be decoupled into the following auxiliary problem at each single SP t: subject to (2) and (7b)-(7d). For more precise transformation, we introduce eorem 1 to prove the relationship between the solutions of P1 and P2.
□ Theorem 1. S β is denoted as the optimal average energy consumption under P2, which is limited by an upper bound.
where S α is the optimal average energy consumption under P1. e optimal value of S β can be arbitrarily close to the optimal value S α by selecting asymptotically large V.
Proof. See Appendix B.
subject to (2) and (7d), To solve the decentralized network problem, the mmW APs are modeled as agents which can make decisions independently. Based on the initialization-free distributed method in [17,19], problem (15) can be rewritten as min X i , i∈I i∈I subject to where f i (X i ) is a convex function which is equivalent to (15) and is the global constraint related to (2), where I � [I T , . . . , g ij � [g ij,1 , . . . , 0, . . . , g ij,J ] T , and g ij,k � n∈I x n,j g a ij,kn g u ji,nk g c jn . Equation (18b) is the local feasibility constraint of agent i associated with (7c), (16a), and (16b).
e local objective function f i (X i ), the interference matrix G i and closed convex set Ω i are the private information not shared with others of agent i. Equation (17) is a distributed optimization problem. In our framework, under the coverage of information cloud, the mmW APs can carry out data calculation and information interaction through the IC. Hence, we need to design a semidistributed joint resource and power allocation strategy for the agents, in which each agent interacts with its neighbor agents through the IC and collaborates to find its own optimal scheme.

e Proposed Semidistributed
Algorithm. First, we defined the Lagrangian duality of (17) with Lagrange multiplier λ as where λ can be updated via IC. Due to the global multiplier λ, we need to solve the subproblems to calculate the gradients in the IC. To avoid this shortage, we can introduce the Laplacian matrix L and the multiplier matrix Λ � col(λ 1 , . . . , λ I ) and reformulate the following constrained optimization problem [15].
subject to where ⊗ denotes the Kronecker product operation and I i is an identity matrix. Before further defining the Laplacian matrix L, we need to introduce the graph theory which can describe the information exchange among the agents through a graph G � (N, E). According to the graph theory [20], the edge set is defined as E ⊂ N × N. If agent k can receive information from agent n, then agent n belongs to agent k's neighbor set, denoted asN k � n| (n, k) ∈ E { }. In this work, we need to assume that the information exchange graph G � (N, E) is ALGORITHM 1: Distributed algorithm for each agent i, i ∈ I.
(1) Initialize ℓ � 1, X(1), Λ(1), Z(1) (2) Set the neighbor sets for all the mmW APs as 2 ) ≥ σ (5) mmW APs obtain the information of Λ i (ℓ) and Z i (ℓ), ∀i ∈ I from the information cloud; (6) Each mmW AP i, ∀i ∈ I updates its power allocation and multiplier parameters independently based on (23); (7) mmW APs i feed back the information of Λ i (ℓ) and Z i (ℓ), ∀i ∈ I to the information cloud; (8) set ℓ � ℓ + 1.   6 Complexity undirected and connected to guarantee that any agent's information can reach any other agents. According to the above, the Laplacian matrix can be defined as L � De g − A, whereA � [a kn ] is the adjacency matrix of graph G. We denote a kn � 1 if n belongs to agent k's neighbor set, and a kn � 0 otherwise. De g � diag ( I n�1 a 1n , . . . , I n�1 a In ) is the degree matrix. Constraint (21a) is defined to guarantee λ 1 � λ 2 � · · · � λ I , which insure the consensus among agents i is reached.
Based on the above conversion, the original distributed optimization problem has been decoupled and each agent can decide its own resource and power allocation strategy.
By introducing the Lagrangian multiplier Z � col (z 1 , . . . , z I ), we denote the augmented Lagrangian duality of (20) as According to [19], (22) can be solved by applying the gradient flow and the distributed algorithm for agent i based on projection dynamics given in Algorithm 1.

Complexity
In Algorithm 1, _ f(·) represents gradient calculation of f, ℓ is the update index, P Ω i (x) � argmin y∈Ω i ‖x − y‖ is the projection operation of x onto the closed convex set Ω i , ∇f means the gradient of f, and N i is the neighbor set of agent i. Based on the distributed algorithm, both the power allocation and multiplier parameters can be updated by where α is the integration step. Algorithm 2 summarizes the detailed procedures of the semidistributed scheme. Specifically, mmW APs achieve the information of Lagrangian multipliers from the information cloud, then calculatetheir power allocation strategy independently, and feed back λ i (ℓ) and z i (ℓ), ∀i ∈ I, to the information cloud.

e Performance Analysis of the Semidistributed Scheme.
e first and most important advantage of Algorithm 2 is that it does not sacrifice the optimality. We can obtain the global optimal solution for the distributed mmW networks to maximize the utilization efficiency of wireless resource. We introduce the following theorem to prove.

Theorem 2.
e proposed semidistributed algorithm can converge to an optimal power allocation solution.
Proof. See Appendix C. e second merit is initialization-free, which means that the semidistributed algorithm will converge to the global optimal solution under any initial condition due to the projection operation.

Simulation and Evaluation
In this section, different contrast graphs were obtained by adjusting various parameters. ere are multiple line segments with different parameters in each experimental result image. We draw images with lines of different colors and shapes for easy differentiation [5]. Under the coverage of IC, we set 4 mmW APs, each equipped with 6 RF chains. And there are 24 densely deployed MRs served by the mmW network. We denote the set of MRs as N � N 1 , N 2 , . . . , N 24 , where the minimum data rate of the MRs is set from 0.5 Mbps to 30 Mbps. e remaining parameters are shown in Table 1. Figure 3 shows the convergence of Algorithm 2 and the effect of integration step on the convergence rate. From the figure, we can observe that the semidistributed power allocation algorithm converges to the optimal solution through iteration, and the larger the integration step setting is, the faster the convergence rate will be. erefore, we select the appropriate integration step on the premise of ensuring the performance of the algorithm. Figure 4 shows the variation of data rate, energy consumption, and average latency with the nonnegative control parameters V, which represents the weight of the objective function in the conversion of Lyapunov drift penalty. From Figure 4(C), we can clearly observe that the time-average latency increases gradually with the gradual increase of parameter V. In addition, mobile robots with higher data rates usually result in higher energy consumption and lower time latency. Figure 5 shows the performance of two different solutions. e semidistributed method refers to the proposed resource and power allocation scheme in this paper, and the centralized method refers to solve optimization problems directly. From the picture, we can observe that the semidistributed method has significantly lower energy consumption but slightly higher time-average latency than the centralized method.

Conclusion
is paper proposes a mmW communication network for distributed MRs in a MEC environment. While minimizing network energy consumption, a joint resource and power distribution scheme is designed. First, Lyapunov optimization is processed by the target function as a single frame problem and then introduced into the semidistributed algorithm which further solves distributed optimization issues on mmW MECs. Experiments prove that the program can effectively reduce network energy consumption, and the semidistributed algorithm can greatly reduce signaling overheads, prevent privacy leakage, and further enhance the performance of distributed MRs communication networks.

Appendix A
Proof. of Lemma 1. Denote . Without expectation, ΔZ j (t) can be written as   Complexity Further, we can derive that Here, (a) is with the fact of ([x] + ) 2 ≤ x 2 , and (b) is proved by the following equation: We assume that the queue data that have not been backlogged and need not to be transmitted; inequation (c) has the causality constraint of According to [18], constraint (A.4) can be ignored. Substituting (A.2) into (11), we can obtain the upper bound of Δ ] Z(t), which ends the proof.
Without loss of generation, we assume that each agent has only one variable, i.e., X k ∈ R. A new vector S � col(X, Λ, Z) is defined, and its feasible region is Ω � Ω × R K × R K . By applying the gradient flow, the updating rules of all agents can be expressed as where the vector function F(S) is denoted as To prove the convergence of the proposed algorithm, the Lyapunov stability can be applied. First, we introduce a Lyapunov function to analyze the Lyapunov stability, which is given by where H(S) � P Ω (S − F(S)), S * � col(X * , Λ * , Z * ), X * is the optimal solution, and Λ * and Z * are the optimal multipliers. To guarantee the Lyapunov stability of the equilibrium point, V g must be constant positive or constant negative. Since any asymmetric variational inequality can be converted to a differentiable optimization problem [21], V g can be expressed as With the assumption that R(X k ) are strictly convex and the information exchanging graph is undirected and connected, the following result can be obtained: Based on (C.9) and (C.10), _ V g can be further analyzed as follows: Since 〈F(S), S − S * 〉 ≥ 0, _ V g ≤ 0, and the equilibrium point is assumed to be S * . erefore, the equilibrium point is Lyapunov stable. Hence, the proposed distributed algorithm can converge to its equilibrium point, which ends the proof.

Data Availability
Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.