# Distributed Optimization for Mobile Robots under Mobile Edge Computing Environment

Authors: Hui Luo; Quan Yin
Journal: Complexity
 (2021)

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## Abstract

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. This paper proposes a mmW communication network architecture for distributed MRs in MEC environment. The 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. Then, 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.


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## Body

## 1. 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. The authors in [4] proposed a scheme of mobile robot object recognition in Internet of Things (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. The 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 small cellular networks and maximize the total utility of the whole system. The authors in [7] proposed a distributed algorithm based on the machine learning framework of echo state networks, aiming at the problems of user association and spectrum allocation in heterogeneous small cellular networks, which allowed small base stations to independently complete the optimal allocation of resources under the condition of limited system state information and greatly reduced the wireless network information exchange. The authors in [8] proposed an adaptive semidistributed algorithm that can jointly allocate power and spectrum resources on licensed and unlicensed bands, which can improve the independent adaptability of the base station and maximize the global spectrum efficiency. Based on the research above, we need to develop a distributed network resource management scheme which is more suitable for MRs network and has stronger autonomy and flexibility, higher security, and cost performance.In this regard, Mobile Edge Computing (MEC) technology can sink elastic resources such as computing, network, and storage in cloud data center to the edge of distributed network, effectively relieving the pressure on core network, improving computing efficiency, and reducing service latency [9–11]. MEC deeply integrates mobile access network with Internet service, which is very suitable for the MRs communication system with distributed structure. Meanwhile, to meet the development needs of high speed, low delay, and large capacity of mobile communication network in the future, human beings are exploring more abundant spectrum resources in higher frequency band. Among them, millimeter wave (mmW) not only has a wide spectrum, which can greatly improve the channel capacity, but also has a good direction, which can effectively reduce channel interference and protect user privacy [12]. Applying mmW technology to MRs communication networks can further improve network performance and user experience, which is an important trend of distributed MRs network development. In the existing work, the authors in [13] combined the computational shunt technology in MEC with mmW communication to conduct joint optimization of computing and communication resources, forming an energy-saving and efficient computing system. Aiming at distributed small cellular networks, a green load management and resource allocation scheme based on MEC is proposed in [14, 15], which is constrained by network cost and communication quality to achieve green computing shunt and reduce the total energy consumption of users to the greatest extent.Based on the current research, we propose a distributed MRs communication network architecture in MEC environment. The mmW access points (APs) are randomly distributed under the coverage of the information cloud (IC) to provide communication services for the densely deployed MRs, in which the IC is responsible for information exchange, storage, and interaction. To minimize the energy consumption of the MRs network and achieve the optimal network performance, we designed a joint resource and power allocation strategy based on MEC under the condition of guaranteeing the network traffic delay, limiting rate, 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.The rest of the paper is organized as follows. In Section2, we describe the system model of the distributed MRs communication network. Section 3 formulates and transforms the optimization problem. The 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 diagx1,…,xK 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.


## 2. 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 toJ MRs within its coverage as shown in Figure 1. The set of mmW APs and MRs is denoted as ℐ=W1,W2,…,WI and J=R1,R2,…,RJ. 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.Figure 1
The network framework for mobile robots in MEC environment.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. The mmW APWi equipped with Ci radio frequency (RF) chains can serve most Ci MRs. Moreover, we assume that the proposed network transmits in a slotted system, which is represented as t = {0, 1, 2, …}. Denote xi,jt=1 if the j-th MR is in the coverage of the mmW AP Wi at time slot t, and xi,jt=0 otherwise. We assume each MR is attached to not more than one mmW AP, i.e., ∑i=1Ixi,jt≤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=1Jxi,jt≤Ci,∀i∈ℐ.


### 2.1. mmW Transmission Model

The 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 θiat andθjrt be the operating beamwidths of mmW AP Wi and MR Rj, respectively. Let ζijat be the angle between the positive x-axis and the direction in which the mmW AP Wi sees MR Rj, and let ζijrt be similarly defined by changing the roles of the mmW AP Wi and MR Rj.. The values of ζijat and ζijrt satisfy ζijat−ζijrt=π. We further denote φijat and φjrt as the boresight angles of the mmW AP Wi and MR Rj relative to the positive x-axis. Let gij,jiat be the transmission gain between the beam from the mmW AP Wi to MR Rj. and gji,ijrt be the reception gain between the beam from MR Rj to the mmW AP Wi, which can be written as follows.Figure 2
Illustration of the angles between the mmW AP and MRs in the transmission.(1)gij,jiat=ε,ifθiat2<φijat−ζijat<2π−θiat2,2π−2π−θiatεθiat,otherwise,gji,ijrt=ε,ifθirt2<φjrt−ζijrt<2π−θirt2,2π−2π−θirtεθirt,otherwise.


### 2.2. Interference Management

According to [16], there are two types of interference in the proposed mmW network that should be managed. The intracell interference exists among devices within a cell and can be mitigated by using proper scheduling and beamforming design. The intercell interference exists among different cells and can be significantly reduced by resource blocks scheduling and omnidirectional communication. Therefore, to ensure that the intercell interference in the mmW network is small enough to be ignored, the following constraints can be introduced:(2)∑i′∈ℐi′≠i∑j′∈Jj′≠jpi′j′atgi′j′,jiatgi′jctgj′i′,ijrt≤IT,where IT is the intercell interference threshold, pijat denotes the transmit power from the mmW AP Wi to MR Rj, andgijct refers the channel gain between the mmW AP Wi to MR Rj.


### 2.3. Data Transmission

LetQjt be the data queue length of MR Rj at time slot t. The data entering the queue of MR Rj at time slot t are denoted as Ajt, whose maximum value is Aj,maxt. The traffic buffer queue Qjt of MR Rj evolves as(3)Qjt+1=Qjt−Rjatτ++Ajtτ,∀j∈J,t,where x+=maxx,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 Qj0=0,∀j∈J for simplicity.The achievable data rate of MRRj served by the mmW AP Wi at time slot t can be written as(4)Rijat=xi,jtBalog21+pijatgij,jiatgijctgji,ijrtBaN0,∀j∈J,i∈ℐ,t,where Ba is the mmW bandwidth and N0 is the noise power spectral density. Based on (4), the achievable data rate of MR Rj served by the mmW network at time slot t is given by(5)Rjat=∑i=1IRijat,∀j∈J,t.


## 2.1. mmW Transmission Model

The 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 θiat andθjrt be the operating beamwidths of mmW AP Wi and MR Rj, respectively. Let ζijat be the angle between the positive x-axis and the direction in which the mmW AP Wi sees MR Rj, and let ζijrt be similarly defined by changing the roles of the mmW AP Wi and MR Rj.. The values of ζijat and ζijrt satisfy ζijat−ζijrt=π. We further denote φijat and φjrt as the boresight angles of the mmW AP Wi and MR Rj relative to the positive x-axis. Let gij,jiat be the transmission gain between the beam from the mmW AP Wi to MR Rj. and gji,ijrt be the reception gain between the beam from MR Rj to the mmW AP Wi, which can be written as follows.Figure 2
Illustration of the angles between the mmW AP and MRs in the transmission.(1)gij,jiat=ε,ifθiat2<φijat−ζijat<2π−θiat2,2π−2π−θiatεθiat,otherwise,gji,ijrt=ε,ifθirt2<φjrt−ζijrt<2π−θirt2,2π−2π−θirtεθirt,otherwise.


## 2.2. Interference Management

According to [16], there are two types of interference in the proposed mmW network that should be managed. The intracell interference exists among devices within a cell and can be mitigated by using proper scheduling and beamforming design. The intercell interference exists among different cells and can be significantly reduced by resource blocks scheduling and omnidirectional communication. Therefore, to ensure that the intercell interference in the mmW network is small enough to be ignored, the following constraints can be introduced:(2)∑i′∈ℐi′≠i∑j′∈Jj′≠jpi′j′atgi′j′,jiatgi′jctgj′i′,ijrt≤IT,where IT is the intercell interference threshold, pijat denotes the transmit power from the mmW AP Wi to MR Rj, andgijct refers the channel gain between the mmW AP Wi to MR Rj.


## 2.3. Data Transmission

LetQjt be the data queue length of MR Rj at time slot t. The data entering the queue of MR Rj at time slot t are denoted as Ajt, whose maximum value is Aj,maxt. The traffic buffer queue Qjt of MR Rj evolves as(3)Qjt+1=Qjt−Rjatτ++Ajtτ,∀j∈J,t,where x+=maxx,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 Qj0=0,∀j∈J for simplicity.The achievable data rate of MRRj served by the mmW AP Wi at time slot t can be written as(4)Rijat=xi,jtBalog21+pijatgij,jiatgijctgji,ijrtBaN0,∀j∈J,i∈ℐ,t,where Ba is the mmW bandwidth and N0 is the noise power spectral density. Based on (4), the achievable data rate of MR Rj served by the mmW network at time slot t is given by(5)Rjat=∑i=1IRijat,∀j∈J,t.


## 3. Problem Formulation and Transformation

### 3.1. 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. The energy aware optimization problem can be formulated as(6)P1:minpalimsupT⟶+∞1T∑t=0T−1E∑j∈Jpjatτ,subject to (2),(7a)limsupT⟶+∞1T∑t=0T−1EQjt<Q¯j,∀j∈J,t,(7b)Rjat≥RjT,∀j∈J,t,(7c)pjat≤PjT,∀j∈J,t,(7d)pijat≥0,∀j∈J,i∈ℐ,t,wherepjat=∑i∈ℐpijat, Q¯j is the predefined average queue backlog tolerance of MR Rj, and RjTt and PjT are the minimum data rate requirement and the maximum power consumption of MR Rj 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.


### 3.2. Lyapunov Drift Penalty

Since the optimization functionP1 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 Zn for constraint (7a) as(8)Zjt+1=Zjt+Qjt+1−Q¯j+,∀j∈J,t.For each slott, the Lyapunov function can be formulated as a sum of virtual queue length squares, given by(9)LZt=12∑j=1JZj2t,∀t,where Zt=Z1t,Z2t,…,ZJt.Moreover, the Lyapunov drift function is expressed as(10)△Zt=ELZt+1−LZt|Zt,∀t.Then, the Lyapunov drift penalty function is defined to minimize the drift-plus-penalty expression boundary, which can be formulated as(11)ΔνZt=E12∑j∈JZj2t+1−Zj2t|Zt+VE∑j∈Jpjatτ|Zt,∀t,where V is the nonnegative control parameter, which represents the weight of the objective function.Lemma 1.
The upper bound for the Lyapunov drift penalty (11) can be written as(12)△νZt≤∑j∈JZj1+Zj22+VE∑j∈Jpjatτ|Zt−E∑j∈JQjt+ZjtRjatτ|Zt,∀t.
Here,Zj1=Qj,max+Aj,max2+Qj,max2+Q¯j2,Zj2=2Zj,maxQj,max+Aj,max, and Qj,max and Zj,maxare upper bounds for Qjt and Zjt.Proof.
See Appendix A.
From Lemma1, P1 can be decoupled into the following auxiliary problem at each single SP t:(13)P2:minpaV∑j∈Jpjat−∑j∈JQjt+ZjtRjat,subject to (2) and (7b)–(7d). For more precise transformation, we introduce Theorem 1 to prove the relationship between the solutions of P1 and P2.Theorem 1.
Sβ is denoted as the optimal average energy consumption under P2, which is limited by an upper bound.(14)Sβ≤∑j∈JZj1+Zj22V+Sα,where Sα is the optimal average energy consumption under P1. The optimal value of Sβ can be arbitrarily close to the optimal value Sα by selecting asymptotically large V.Proof.
See Appendix B.


### 3.3. Distributed Optimization Problem

In the following, we omit the time slot indext for ease of exposition. Based on P2, we introduce a block of slack variables c=c1,c2,…,cJ and d=d1,d2,…,dJ and get the decentralized mmW network problem as follows:(15)minpaV∑i∈ℐ∑j∈Jpija−∑i∈ℐ∑j∈JQi+ZiRija,subject to (2) and (7d),(16a)Rija≥xi,jRjT−cj,∀j∈J,i∈ℐ,(16b)pija≤xi,jPjT−dj,∀j∈J,i∈ℐ.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(17)minXi,i∈ℐ∑i∈ℐfiXi,subject to(18a)∑i∈ℐGiXi=∑i∈ℐI,(18b)Xi∈Ωi,∀i∈ℐ.where fiXi is a convex function which is equivalent to (15) and Xi=p1ia,…,pJiaT. Equation (18a) is the global constraint related to (2), where I=IT,…,ITT, Gi=gi1,…,giJ, gij=gij,1,…,0,…,gij,JT, and gij,k=∑n∈ℐxn,jgij,knagji,nkugjnc. Equation (18b) is the local feasibility constraint of agent i associated with (7c), (16a), and (16b).The local objective functionfiXi, the interference matrix Gi and closed convex set Ωi 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.


## 3.1. 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. The energy aware optimization problem can be formulated as(6)P1:minpalimsupT⟶+∞1T∑t=0T−1E∑j∈Jpjatτ,subject to (2),(7a)limsupT⟶+∞1T∑t=0T−1EQjt<Q¯j,∀j∈J,t,(7b)Rjat≥RjT,∀j∈J,t,(7c)pjat≤PjT,∀j∈J,t,(7d)pijat≥0,∀j∈J,i∈ℐ,t,wherepjat=∑i∈ℐpijat, Q¯j is the predefined average queue backlog tolerance of MR Rj, and RjTt and PjT are the minimum data rate requirement and the maximum power consumption of MR Rj 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.


## 3.2. Lyapunov Drift Penalty

Since the optimization functionP1 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 Zn for constraint (7a) as(8)Zjt+1=Zjt+Qjt+1−Q¯j+,∀j∈J,t.For each slott, the Lyapunov function can be formulated as a sum of virtual queue length squares, given by(9)LZt=12∑j=1JZj2t,∀t,where Zt=Z1t,Z2t,…,ZJt.Moreover, the Lyapunov drift function is expressed as(10)△Zt=ELZt+1−LZt|Zt,∀t.Then, the Lyapunov drift penalty function is defined to minimize the drift-plus-penalty expression boundary, which can be formulated as(11)ΔνZt=E12∑j∈JZj2t+1−Zj2t|Zt+VE∑j∈Jpjatτ|Zt,∀t,where V is the nonnegative control parameter, which represents the weight of the objective function.Lemma 1.
The upper bound for the Lyapunov drift penalty (11) can be written as(12)△νZt≤∑j∈JZj1+Zj22+VE∑j∈Jpjatτ|Zt−E∑j∈JQjt+ZjtRjatτ|Zt,∀t.
Here,Zj1=Qj,max+Aj,max2+Qj,max2+Q¯j2,Zj2=2Zj,maxQj,max+Aj,max, and Qj,max and Zj,maxare upper bounds for Qjt and Zjt.Proof.
See Appendix A.
From Lemma1, P1 can be decoupled into the following auxiliary problem at each single SP t:(13)P2:minpaV∑j∈Jpjat−∑j∈JQjt+ZjtRjat,subject to (2) and (7b)–(7d). For more precise transformation, we introduce Theorem 1 to prove the relationship between the solutions of P1 and P2.Theorem 1.
Sβ is denoted as the optimal average energy consumption under P2, which is limited by an upper bound.(14)Sβ≤∑j∈JZj1+Zj22V+Sα,where Sα is the optimal average energy consumption under P1. The optimal value of Sβ can be arbitrarily close to the optimal value Sα by selecting asymptotically large V.Proof.
See Appendix B.


## 3.3. Distributed Optimization Problem

In the following, we omit the time slot indext for ease of exposition. Based on P2, we introduce a block of slack variables c=c1,c2,…,cJ and d=d1,d2,…,dJ and get the decentralized mmW network problem as follows:(15)minpaV∑i∈ℐ∑j∈Jpija−∑i∈ℐ∑j∈JQi+ZiRija,subject to (2) and (7d),(16a)Rija≥xi,jRjT−cj,∀j∈J,i∈ℐ,(16b)pija≤xi,jPjT−dj,∀j∈J,i∈ℐ.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(17)minXi,i∈ℐ∑i∈ℐfiXi,subject to(18a)∑i∈ℐGiXi=∑i∈ℐI,(18b)Xi∈Ωi,∀i∈ℐ.where fiXi is a convex function which is equivalent to (15) and Xi=p1ia,…,pJiaT. Equation (18a) is the global constraint related to (2), where I=IT,…,ITT, Gi=gi1,…,giJ, gij=gij,1,…,0,…,gij,JT, and gij,k=∑n∈ℐxn,jgij,knagji,nkugjnc. Equation (18b) is the local feasibility constraint of agent i associated with (7c), (16a), and (16b).The local objective functionfiXi, the interference matrix Gi and closed convex set Ωi 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.


## 4. Semidistributed Resource and Power Allocation

### 4.1. The Proposed Semidistributed Algorithm

First, we defined the Lagrangian duality of (17) with Lagrange multiplier λ as(19)maxλqλ=∑i∈ℐqiλ=∑i∈ℐinfXi∈ΩifiXi+λTGi−I,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=colλ1,…,λI and reformulate the following constrained optimization problem [15].(20)maxΛQΛ=∑i∈ℐqiλi,subject to(21a)L⊗IiΛ=0,where ⊗ denotes the Kronecker product operation and Ii is an identity matrix.Before further defining the Laplacian matrixL, 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 agentn belongs to agent k’s neighbor set, denoted asNk=n|n,k∈E. In this work, we need to assume that the information exchange graph G=N,E is 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 asL=Deg−A, whereA=akn is the adjacency matrix of graph G. We denote akn=1 if n belongs to agent k’s neighbor set, and akn=0 otherwise. Deg=diag∑n=1Ia1n,…,∑n=1IaIn is the degree matrix. Constraint (21a) is defined to guarantee λ1=λ2=⋯=λI, 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 multiplierZ=colz1,…,zI, we denote the augmented Lagrangian duality of (20) as(22)minZmaxΛQΛ,Z=∑i∈ℐqiλi−ZTL⊗IiΛ−12ΛTL⊗IiΛ.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.Algorithm 1: Distributed algorithm for each agent i, i∈ℐ.
(1)
X⋅iℓ+1=PΩiXiℓ−∇fiXiℓ+λiℓGi−Xiℓ;(2)
λi⋅ℓ+1=−∑k∈Niλiℓ−λkℓ−∑k∈Nkziℓ−zkℓ+I/K−XiℓGi;(3)
z⋅iℓ+1=∑k∈Niλiℓ−λkℓ;In Algorithm1, f˙⋅ represents gradient calculation of f, ℓ is the update index, PΩix=argminy∈Ωix−y is the projection operation of x onto the closed convex set Ωi, ∇f means the gradient of f, and Ni is the neighbor set of agent i. Based on the distributed algorithm, both the power allocation and multiplier parameters can be updated by(23)piaℓ+1=piaℓ+αpia⋅ℓ+1,λiℓ+1=λiℓ+αλi⋅ℓ+1,ziℓ+1=ziℓ+αzi⋅ℓ+1,where α is the integration step.Algorithm2 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 λiℓ and ziℓ,∀i∈ℐ, to the information cloud.Algorithm 2: Semidistributed power allocation scheme.
(1)
Initializeℓ=1,X1,Λ1,Z1(2)
Set the neighbor sets for all the mmW APs asNi,∀i∈ℐ(3)
Letσ⟶0(4)
whileX˙ℓ22+Λ˙ℓ22+Z˙ℓ22≥σ(5)
mmW APs obtain the information ofΛiℓ and Ziℓ,∀i∈ℐ from the information cloud;(6)
Each mmW APi,∀i∈ℐ updates its power allocation and multiplier parameters independently based on (23);(7)
mmW APsi feed back the information of Λiℓ and Ziℓ,∀i∈ℐ to the information cloud;(8)
setℓ=ℓ+1.(9)
Endwhile


### 4.2. The Performance Analysis of the Semidistributed Scheme

The first and most important advantage of Algorithm2 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.
The proposed semidistributed algorithm can converge to an optimal power allocation solution.Proof.
See Appendix C.
The 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.


## 4.1. The Proposed Semidistributed Algorithm

First, we defined the Lagrangian duality of (17) with Lagrange multiplier λ as(19)maxλqλ=∑i∈ℐqiλ=∑i∈ℐinfXi∈ΩifiXi+λTGi−I,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=colλ1,…,λI and reformulate the following constrained optimization problem [15].(20)maxΛQΛ=∑i∈ℐqiλi,subject to(21a)L⊗IiΛ=0,where ⊗ denotes the Kronecker product operation and Ii is an identity matrix.Before further defining the Laplacian matrixL, 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 agentn belongs to agent k’s neighbor set, denoted asNk=n|n,k∈E. In this work, we need to assume that the information exchange graph G=N,E is 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 asL=Deg−A, whereA=akn is the adjacency matrix of graph G. We denote akn=1 if n belongs to agent k’s neighbor set, and akn=0 otherwise. Deg=diag∑n=1Ia1n,…,∑n=1IaIn is the degree matrix. Constraint (21a) is defined to guarantee λ1=λ2=⋯=λI, 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 multiplierZ=colz1,…,zI, we denote the augmented Lagrangian duality of (20) as(22)minZmaxΛQΛ,Z=∑i∈ℐqiλi−ZTL⊗IiΛ−12ΛTL⊗IiΛ.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.Algorithm 1: Distributed algorithm for each agent i, i∈ℐ.
(1)
X⋅iℓ+1=PΩiXiℓ−∇fiXiℓ+λiℓGi−Xiℓ;(2)
λi⋅ℓ+1=−∑k∈Niλiℓ−λkℓ−∑k∈Nkziℓ−zkℓ+I/K−XiℓGi;(3)
z⋅iℓ+1=∑k∈Niλiℓ−λkℓ;In Algorithm1, f˙⋅ represents gradient calculation of f, ℓ is the update index, PΩix=argminy∈Ωix−y is the projection operation of x onto the closed convex set Ωi, ∇f means the gradient of f, and Ni is the neighbor set of agent i. Based on the distributed algorithm, both the power allocation and multiplier parameters can be updated by(23)piaℓ+1=piaℓ+αpia⋅ℓ+1,λiℓ+1=λiℓ+αλi⋅ℓ+1,ziℓ+1=ziℓ+αzi⋅ℓ+1,where α is the integration step.Algorithm2 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 λiℓ and ziℓ,∀i∈ℐ, to the information cloud.Algorithm 2: Semidistributed power allocation scheme.
(1)
Initializeℓ=1,X1,Λ1,Z1(2)
Set the neighbor sets for all the mmW APs asNi,∀i∈ℐ(3)
Letσ⟶0(4)
whileX˙ℓ22+Λ˙ℓ22+Z˙ℓ22≥σ(5)
mmW APs obtain the information ofΛiℓ and Ziℓ,∀i∈ℐ from the information cloud;(6)
Each mmW APi,∀i∈ℐ updates its power allocation and multiplier parameters independently based on (23);(7)
mmW APsi feed back the information of Λiℓ and Ziℓ,∀i∈ℐ to the information cloud;(8)
setℓ=ℓ+1.(9)
Endwhile


## 4.2. The Performance Analysis of the Semidistributed Scheme

The first and most important advantage of Algorithm2 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.
The proposed semidistributed algorithm can converge to an optimal power allocation solution.Proof.
See Appendix C.
The 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.


## 5. Simulation and Evaluation

In this section, different contrast graphs were obtained by adjusting various parameters. There 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=N1,N2,…,N24, where the minimum data rate of the MRs is set from 0.5 Mbps to 30 Mbps. The remaining parameters are shown in Table 1.Table 1
Parameter setting.
ParametersValuesMinimum data rateRT0.5 to 30 mbpsMaximum power consumptionPT27 dBmQueue backlog toleranceQj1 to 10 msNoise power spectral densityN0−134 dBm/MHzThe mmW bandwithBa2 GHzOperating beamwidths between mmW AP and MRθ10°The intercell interference thresholdIT10−10 WThe duration of each time slotτ1 msFigure3 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. Therefore, we select the appropriate integration step on the premise of ensuring the performance of the algorithm.Figure 3
Convergence performance of Algorithm2 under asynchronous integration step.Figure4 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 4
The effects of nonnegative control parameter on mmW network performance: (a) data rate; (b) energy consumption; (c) average latency.
(a)(b)(c)Figure5 shows the performance of two different solutions. The 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.Figure 5
The performance between two different solutions.


## 6. Conclusion

This 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.



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*Source: 8342610-2021-10-21.xml*