Intelligent Reflecting Surface Meets Mobile Edge Computing: Enhancing Wireless Communications for Computation Offloading
Yang Liu, Jun Zhao, Zehui Xiong, Dusit Niyato, Chau Yuen, Cunhua Pan, Binbin Huang
aa r X i v : . [ ee ss . SP ] J a n Intelligent Reflecting Surface Meets Mobile EdgeComputing: Enhancing Wireless Communicationsfor Computation Offloading
Yang Liu , Jun Zhao , Zehui Xiong , Dusit Niyato , Chau Yuen , Cunhua Pan , Binbin Huang { yang-liu,junzhao,zxiong002,dniyato } @ntu.edu.sg, [email protected], [email protected], [email protected]
1: Nanyang Technological University2: Singapore University of Technology and Design3: Queen Mary University of London4: Hangzhou Dianzi University
Abstract —We consider computation offloading for edge com-puting in a wireless network equipped with intelligent reflectingsurfaces (IRSs). IRS is an emerging technology and has recentlyreceived great attention since they can improve the wirelesspropagation environment in a configurable manner and enhancethe connections between mobile devices (MDs) and access points(APs). At this point not many papers consider edge computing inthe novel context of wireless communications aided by IRS. In ourstudied setting, each MD offloads computation tasks to the edgeserver located at the AP to reduce the associated comprehensivecost, which is a weighted sum of time and energy. The edgeserver adjusts the IRS to maximize its earning while maintainingMDs’ incentives for offloading and guaranteeing each MD acustomized information rate. This problem can be formulatedinto a difficult optimization problem, which has a sum-of-ratioobjective function as well as a bunch of nonconvex constraints.To solve this problem, we first develop an iterative evaluationprocedure to identify the feasibility of the problem when con-fronting an arbitrary set of information rate requirement. Thismethod serves as a sufficient condition for the problem beingfeasible and provides a feasible solution. Based on that we developan algorithm to optimize the objective function. Our numericalresults show that the presence of IRS enables the AP to guaranteehigher information rate to all MDs and at the same time improvethe earning of the edge server.
Index Terms —Computation offloading, edge computing,intelligent surfaces, wireless networks.
I. I
NTRODUCTION
Edge computing . As an emerging computing paradigm, edge computing [1], [2] pushes computing tasks to the networkedges such as base stations and access points. Both edgecomputing and cloud computing can support computation-intensive applications. Yet, since edge servers are closer tomobile devices than clouds, edge computing is more suitableto enable latency-critical applications at mobile devices [3].Many companies are now leveraging edge computing to en-sure that their services are available at the edges with highspeed [4]. In July 2019, AT&T signed a deal of US$2 billionwith Microsoft to use the latter’s capabilities related to edgeand cloud computing [5].
Computation offloading for edge computing.
An impor-tant topic studied in edge computing is computation offload-ing [6], where a mobile device offloads intensive computationtasks to the edge server with stronger computational resources.In the seminal model for computation offloading proposedby [7], [8] by Chen and his co-authors, mobile devices choosecomputation offloading to the edge cloud or local computingon the devices by comparing the utilities under the twosettings, where the utility is computed as a weighted functionof the time and energy required to complete the computationtask. This model, which our paper will use, has been adopted(with possible refinements) in many studies [9], [10].
Intelligent reflecting surfaces (IRSs) and IRS-aided com-munications . An intelligent reflecting surface (IRS) can intel-ligently control the wireless environment to improve signalstrength received at the destination. This is vastly differentfrom prior techniques which improve wireless communicationsvia optimizations at the sender or receiver. Specifically, anIRS consists of many IRS units, each of which can reflect theincident signal at a reconfigurable angle. In such
IRS-aidedcommunications , the wireless signal travels from the source tothe IRS, is optimized at the IRS, and then travels from the IRSto the destination. Such communication method is particularlyuseful when the source and destination have a weak wirelesschannel in between due to obstacles or poor environmentalconditions, or they do not have direct line of sights.Because of the capability of configuring wireless envi-ronments, IRSs are envisioned by many experts in wirelesscommunications to play an important role in 6G networks. InNovember 2018, the Japanese mobile operator NTT DoCoMoand a startup MetaWave demonstrated the use of IRS-liketechnology for assisting wireless communications in 28GHzband [11]. IRSs have been compared with the massive MIMOtechnology used in 5G communications. IRSs reflect wirelesssignals and hence consume little power, whereas massiveMIMO transmits signals and needs much more power [12].
Problem studied in this paper: Computation offloadingfor edge computing in IRS-aided communications.
Weackle the problem of mobile devices offloading computationtasks to a base station equipped with an edge server, in thecontext of IRS-aided wireless communications. In the studiedsetting, mobile devices intend to send computation tasks tothe base station via wireless channels, and communicationsbetween them are assisted by an IRS. Fig. 1 provides anillustration of the system model.
Contributions.
The contributions of this paper are summa-rized as follows:1) To the best of our knowledge, there exists no paperconsidering the computation offloading problem in thepresence of IRS except that the very recent paper [13]by Bai et al. considers the latency minimization problemwith the aid of IRS in edge computing system. In thispaper, we extend the cost metric of mobile computingto a more comprehensive consideration, which subsumesthe one used in [13] as a special case. Besides we alsotake into account the rate constraints, which makes ourproblem more meaningful and much more challenging.2) To solve the problem, We first develop an algorithmto detect the feasibility when given an arbitrary setof information rate constraints. The algorithm can alsoprovide a feasible solution when we identify the problemas feasible. Then we develop an efficient solution whichcan maximize the earning of the edge server.3) Through substantial numerical results, we demonstratethe significance of IRS devices. Our experimental resultsshow that IRS can significantly boost the feasibilityprobability for information rate constraints. Besides, nu-merical results suggest that our algorithm converges fastand can effectively improve the edge server’s earning.
Notation.
Scalars are denoted by italic letters, while vectorsand matrices are denoted by bold-face lower-case and upper-case letters, respectively. C denotes the set of all complexnumbers. For a vector x , k x k denotes its Euclidean norm. Fora matrix M , its transpose and conjugate transpose are denotedby M T and M H , while M i,j means the element in the i -th row and j -th column of M . For a vector x , its transpose,conjugate transpose, and Euclidean norm are denoted by x T , x H , and k x k , while x i means the i -th element of x .II. R ELATED W ORK
Since our paper combines edge computing and wirelesscommunications aided by intelligent reflecting surfaces (IRSs),we discuss both related studies in edge computing and IRS-aided communications.
Edge computing and computation offloading . Edge com-puting pushes computations to the network edges in order tosupport latency-critical applications at mobile devices [14].In many studies [2], edge computing is also referred to as fog computing , a term introduced by Bonomi et al. [15] ofCisco. In 2018, IEEE integrated the fog computing referencearchitecture developed by the
OpenFog Consortium (now the
Industrial Internet Consortium ) as a standard [16].Among many references on computation offloading for edgecomputing, two studies most related to our work are the … Base Stationwith Edge Server Reconfigurable Intelligent Surface (RIS) RIS Unit (cid:1837)
Mobile Users possibly weak connections due to obstacles or environmental conditions
Fig. 1: An Illustration of IRS-Aided Communication Systemseminal work [7], [8] by Chen and his co-authors. Chen [7]considers a network of many mobile device users and onebase station equipped with edge cloud. Mobile devices selectcomputation offloading to edge cloud or local computing onthe devices by comparing the obtained utilities under the twosettings. All communications in the network use the samewireless channel and interfere each other. Hence, the datauploading rate of a mobile device selecting computation of-floading is related to the offloading decisions of other devices.Then the dynamics in offloading decision making of mobiledevices are modeled as a game, which is analyzed usingpotential game theory. Improving [7] to a more general setting,Chen et al. [8] address a wireless interference environment ofmultiple channels instead of only one channel. Thus, the datauploading rate of a mobile device selecting computation of-floading is related to other devices which use the same wirelesschannel. Then a game is used to decide local/edge computingchoices of mobile devices and their channel selection in thecase of edge computing.In addition to [7], [8] discussed above, many studies oncomputation offloading for edge computing have appearedrecently. We discuss some of them below and refer readersto [9], [17] for more comprehensive reviews. Mao et al. [10]consider computation offloading of an energy-harvesting de-vice to the edge server, where the incorporation of energyharvesting makes the design of computation offloading policychallenging. Partial computation offloading instead of full of-floading is investigated in [9], [18]–[20]. Sardellitti et al. [10]tackle computation offloading for edge computing in MIMOmulticell system so that the optimization variables to minimizemobile devices’ energy consumption include the transmit pre-coding matrices of mobile devices in addition to computationalresources assigned by the edge cloud to the devices. Chen andHao [21] analyze computation offloading in the framework ofsoftware defined networking (SDN), where the SDN controllerdecides for each offloaded task which edge cloud and howmuch computing resource to use.
Intelligent reflecting surface-aided communications .ince IRSs can be controlled to reflect incident wireless signalsin a desired way, IRS-aided communications have recentlyreceived much attention in the literature [22]–[28]. The studiesinclude analyses of data rates [22], [23], optimizations ofpower or spectral efficiency [24], [26], and channel estima-tion [27], [28]. In these studies, IRSs are also referred toas large intelligent surface [12], [23], intelligent reflectingsurface [27], [29], [30], software-defined surface [31], and passive intelligent mirrors [32] [33].We now first discuss IRS studies analyzing uplink commu-nications, where IRSs help transmissions from mobile devicesto base stations, since our work also focuses on uplinks.Jung et al. [22], [23] analyze the impact of channel estimationerrors on the uplink data rates.For IRSs assisting communications between mobile devicesand base stations, in addition to uplink studies discussedabove, downlinks are investigated in [24], [26], [34]–[36].Regarding that there seems to be more downlink studies thanuplink ones, a possible explanation is that in the former case,the base station can optimize the transmit beamforming in acentralize manner.In addition to the above settings where IRSs aid communi-cations between mobile devices and base stations, direct com-munications between mobile devices and IRSs are analyzedin [12], [37]–[39]. III. S
YSTEM M ODEL
We consider a wireless network consisting of a multi-antenna base station (BS)/access point (AP) and K single-antenna mobile devices (MDs). An illustration of the systemmodel has been given in Fig. 1. The BS/AP hosts an edgeserver for providing edge computing via wireless commu-nications. The mobile devices are numbered from to K .For k ∈ K , { , , . . . , K } , the k -th mobile device has acomputation task J k := ( b k , d k ) which it intends to offloadto the edge server, where b k denotes the data size of the task,and d k denotes the number of CPU cycles to execute the task.We now introduce the computing model and communicationmodel used in this paper. Mathematical details for them aredeferred to Sections IV-A and IV-C. Computing model.
Our computing model mainly followsthat of the seminal work [7], [8] discussed in Section II.The same as [7], [8], mobile devices choose computationoffloading to the edge cloud or local computing on the devicesby comparing the utilities under the two settings, where theutility in each setting depends on the time and energy requiredto complete the computation task. Compared with [7], [8],our model also requires the mobile device to issue a paymentto the edge cloud if a weighted function of the time andenergy required to complete the computation task has a largervalue under edge computing compared with local computing.Formal details of the computing model will be presented inSection IV-A.
Communication model.
The communications between mo-bile devices and the BS/AP are aided by IRSs. The signalsalso interfere each other when arriving at the base station. Formal details of the communication model will be presentedin Section IV-C.IV. O
PTIMIZATION P ROBLEM F ORMULATION
A. A Most Generic Model: Local Computing versus EdgeComputing
Utility of a mobile device under local computing.
Belowwe compute the utility of the k -th mobile device when itcompletes the task J k under local computing, where k ∈ K .Let the k -th mobile device’s computation capability be c ( m ) k CPU cycles per unit time, where the superscript “ ( m ) ”throughout the paper is used for quantities with a mobile de-vice. Then under local computing, the computation executiontime of the task J k with d k CPU cycles is t ( m ) k := d k c ( m ) k . (1)Let µ k be the energy per CPU cycle. The energy to completethe task J k with d k CPU cycles is e ( m ) k := µ k d k . (2)We consider that the k -th mobile device combines the timeand energy to complete the task J k in a weighted mannerto quantify the cost. Specifically, with the time and energyweighted by coefficients w ( t ) k and w ( e ) k respectively, the com-bined cost for the k -th mobile device is w ( t ) k t ( m ) k + w ( e ) k e ( m ) k .Let the benefit (without subtracting the cost) of completingthe task J k be f k ( b k , d k ) . Then the gross utility for the k -thmobile device to complete the task J k under local computingis U ( m ) k ( b k , d k ) := f k ( b k , d k ) − w ( t ) k t ( m ) k − w ( e ) k e ( m ) k . (3)Substituting Eq. (1) and Eq. (2) into Eq. (3), we obtain U ( m ) k ( b k , d k ) = f k ( b k , d k ) − w ( t ) k d k c ( m ) k − w ( e ) k µ k d k . (4) Utility of a mobile device under edge computing.
Wenow compute the utility of the k -th mobile device when itstask J k is completed under edge computing, where k ∈ K .First, we analyze the time needed for the k -th mobile deviceto send the task J k with b k bytes to the edge server. Thisdepends on the transmission rate, which further depends onwhich of the other K − users choose edge computingand hence are transmitting. Thus, we introduce an indicatorvariable a k to represent the k -th mobile device’s choice ofedge computing or local computing; specifically, a k = (cid:26) , if the k -th mobile device selects edge computing,(5a) , if the k -th mobile device selects local computing.(5b)Let [ − k ] be the indices of mobile devices other than the k -th one; i.e., [ − k ] , K \ { k } . Let a vector a − k represent thechoices made by mobile devices in [ − k ] ; i.e., a − k , [ a i : i ∈K \ { k } ] T .or an IRS with N IRS units, the IRS phase shift matrixas a diagonal matrix Φ ∈ C N × N is given by Φ := Diag ( φ ) , for φ , [ κ e j θ , . . . , κ N e j θ N ] T , (6)with κ n and θ n denoting the reflecting amplitude and angle ofthe n -th IRS unit respectively, for n ∈ N , { , , . . . , N } . Tofully exploit the reflecting surface’s ability to adjust the wire-less environment, here we assume that the reflecting amplitudeand reflecting angle could take values within [0 , and [0 , π ) respectively. The data transmission rate of the k -th mobiledevice depends on a − k and the IRS reflecting matrix Φ (orthe vector φ ), and we denote it by R k ( a − k , φ ) . The specificexpression of R k ( a − k , φ ) will be clear in Section IV-C. Thenthe time to transmit the data of b k bytes to the edge server is t ( e ) k,send ( a − k , φ ) = b k R k ( a − k , φ ) , (7)where the superscript “ ( e ) ” throughout the paper is used forquantities associated with the edge server.In the transmission of the k -th mobile device for choos-ing edge computing, let ν k be the energy consumed perunit time. Then after time t ( e ) k,send ( a − k , φ ) , an amount of ν k t ( e ) k,send ( a − k , φ ) energy is consumed. The same as [8], wealso consider that the k -th mobile device commits a tail energyof L k after transmitting the data of b k bytes. Then the totalenergy spent by the k -th mobile device is e ( e ) k ( a − k , φ ) = ν k t ( e ) k,send ( a − k , φ ) + L k . (8)After the data of b k bytes is fully transmitted to the edgeserver, with the edge server computation capability denotedby c ( e ) CPU cycles per unit time, the time for the edge serverto complete the task J k with d k CPU cycles is t ( e ) k,exe = d k c ( e ) . (9)As in [7], [8], we ignore the time required for the basestation to send back the result to a mobile device. Hence,our study includes only uplinks but not downlinks in Fig. 1.Then under edge computing, the total time to complete thetask J k is t ( e ) k,send ( a − k , φ ) + t ( e ) k,exe . With w ( t ) k and w ( e ) k beingthe weights for the time and energy in the cost computation,the combined cost is w ( t ) k · h t ( e ) k,send ( a − k , φ ) + t ( e ) k,exe i + w ( e ) k · e ( e ) k ( a − k , φ ) . Under edge computing, we consider that the k -thmobile device also needs to have a payment of P k to the edgeserver. Then with the benefit (without subtracting the cost) ofcompleting the task J k being f k ( b k , d k ) , the gross utility forthe k -th mobile device to complete the task J k under edgecomputing is U ( e ) k ( a − k , φ , b k , d k )= f k ( b k , d k ) − w ( t ) k · h t ( e ) k,send ( a − k , φ ) + t ( e ) k,exe i − w ( e ) k · e ( e ) k ( a − k , φ ) − P k . (10) Substituting Eq. (7) (8) (9) into Eq. (10), we obtain U ( e ) k ( a − k , φ , b k , d k )= f k ( b k , d k ) − w ( t ) k · (cid:20) b k R k ( a − k , φ ) + d k c ( e ) (cid:21) − w ( e ) k · (cid:20) ν k b k R k ( a − k , φ ) + L k (cid:21) − P k . (11)In our studied model, for the k -th mobile device, if the grossutility obtained under edge computing is at least the grossutility obtained under local computing, then the k -th mobiledevice will have a k = 1 to choose edge computing over localcomputing; otherwise, a k = 0 . Formally, a k = ( , if U ( e ) k ( a − k , φ , b k , d k ) ≥ U ( m ) k ( b k , d k ) , , otherwise. (12)From Eq. (4) and Eq. (11), the inequality U ( e ) k ( a − k , φ , b k , d k ) ≥ U ( m ) k ( b k , d k ) is equivalent to P k ≤ f k ( b k , d k ) − w ( t ) k · (cid:20) b k R k ( a − k , φ ) + d k c ( e ) (cid:21) − w ( e ) k · (cid:20) ν k b k R k ( a − k , φ ) + L k (cid:21) − " f k ( b k , d k ) − w ( t ) k d k c ( m ) k − w ( e ) k µ k d k . (13)Defining C k and A k ( b k ) by C k := w ( t ) k d k c ( m ) k + w ( e ) k µ k d k − w ( t ) k d k c ( e ) − w ( e ) k L k . (14)and A k ( b k ) := w ( t ) k b k + w ( e ) k ν k b k , (15)we write the above Inequality (13) as P k ≤ C k − A k ( b k ) R k ( a − k , φ ) , (16)which converts Eq. (12) into a k = ( , if P k ≤ C k − A k ( b k ) R k ( a − k , φ ) , , otherwise. (17)From Eq. (17), we have the following: • If C k − A k ( b k ) R k ( a − k , φ ) ≥ , the edge server can set P k ∈ [0 , C k − A k ( b k ) R k ( a − k , φ ) ] , and the k -th mobile device will have a k = 1 tochoose edge computing over local computing. • If C k − A k ( b k ) R k ( a − k , φ ) < , in case that the edge server doesnot accept a negative payment from the k -th mobile device(i.e., the edge server does not compensate the mobile devicefor choosing edge computing), then the k -th mobile devicewill have a k = 0 to choose local computing over edgecomputing, and the edge server will receive no paymentfrom the k -th mobile device so that P k = 0 .Besides, the information rate for each MD should be nosmaller than a requirement r k to ensure that each MD couldulfill a basic communication and/or computation task.From the above analysis, the maximal payment thatthe edge server can get from the k -th mobile device is max n C k − A k ( b k ) R k ( a − k , φ ) , o . Then to maximize the total pay-ment from all K mobile devices, the edge server solves thefollowing optimization Problem (P1):(P1): max φ K X k =1 max (cid:26) C k − A k ( b k ) R k ( a − k , φ ) , (cid:27) (18a) s . t . a k = ( , if C k − A k ( b k ) R k ( a − k , φ ) ≥ , , otherwise , ∀ k ∈ K ,R k ( a − k , φ ) ≥ r k , ∀ k, (18b) | φ n | ≤ , ∀ n ∈ N . (18c)The above constraint (18c) holds since φ n = κ n e jθ n fromEq. (6). B. A Simplified Case
Maximizing the non-concave objective function in (18a)even without considering the constraints (18b) and (18c) isdifficult. Hence, we consider that if C k − A k ( b k ) R k ( a − k , φ ) < , theedge server accepts a negative payment from the k -th mobiledevice (i.e., the edge server compensates the mobile device forchoosing edge computing). Then regardless of the relationshipbetween C k − A k ( b k ) R k ( a − k , φ ) and , the maximal payment that theedge server can get from the k -th mobile device is always C k − A k ( b k ) R k ( a − k , φ ) . Also, the k -th mobile device for each k ∈ K will always have a k = 1 to choose edge computing overlocal computing. Now with a i = 1 for i ∈ K , a − k becomesa vector of all s, and we write R k ( a − k , φ ) as R k ( φ ) fornotation simplicity. Thus, to maximize the total payment fromall K mobile devices, the edge server solves the followingoptimization Problem (P2):(P2): max φ K X k =1 (cid:20) C k − A k ( b k ) R k ( φ ) (cid:21) (19a)s.t. R k ( φ ) ≥ r k , ∀ k ∈ K , (19b) | φ n | ≤ , ∀ n ∈ N . (19c)Later we will show that the problem (P2) can be efficientlysolved. Hence, Problem (P2) of (19) significantly reduces thedifficulty of solving Problem (P1) of (18).Problem (P2) provides a lower bound for Problem (P1).Moreover, considering Problem (P2) also has practical interest.As discussed above, we consider that the edge server is willingto compensate a mobile device for choosing edge computingif providing edge computing to the mobile device for somecomputation task does not bring immediate earning. Doing somay encourage the mobile device to adopt edge computing forfuture computation tasks, and the edge server can profit fromthe mobile device in these future tasks. In addition, the edgeserver’s compensation may help the edge server to increasethe user base of its edge computing and hence improve thelong-term earning. C. Signal Model
In this paper we consider deploying the IRS to assist atypical wireless communication system, where a multi-antennaAP is communicating with K single-antenna MDs. Supposethat the AP has M antennas and the IRS equipment has N reflecting elements. We use h H r ,k ∈ C × N to denote the wire-less channel from the k -th MD to the IRS. Let G ∈ C M × N be the channel from the IRS to the AP. Define h H d ,k ∈ C × M to be the channel from the k -th MD to the AP.Here we consider the system experiences quasi-static flat-fading channel. Therefore the AP can obtain the channelstatus information (CSI) h H r ,k , G , h H d ,k via the standard channelestimation and feedback technique (e.g. in a classical time-division duplexing (TDD) system). Based on knowledge of h H r ,k , G , h H d ,k , the AP manipulates the amplitude and phase ofthe IRS elements to improve its earnings. That is AP tries tosolve the Problem (P2) with respect to φ .In view of Φ = diag ( φ ) , we define the effective channel h k ( φ ) ∈ C M × of the k -th MD by h k ( φ ) := G diag ( φ ) h r ,k + h d ,k . (20)The received signal at AP can be represented as r ( φ ) = X j √ q j h j ( φ ) s j + n , (21)where the vector n ∈ C M denotes the local thermal noise atthe receiver and is usually modeled as Gaussian distribution CN ( , σ I M ) .Assume that linear receiver u Hk ∈ C × M is utilized at theAP to decode signals associated with the k -th MD. Then thesignal-to-interference-plus-noise ratio (SINR) is a function of φ given as γ k (cid:0) φ (cid:1) = q k | u Hk h k ( φ ) | u Hk W k ( φ ) u k . (22)with interference-plus-noise spatial covariance matrix W k ( φ ) = σ I M + X i = k q i h i ( φ ) h Hi ( φ ) . (23)To improve receiving quality, the optimal u ⋆k [40], [41] u ⋆k = [ W k ( φ )] − h k ( φ ) k [ W k ( φ )] − h k ( φ ) k , (24)should be used to yield the maximum γ ⋆k (cid:0) φ (cid:1) given as γ ⋆k (cid:0) φ (cid:1) = q k h Hk ( φ )[ W k ( φ )] − h k ( φ ) . (25)Assuming that unit bandwidth is used, then by Shannon’sformula [42] the allowed information rate of the k -th MD R k ( φ ) is evaluated as R k ( φ ) = log(1 + γ ⋆k ( φ )) . (26) Here we somewhat abuse the notation R k ( · ) . Our R k (cid:0) φ (cid:1) is actuallydefined as the information rate associated with SINR maximizing receivingvector, not for generic receiving vectors. onsidering result in (25) we obtain R k ( φ ) = log(1 + q k h Hk ( φ )[ W k ( φ )] − h k ( φ )) . (27) D. Problem Formulation
Putting the above discussions together, our goal is to solve(P2) in (19), where R k ( φ ) is given by (27).In (19a), since b k does not depend on the optimizationvariable φ , we just abbreviate A k ( b k ) to A k in the followingdiscussion. Maximizing the term in (19a) is the same asminimizing P Kk =1 A k R k ( φ ) , so we can write Problem (P2) asthe following Problem (P3):(P3): min φ K X k =1 A k R k ( φ ) (28a)s.t. R k ( φ ) ≥ r k , ∀ k ∈ K , (28b) | φ n | ≤ , ∀ k ∈ N . (28c)We will discuss the feasibility and solution of (P3) in thesubsequent sections in details.V. F EASIBILITY P ROBLEM
The problem (P3) is difficult due to both of its complexobjective as well as the group of rate constraints, which areall nonconvex. Putting aside the highly nonconvex objective,one first question comes to our mind is the feasibility ofthe problem. That is, with an arbitrary set of rate constraints { r k } assigned, is the problem (P3) feasible? Determining thefeasibility of (P3) is itself hard but at the same time highlymeaningful. Since it is the base for scheming a reasonablerate requirement and serves as a starting point to solve (P3)(as will become clear in the next section). In this section wedevelop an iterative evaluation procedure, which i) can serveas a sufficient condition for the feasibility of (P3) and ii) canprovide a feasible solution when (P3) is identified as feasible.Noticing the relation in (26), the feasibility check of (P3),can be equivalently written as the following problem:(P4): Find φ (29a) s . t .γ ⋆k ( φ ) ≥ e r k − , ∀ k ∈ { , · · · , K } , (29b) | φ n | ≤ , ∀ n ∈ { , · · · , N } . (29c)To further simplify the above SINR constraints, which havefractional nature, we consider a closely related metric—meansquare error (MSE), which is defined as ε k ( φ , w k ) , E n(cid:12)(cid:12) s k − w Hk (cid:0) X j √ q j h j s j + n (cid:1)(cid:12)(cid:12) o (30)where the vector w k is the linear receiver utilized at AP tosuppress the interference and noise. The expectation in (30) istaken over the distribution of noise n and information symbols { s j } . When linear receiver is deployed, the following identityconnects the minimal MSE ε ⋆k ( φ ) and maximal SINR γ ⋆k ( φ ) : ε ⋆k ( φ ) = 11 + γ ⋆k ( φ ) . (31) In fact, it can be proved that optimal linear MSE receiver isgiven as w ⋆k = f W ( φ ) − √ q k h k ( φ ) , (32)with f W ( φ ) being f W ( φ ) = σ I M + X j q i h j ( φ ) h j ( φ ) H . (33)and the minimal ε ⋆k ( φ ) can be readily obtained as ε ⋆k ( φ ) =1 − q k h k ( φ ) H f W k ( φ ) − h k ( φ ) . Combined with (25), (31) canbe verified.Based on the above observation in (31), (P4) can equiva-lently transformed into(P5): Find φ (34a) s . t . ε ⋆k ( φ ) ≤
11 + e r k − e − r k , ∀ k ∈ K , (34b) | φ n | ≤ , ∀ n ∈ N . (34c)The above optimization problem can be further transformedas follows(P6): Find φ , { w k } (35a) s . t . ε k ( φ , { w k } ) ≤ e − r k , ∀ k ∈ K , (35b) | φ n | ≤ , ∀ n ∈ N . (35c)The equivalence between (P5) and (P6) can be seen by noticingthe fact that ε ⋆k ( φ ) = min w k ε k ( φ , w k ) and the optima isachieved when choosing w k = w ⋆k given in (32). In fact, if ε ⋆k ( φ ) ≤ e r k , then ε k ( φ , w ⋆k ) = ε ⋆k ( φ ) ≤ e r k . Conversely, ifthere exists ( φ , w k ) such that ε k ( φ , w k ) ≤ e r k , then ε ⋆k ( φ ) ≤ ε k ( φ , w k ) ≤ e − r k .Now we consider the following optimization problem(P7): min φ , { w k } ,α α (36a) s . t . e r k ε k ( φ , w k ) ≤ α, ∀ k ∈ K , (36b) | φ n | ≤ , ∀ n ∈ N . (36c)If the optimal value of (P7) is no greater than , then(P6), and consequently the original problem (P3), is feasible.Otherwise the problem is infeasible. The intention behind thetransformation from (P4) to (P7) lies in the fact that ε k ( φ , w k ) has a quadratic form and is bi-convex with respect to φ and { w k } , which is much simpler to tackle compared to γ ⋆k ( φ ) .Since the problem is still nonconvex with respect to φ and { w k } jointly, we can use block coordinate descent (BCD)method to solve it. Specifically we can alternatively update ( φ , α ) and ( { w k } , α ) . When φ is fixed, the optimal { w k } isgiven in (32) as discussed previously and α can be accordinglydetermined. When { w k } are fixed, φ should be obtained viasolving the following problem:(P8): min φ ,α α (37a) s . t . φ H Q k φ + 2Re { q Hk φ } + d k ≤ α, ∀ k ∈ K , (37b) | φ n | ≤ , ∀ n ∈ N . (37c) lgorithm 1 Feasibility Check Algorithm Randomly initialize φ (0) with φ (0) n = κ n e jθ n with κ n and θ n uniformly distributed among [0 , and [0 , π ) respectively, ∀ n ∈ N ; Initialize { w (0) k } via (32); Initialize α (0) := max k ε k ( φ (0) , w (0) k ) ; repeat update w ( t +1) k via (32); update α ( t +1) and φ ( t +1) by solving (P8); t + + ; until convergence or α ( t ) ≤ if α ( t ) < = 1 then claim Feasible , output φ ( t ) as feasible solution; else claim Infeasible end if with the parameters defined as follows: F k , G Diag ( h r , k ) , (38a) Q k , e r k X j q j F Hj w k w Hk F j , (38b) q k , e r k X j q j w Hk h d , j F Hj w k − e r k √ q k F Hk w k (38c) d k , e r k (cid:0) X j q j (cid:12)(cid:12) w Hk h d , j (cid:12)(cid:12) − √ q k Re { w Hk h d , k } (38d) + σ k w k k +1 (cid:1) . According to (38b), Q k is obvioulsy positive-semidefinite ∀ k and therefore (P8) is convex. In fact (P8) can be trans-formed into standard second-order-cone-programming (SOCP)problem and can be efficiently solved via standard numericalsolver like CVX.Since the BCD procedure alternatively updates φ and { w k } minimizing α , α monotonically decreases until convergence.Once the value of α is found to be smaller than , then theoriginal problem is feasible and a feasible φ has been foundand BCD procedure could be stop. Otherwise, α converges toa value larger than , and we have to claim that the problemis infeasible. The above discussion is summarized in Alg. 1.It should be noted that (P7) is itself nonconvex and itsglobal optimal solution can hardly be obtained. Alg. 1 findsa suboptimal solution of (P7). Therefore Alg. 1 finding anobjective value no greater than is a sufficient condition forfeasibility of the original problem.VI. S OLVING THE O PTIMIZATION
In this section we develop an algorithm to solve the Prob-lem (P3), whose objective has a sum-of-ratios form and isdifficult to solve. Based on the discussion in previous section,we assume that the problem (P3) is feasible and we can finda feasible solution of (P3).To attack (P3), we firstly introduce the an equivalent trans-formation of the rate function R k ( φ ) [43], [44], which will significantly simplify the optimization procedure, as will beshown later. The following identities hold: R k ( φ ) = log(1 + γ ⋆k ( φ )) (39a) = − log h(cid:0) q − k + h Hk W − k h k (cid:1) − i + log( q k )) (39b) ( a ) = max ̟ k ≥ n log( ̟ k ) − ̟ k h(cid:0) q − k + h Hk W − k h k (cid:1) − i + 1 + log( q k ) o (39c) ( b ) = max ̟ k ≥ , v k n − ̟ k h q k (cid:12)(cid:12) − v Hk h k (cid:12)(cid:12) + v Hk W k v k i + log( ̟ k ) + 1 + log( q k ) o (39d) , max ̟ k ≥ , v k e R k (cid:0) φ , v k , ̟ k (cid:1) . (39e)In the above step (a) can be readily verified by realizing thefunction in the maximization is concave with respect to ̟ k and its optima is obtained via ̟ ⋆k = q − k + h k ( φ ) H W k ( φ ) − h k ( φ ) . (40)The transformation (b) holds since the function in (39d) isquadratic concave in v k and its optima is achieved at v ⋆k = W k ( φ ) − q k h k ( φ ) . (41)Abbreviating the notations ̟ , [ ̟ , · · · , ̟ K ] and V , [ v , · · · , v K ] , we transform the problem (P3) as follows(P9): min φ , V , ̟ K X k =1 A k e R k ( φ , v k , ̟ k ) (42a)s.t. e R k ( φ , v k , ̟ k )) ≥ r k , ∀ k ∈ K , (42b) | φ n | ≤ , ∀ n ∈ N . (42c)Introducing an intermediate variable µ , [ µ , . . . , µ K ] , (P9)can be equivalently written as(P10): min φ , V , ̟ , µ K X k =1 µ k (43a)s.t. A k e R k ( φ , v k , ̟ k ) ≤ µ k , ∀ k ∈ K , (43b) e R k ( φ , v k , ̟ k ) ≥ r k , ∀ k ∈ K , (43c) | φ n | ≤ , ∀ n ∈ N , . (43d)To solve (P10), we have the following lemmas, which areproved in the Appendix of the online full version [45] (i.e.,this paper): Lemma 1. If ( φ ⋆ , V ⋆ , ̟ ⋆ , µ ⋆ ) is a solution to (P10), thenthere exists λ ⋆ = [ λ ⋆ , . . . , λ ⋆K ] such that ( φ ⋆ , V ⋆ , ̟ ⋆ ) satis-fies the Karush-Kuhn-Tucker (KKT) condition of the followingroblem (P11) when λ = λ ⋆ and µ = µ ⋆ : (P11): min φ , V , ̟ K X k =1 λ k (cid:0) A k − µ k e R k ( φ , v k , ̟ k ) (cid:1) (44a) s . t . e R k ( φ , v k , ̟ k ) ≥ r k , ∀ k ∈ K , (44b) | φ n | ≤ , ∀ n ∈ N . (44c) Also, we have λ ⋆k e R k ( φ ⋆ , v ⋆k , ̟ ⋆k ) = 1 , ∀ k ∈ K , (45) µ ⋆k e R k ( φ ⋆ , v ⋆k , ̟ ⋆k ) = A k , ∀ k ∈ K . (46) Lemma 2.
If both ① and ② below hold: ① ( φ ⋆ , V ⋆ , ̟ ⋆ ) solves (P11) when λ = λ ⋆ and µ ⋆ = µ ⋆ ; ② ( φ ⋆ , V ⋆ , ̟ ⋆ ) , λ ⋆ , and µ ⋆ satisfy (45) and (46),then ( φ ⋆ , V ⋆ , ̟ ⋆ , µ ⋆ ) satisfies the KKT condition of (P10). According to Lemmas 1 and 2, we can solve (P10) viatackling (P11). Specifically, we adopt the approach of [46]to find ( φ ⋆ , V ⋆ , ̟ ⋆ ) , λ ∗ , and µ ∗ satisfying ① and ② ofLemma 2 via alternatively performing the following two steps: • Step-1: Solve (P11)
Given λ ( t − and µ ( t − , we solve(P10) and denote the solution by (cid:0) φ ( t ) , V ( t ) , ̟ ( t ) (cid:1) . • Step-2: Update λ and µ Given (cid:0) φ ( t ) , V ( t ) , ̟ ( t ) (cid:1) , weutilize the modified Newton (MN) method of [46] toobtain λ ( t ) and µ ( t ) .For the update in Step-1 , (P11) is nonconvex in ( φ , V , ̟ ) jointly. Therefore we still adopt the BCD method to solveit. Recall the discussion explaining (39), the optimal ̟ isobtained in (40) when ( φ , V ) is fixed, and the optimal V can be determined in (41) when ( φ , ̟ ) is fixed. Thereforewe need to investigate the optimization of φ . When ( V , ̟ ) ,the optimiztion of (P11) with respect to φ can be written as(P12): max φ φ H T φ + 2Re { t H φ } + c (47a) s . t . φ H T k φ + 2Re { t Hk φ } + c k ≥ r k , ∀ k ∈ K , (47b) | φ n | ≤ , ∀ n ∈ N (47c)with the parameters defined as follows: T k , − ̟ k h X j q j F Hj v k v Hk F j i , (48a) t k , − ̟ k h q k (cid:0) v Hk h d ,k − (cid:1) F Hk v k + X j = k q j v Hk h d ,j F Hj v k i c k , log ̟ k − ̟ k q k +1+log q k − σ ̟ k k v k k (48b) − ̟ k X j q j (cid:12)(cid:12) h H d ,j v k (cid:12)(cid:12) + 2 ̟ k q k Re (cid:8) h H d ,k v k (cid:9) T , X k λ k µ k T k , t , X k λ k µ k t k , c , X k λ k µ k c k . (48c)The above problem is convex and can be conveniently solvednumerically.For the update in Step-2 , we introduce the following no-tations to simplify the subsequent expositions. According to (45) and (46) define Λ ( t ) k ( λ k ) , λ k e R k ( φ ( t ) , v ( t ) k , ̟ ( t ) k ) − , ∀ k ∈ K , (49) Γ ( t ) k ( µ k ) , µ k e R k ( φ ( t ) , v ( t ) k , ̟ ( t ) k ) − A k , ∀ k ∈ K . (50)Then according to the modified Newton’s (MN) method in[46], we update λ and µ in the following way λ ( t ) k,i := λ ( t − k − ξ i Λ ( t ) k ( λ ( t − k ) d Λ ( t ) k ( λ k ) d λ k (cid:12)(cid:12) λ k = λ ( t − k = λ ( t − k − ξ i Λ ( t ) k ( λ ( t − k ) e R k ( φ ( t ) , v ( t ) k , ̟ ( t ) k ) , ∀ k ∈ K (51) µ ( t ) k,i := µ ( t − k − ξ i Γ ( t ) k ( µ ( t − k ) d Γ ( t ) k ( µ k ) d µ k (cid:12)(cid:12) µ k = µ ( t − k = µ ( t − k − ξ i Γ ( t ) k ( µ ( t − k ) e R k ( φ ( t ) , v ( t ) k , ̟ ( t ) k ) , ∀ k ∈ K . (52)Clearly, (45) and (46) will reduce to the standard Newton’smethod update when ξ i above is . In the modified Newton’smethod, a suitable i should be found. Let i ( t ) denote thesmallest integer among i ∈ { , , . . . } satisfying that K X k =1 ( (cid:16) Λ ( t ) k ( λ ( t ) k,i ) (cid:17) + K X k =1 (cid:16) Γ ( t ) k ( µ ( t ) k,i ) (cid:17) ) (53) ≤ (cid:0) − ξ i ǫ (cid:1) " K X k =1 (cid:16) Λ ( t ) k ( λ ( t − k ) (cid:17) + K X k =1 (cid:16) Γ ( t ) k ( µ ( t − k ) (cid:17) , where ǫ is a predefined precision parameter and ξ is a positiveconstant smaller than . Once i ( t ) is determined, λ ( t ) and µ ( t ) should be updated as λ ( t ) k ← λ ( t ) k,i ( t ) in (51) with i being i ( t ) , (54) µ ( t ) k ← µ ( t ) k,i ( t ) in (52) with i being i ( t ) , (55)According to the analysis in [46] the value δ ( t ) , K X k =1 (cid:16) Λ ( t ) k ( λ ( t ) k ) (cid:17) + K X k =1 (cid:16) Γ ( t ) k ( γ ( t ) k ) (cid:17) (56)converges to zero as t → ∞ , and the rate of convergence isglobal linear and local quadratic. When δ ( t ) in (56) convergesto zero, according to (49) and (50) Λ ( t ) k ( λ ( t ) k ) = 0 ⇒ λ ( t ) k e R k ( φ ( t ) , v ( t ) k , ̟ ( t ) k )=1 , ∀ k ∈ K , (57) Γ ( t ) k ( µ ( t ) k ) = 0 ⇒ µ ( t ) k e R k ( φ ( t ) , v ( t ) k , ̟ ( t ) k )= A k , ∀ k ∈ K . (58)Hence we will arrive at (45) and (46) and consequently, byLemmas 1 and 2, the KKT condition of (P10). Therefore (56),i.e. (57) and (58), performs as a termination condition for theiterative process. VII. S IMULATION
In this section, we present numerical results to validate ourproposed algorithms. In the experiment, we adopt a similarsystem setting-up reported in [35]. As shown in Fig. 2, the lgorithm 2
Optimizing (P10) Set ξ ∈ (0 , ; set the positive ǫ , ρ sufficiently small; Invoke Alg. 1 to obtain a feasible φ (0) ; t = 0 ; Initialize ̟ (0) and V (0) via (40) and (41) respectively; Set λ (0) k = R k ( φ (0) ) − and µ (0) k = A k /R k ( φ (0) ) ; repeat repeat update ̟ ( t ) by (40); update V ( t ) by (41); update φ ( t ) by solving (P12); until convergence update λ ( t +1) k and µ ( t +1) k by (54) and (55) respectively; Evaluate δ ( t ) by (53); t ← t + 1 ; until δ (0) < ρ IRSAP
MD1 MD2 MD3 MD4
50m 5m 5m 5m 2m
Fig. 2: Simulation Setting-Upsystem comprises one AP with M = 4 antennas, 4 single-antenna mobile users and one IRS device. The number ofreflecting elements N of the IRS will take different valuesfrom { , , , } . The distance between the AP andIRS is m and it is assumed that the signal propagationenvironment between the AP and the IRS is dominated bythe line-of-sight (LoS) link. Considering the mobility of theMDs will deteriorate the wireless propagation environment,we assume that the signals sent by MDs to both the APand the IRS experience dB penetration loss, independentRayleigh fading and the pathloss exponent of . The noisehas - dBm/Hz and the channel bandwidth is KHz, so σ k = 10 − mW. We set the transmission power of each MDas dBm. Besides we assumed the antenna gain of both theAP and user is dBi and that of each reflecting element atthe IRS is dBi [47]. The distance between the AP and theIRS is m and the 4 MDs are located in a row parallel tothe AP-IRS line and with a m interval between the adjacentpeers.Fig. 3 shows us the behavior of the Alg. 1 for checkingthe feasibility associated with a given set of information raterequirements. In our experiment, we set N = 30 . To ensurefairness between different MDs, we just assume that all MDswill be guaranteed one identical information rate. We vary thiscommon information rate requirement from . Nats/s/channel-
Iteration Index V a l ue o f Convergence of Feasibility Check Alg. rate-rqrmnt=2.1rate-rqrmnt=2.3rate-rqrmnt=2.5rate-rqrmnt=2.7rate-rqrmnt=2.9
Fig. 3: Convergence of Feasibility Check Alg. 1use to . Nats/s/channel-use and perform Alg. 1. Note that inour experiment one random channel realization is generatedaccording to system setting explained above and then fixed.For each specific information rate requirement, φ is startedfrom a common initial point, which is randomly chosen.According to Fig. 3, Alg. 1 generally converges within 10iterations. The convergent α value increases when the infor-mation rate requirement inflates. It can be inferred from Fig. 3that maximal feasible common information rate requirementshould lie between . to . Nats/s/channel-use.
Info-Rate Requirement for Each MD (in Nat/s/chn-use) F ea s i b ili t y P r obab ili t y ( % ) IRS Impact on Feasibility opt-IRS,N=30opt-IRS,N=60opt-IRS,N=90opt-IRS,N=120rnd-IRS,N=30rnd-IRS,N=60rnd-IRS,N=90rnd-IRS,N=120no-IRS
Fig. 4: The Impact of IRS on Feasibility of Rate RequirementsFig. 4 illustrates the impact of IRS on enhancing theguaranteed information rate to MDs. In our experiment, westill assume that all MDs require one common informationrate requirement. For each specific N and a predefined raterequirement, channel realizations are randomly generatedccording to the above settings. For each channel realization,we generate a batch of random IRS elements and then invokeAlg. 1 to optimize the IRS elements trying to reach feasibility.We compare the probability of getting feasibility for the caseof no-IRS, random IRS and optimized IRS in Fig. 4. Theresult in Fig. 4 undoubtedly convince us the significance ofIRS. Among the range of rate requirement in the figure, theoptimized IRS can improve the feasibility probability by to . Besides, more improvement could be obtained withmore number of IRS elements being deployed. Iteration Index O b j e c t i v e V a l ue o f ( P ) Convergence of Alg.2 channel realization 1channel realization 2channel realization 3channel realization 4channel realization 5
Fig. 5: Convergence of Obj. of (P3) by Alg. 2
Iteration Index -12-10-8-6-4-202 V a l ue o f l og () Convergence of Alg.2 channel realization 1channel realization 2channel realization 3channel realization 4channel realization 5
Fig. 6: Convergence of δ ( t ) by Alg. 2Fig. 5 and Fig. 6 show the convergence behavior of Alg. 2.In the experiment we generate a group (5 in the figure)of random channel realizations, for which a feasible pointcan be found (e.g. by Alg. 1). For each specific channelrealization, we perform Alg. 2 starting from the feasible point. The objective value and the δ ( t ) defined in (56) are presentedin Fig. 5 and Fig. 6 respectively. As shown in Fig. 5, Alg. 2usually converges in several outer loops. Fig. 6 convince usthat Alg. 2 exhibits superlinear convergence rate. The resultsin Fig. 5 and Fig. 6 suggest that Alg. 2 has fast converge andis therefore very efficient. O b j . IRS,N=30 rnd-IRSopt-IRS1 2 3 4 5 6 7 8 9 1011.5 O b j . IRS,N=60 O b j . IRS,N=90
Optimizing Obj. of (P3) by Alg.2
Channel Realization Index O b j . IRS,N=120
Fig. 7: Optimizing (P3)Finally Fig. 7 illustrates the effect of Alg. 2 in optimizing(P3). Taking into account of fairness of different MDs, we justassume that A k are equal to each MDs and consequently set A k = 1 ∀ k ∈ K . For each specific N , we generate a bunch (10in the figure) of random channel realizations and their feasiblesolutions which are found via randomization. We comparethe objective values of (P3) obtained via Alg. 2 with thoseassociated with those random IRS elements. As presented inFig. 7, for various values of N , Alg. 2 can effectively decreasethe objective value of (P3) and therefore equivalently increasethe earnings of the edge server.VIII. C ONCLUSION
In this paper we study the mobile edge computing problemwith the aid of the novel technology of IRS. In details, weformulate the earning maximization problem with a group ofinformation rate constraints in the presence of IRS device. Wedevelop an iterative algorithm which can efficiently identifythe feasibility of the information rate requirements and finda feasible solution. Besides we also develop an algorithm tooptimize the earning of the edge server for loading computing.Substantial numerical results have shown that our proposedIRS aided scheme can significantly improve the guaranteedinformation rate to MDs by the AP and also effectively enlargethe earning of the edge server.
PPENDIX
A. Proof of Lemma 1Proof.
To see the statements hold, we first give out the KKTconditions of (P10). To simplify the following expositions, wedefine x k , (cid:0) φ , v k , ̟ k (cid:1) . Then the Lagrangian function isgiven as L (cid:0) { x k } , µ , λ , ζ , ν (cid:1) , X k µ k + X k λ k (cid:16) A k − µ k e R k ( x k ) (cid:17) + X k ζ k (cid:16) r k − e R k ( x k ) (cid:17) + X k ν k (cid:16) | φ n | − (cid:17) , (59)with parameters { λ k } , { ζ k } and { ν k } being the Lagrangianmultipliers associated with the constraints (43b), (43c) and(43d) respectively. The KKT conditions are listed as follows: ∂ L ∂ x ∗ k = , ∀ k ∈ K , (60a) ∂ L ∂µ k = 1 − λ k e R k (cid:0) x k (cid:1) = 0 , ∀ k ∈ K , (60b) ∂ L ∂λ k = A k − µ k e R k (cid:0) x k (cid:1) = 0 , ∀ k ∈ K , (60c) ∂ L ∂ζ k = 0 , ∀ k ∈ K , (60d) ∂ L ∂ν k = 0 , ∀ k ∈ K , (60e) λ k ≥ , ζ k ≥ , ν k ≥ , ∀ k ∈ K . (60f)Assume that ( φ ⋆ , V ⋆ , ̟ ⋆ , µ ⋆ ) is an optimal solution to (P10).Then there exist Lagrangian multipliers λ ⋆ , ζ ⋆ and ν ⋆ suchthat the KKT conditions are satisfied. By (60b) and (60c) weobtain the results in (45) and (46). Besides, it can be readilyseen that the KKT conditions (60a), (60d), (60e) and (60f)with x ⋆ , ζ ⋆ and ν ⋆ being substituted therein are just the KKTconditions of (P11) with its parameters setting as λ = λ ⋆ and µ = µ ⋆ . B. Proof of Lemma 2Proof.
Here we follow the notations used in the proof ofLemma 1. Assume that ( φ ⋆ , V ⋆ , ̟ ⋆ ) solves the problem(P11) when its parameters are set as λ = λ ⋆ and µ = µ ⋆ .Then by optimality conditions, the KKT conditions of (P11)should be satisfied with by ( φ ⋆ , V ⋆ , ̟ ⋆ ) together with the La-grangian multipliers ζ ⋆ and ν ⋆ associated with the constraints(44b) and (44c) respectively. That is the equations (60a),(60d), (60e) and (60f) hold. Taking into account the conditionsin ② , the equations (60b) and (60c) also stand. Therefore(60) are satisfied, which are indeed the KKT conditions of(P10) associated with the optimal solution ( φ ⋆ , V ⋆ , ̟ ⋆ , µ ⋆ ) .Therefore the proof is complete.R EFERENCES[1] M. Satyanarayanan, “The emergence of edge computing,”
Computer ,vol. 50, no. 1, pp. 30–39, 2017.[2] Y. Mao, C. You, J. Zhang, K. Huang, and K. B. Letaief, “A surveyon mobile edge computing: The communication perspective,”
IEEECommunications Surveys & Tutorials , vol. 19, no. 4, pp. 2322–2358,2017. [3] X. Hu, K.-K. Wong, and K. Yang, “Wireless powered cooperation-assisted mobile edge computing,”
IEEE Transactions on Wireless Com-munications , vol. 17, no. 4, pp. 2375–2388, 2018.[4] O. Vaughan, “Working on the edge,”
Nature Electronics , vol. 2, no. 1,p. 2, 2019.[5] “AT&T and Microsoft announce a strategic allianceto deliver innovation with cloud, AI and 5G,”https://techblog.comsoc.org/2019/07/17/att-announced-cloud-partnerships-with-microsoft-1-day-after-similar-deal-with-ibm/,accessed: 2019-07-19.[6] P. Mach and Z. Becvar, “Mobile edge computing: A survey on archi-tecture and computation offloading,”
IEEE Communications Surveys &Tutorials , vol. 19, no. 3, pp. 1628–1656, 2017.[7] X. Chen, “Decentralized computation offloading game for mobile cloudcomputing,”
IEEE Transactions on Parallel and Distributed Systems ,vol. 26, no. 4, pp. 974–983, 2014.[8] X. Chen, L. Jiao, W. Li, and X. Fu, “Efficient multi-user computationoffloading for mobile-edge cloud computing,”
IEEE/ACM Transactionson Networking , vol. 24, no. 5, pp. 2795–2808, 2015.[9] X. Tao, K. Ota, M. Dong, H. Qi, and K. Li, “Performance guaranteedcomputation offloading for mobile-edge cloud computing,”
IEEE Wire-less Communications Letters , vol. 6, no. 6, pp. 774–777, 2017.[10] Y. Mao, J. Zhang, and K. B. Letaief, “Dynamic computation offloadingfor mobile-edge computing with energy harvesting devices,”
IEEEJournal on Selected Areas in Communications
IEEE Transactionson Signal Processing , vol. 66, no. 10, pp. 2746–2758, 2018.[13] T. Bai, C. Pan, Y. Deng, M. Elkashlan, and A. Nallanathan, “Latencyminimization for intelligent reflecting surface aided mobile edge com-puting,” arXiv preprint arXiv:1910.07990v1 , 2019.[14] Y. C. Hu, M. Patel, D. Sabella, N. Sprecher, and V. Young, “Mobile edgecomputinga key technology towards 5G,”
ETSI white paper , vol. 11,no. 11, pp. 1–16, 2015.[15] F. Bonomi, R. Milito, J. Zhu, and S. Addepalli, “Fog computing and itsrole in the internet of things,” in
ACM Mobile Cloud Computing (MCC) ,2012, pp. 13–16.[16] “IEEE standard for adoption of OpenFog reference architecture forfog computing,” https://standards.ieee.org/standard/1934-2018.html, ac-cessed: 2019-07-19.[17] J. Wang, J. Pan, F. Esposito, P. Calyam, Z. Yang, and P. Mohapatra,“Edge cloud offloading algorithms: Issues, methods, and perspectives,”
ACM Computing Surveys (CSUR) , vol. 52, no. 1, p. 2, 2019.[18] Y. Wang, M. Sheng, X. Wang, L. Wang, and J. Li, “Mobile-edge com-puting: Partial computation offloading using dynamic voltage scaling,”
IEEE Transactions on Communications , vol. 64, no. 10, pp. 4268–4282,2016.[19] Z. Ning, P. Dong, X. Kong, and F. Xia, “A cooperative partial computa-tion offloading scheme for mobile edge computing enabled Internet ofThings,”
IEEE Internet of Things Journal , 2018.[20] S. Bi and Y. J. Zhang, “Computation rate maximization for wirelesspowered mobile-edge computing with binary computation offloading,”
IEEE Transactions on Wireless Communications , vol. 17, no. 6, pp.4177–4190, 2018.[21] M. Chen and Y. Hao, “Task offloading for mobile edge computing insoftware defined ultra-dense network,”
IEEE Journal on Selected Areasin Communications arXiv preprintarXiv:1810.05667 , 2018.[24] C. Huang, A. Zappone, G. C. Alexandropoulos, M. Debbah, andC. Yuen, “Reconfigurable intelligent surfaces for energy efficiency inwireless communication,”
IEEE Transactions on Wireless Communica-tions , 2019.25] M. Fu, Y. Zhou, and Y. Shi, “Intelligent reflecting surface fordownlink non-orthogonal multiple access networks,” arXiv preprintarXiv:1906.09434 , 2019.[26] X. Yu, D. Xu, and R. Schober, “MISO wireless communication systemsvia intelligent reflecting surfaces,” arXiv preprint arXiv:1904.12199 ,2019.[27] Q.-U.-A. Nadeem, A. Kammoun, A. Chaaban, M. Debbah, and M.-S. Alouini, “Intelligent reflecting surface assisted multi-user MISOcommunication,” arXiv preprint arXiv:1906.02360 , 2019.[28] D. Mishra and H. Johansson, “Channel estimation and low-complexitybeamforming design for passive intelligent surface assisted MISO wire-less energy transfer,” in
IEEE International Conference on Acoustics,Speech and Signal Processing (ICASSP) , 2019, pp. 4659–4663.[29] T. Jiang and Y. Shi, “Over-the-air computation via intelligent reflectingsurfaces,” arXiv preprint arXiv:1904.12475 , 2019.[30] W. Qingqing and Z. Rui, “Towards smart and reconfigurable envi-ronment: Intelligent reflecting surface aided wireless network,” arXivpreprint arXiv:1905.00152 , 2019.[31] E. Basar, “Large intelligent surface-based index modulation: A newbeyond MIMO paradigm for 6G,” arXiv preprint arXiv:1904.06704 ,2019.[32] C. Huang, A. Zappone, M. Debbah, and C. Yuen, “Achievable ratemaximization by passive intelligent mirrors,” in
IEEE InternationalConference on Acoustics, Speech and Signal Processing (ICASSP) , 2018,pp. 3714–3718.[33] M. Di Renzo and J. Song, “Reflection probability in wireless networkswith metasurface-coated environmental objects: An approach based onrandom spatial processes,” arXiv preprint arXiv:1901.01046 , 2019.[34] Q.-U.-A. Nadeem, A. Kammoun, A. Chaaban, M. Debbah, and M.-S.Alouini, “Asymptotic analysis of large intelligent surface assisted MIMOcommunication,” arXiv preprint arXiv:1903.08127 , 2019.[35] Q. Wu and R. Zhang, “Intelligent reflecting surface enhanced wirelessnetwork: Joint active and passive beamforming design,” in
IEEE GlobalCommunications Conference (GLOBECOM) , 2018, pp. 1–6.[36] ——, “Intelligent reflecting surface enhanced wireless network via jointactive and passive beamforming,” arXiv preprint arXiv:1810.03961 ,2018.[37] M. Jung, W. Saad, and G. Kong, “Performance analysis of largeintelligent surfaces (LISs): Uplink spectral efficiency and pilot training,” arXiv preprint arXiv:1904.00453 , 2019.[38] S. Hu, F. Rusek, and O. Edfors, “The potential of using large antennaarrays on intelligent surfaces,” in
IEEE 85th Vehicular TechnologyConference (VTC Spring) , 2017, pp. 1–6.[39] ——, “Capacity degradation with modeling hardware impairment inlarge intelligent surface,” in
IEEE Global Communications Conference(GLOBECOM) , 2018, pp. 1–6.[40] M. Schubert and H. Boche, “Iterative multiuser uplink and downlinkbeamforming under SINR constraints,”
IEEE Transactions on SignalProcessing , vol. 53, no. 7, pp. 2324–2334, 2005.[41] R. A. Monzingo and T. W. Miller,
Introduction to Adaptive Arrays . NewYork, Wiley-Interscience, 1980.[42] C. E. Shannon, “A mathematical theory of communication,”
Bell SystemTechnical Journal , vol. 27, no. 3, pp. 379–423, 1948.[43] S. Christensen, R. Argawal, E. de Carvalho, and J. M. Cioffi, “Weightedsum-rate maximization using weighted mmse for mimo-bc beamformingdesign,”
IEEE Transactions on Wireless Communication , vol. 7, no. 12,pp. 1–7, 2008.[44] Q. Shi, M. Razaviyayn, Z.-Q. Luo, and C. Chen, “An iteratively weightedmmse approach to distributed sum-utility maximization for a mimointerfering broadcast channel,”
IEEE Transactions on Signal Processing