UAV-Enabled Mobile Edge Computing: Offloading Optimization and Trajectory Design

U A V -Enabled Mobile Edge Computing: Of floading Optimization and T rajectory Design Fuhui Zhou §‡ , Y ongpeng W u ∗ , Haijian Sun § , Zheng Chu † § Utah State Uni versity , USA, ‡ Nanchang Uni versity , Nanchang, China, ∗ Shanghai Jiao T ong Univ ersity , China, † Uni versity of Surre y , Guildford, U.K Email: { zhoufuhui@ieee.or g, yongpeng.wu2016@gmail.com, h.j.sun@ieee .org , andr ew .chuzheng7@gmail.com } Abstract —With the emergence of diverse mobile applications (such as augmented reality), the quality of experience of mobile users is greatly limited by their computation capacity and finite battery lifetime. Mobile edge computing (MEC) and wireless power transfer are promising to address this issue. Ho wever , these two techniques are susceptible to propagation delay and loss. Motivated by the chance of short-distance line-of-sight achieved by le veraging unmanned aerial vehicle (U A V) communications, an U A V -enabled wireless power ed MEC system is studied. A power minimization pr oblem is f ormulated subject to the constraints on the number of the computation bits and energy harvesting causality . The problem is non-con vex and challenging to tackle. An alternativ e optimization algorithm is proposed based on sequential con vex optimization. Simulation results show that our proposed design is superior to other benchmark schemes and the proposed algorithm is efficient in terms of the conv ergence. Index T erms —Mobile edge computing, resour ce allocation, un- manned aerial vehicle communications, trajectory optimization, wireless power transfer . I . I N T RO D U C T I O N W ITH the dev elopment of Internet of Things (IoT), the emerging div erse mobile applications (augmented reality , f ace recognition, mobile online gaming, etc.) enable mobile users to enjoy a high quality of experience [1]. How- ev er, these applications are latency-sensiti ve and need a high computation capability . Due to the limited battery and low computation capability , it is challenging for mobile devices to execute these applications [2], [3]. F ortunately , mobile edge computing (MEC) has been recognized as a promising technique to tackle this challenge [4]. It provides the edge net- work with cloud computing service. Mobile users can offload their computation tasks into the edge network. Unlike mobile cloud computing (MCC), network edge devices in MEC, such as access points, can perform cloud-like computing and are deployed in close proximity to users. MEC has received increasing attention in both academia and industry since it has advantages of saving energy for users, providing low latency services and achieving security for mobile applications [5]-[9]. On the other hand, wireless power transfer (WPT) tech- niques are promising for prolonging the operational time of The research was supported by the U.S. National Science Foundation grant EARS-1547312, the National Natural Science Foundation of China (61701214, 61701301, 61661028, 61631015, and 61561034), the Y oung Natural Science Foundation of Jiangxi Province (20171BAB212002), the China Postdoctoral Science Foundation (2017M610400) and the Postdoctoral schedule fund of Jiangxi Province(2017RC17). energy-limited mobile devices [10], [11]. Particularly , radio frequency (RF) signals are used as energy sources for energy harvesting (EH). Compared to the con ventional EH techniques, such as solar charging, WPT techniques can provide a control- lable and stable power . They are important for energy-limited mobile devices, which are required to e xecute local low- computation tasks and offload computation-extensiv e tasks [6]- [9]. Howe ver , the harvested power level is greatly influenced by the sev ere propagation loss. Recently , an unmanned aerial vehicle (UA V)-enabled WPT architecture has been proposed to improve the energy transfer ef ficiency [12]. It utilizes an U A V as an energy transmitter for powering the ground mobile users. It was sho wn that the harvested power le vel can be significantly improv ed due to the higher chance of short- distance line-of-sight (LoS) energy transmit links [13]. Motiv ated by the U A V -enabled WPT architecture, a U A V - enabled wireless powered MEC system is studied in this paper . In the system, the U A V transmits energy to multiple ground users and the ground users e xploit the harvested energy for local computing and computation tasks offloading. T o the authors’ best knowledge, this is the first work that establishes an U A V -enabled wir eless powered MEC system and studies the joint optimization of computation offloading and trajectory design . The related works are summarized as follows. In [5], the re venue of the wireless cellular networks with MEC was maximized by jointly optimizing the computation offloading decision and resource allocation. The authors in [6]-[9] extended the resource allocation problems into wire- less po wered MEC systems. Specifically , in [6], an energy- efficient resource allocation strategy was proposed by jointly optimizing the number of the local computation bits and the offloading computation bits under the causal energy har- vesting constraint. The authors in [7] proposed an efficient reinforcement learning-based resource management scheme in an MEC system with energy harvesting. It was shown that the computation performance can be significantly improved by using the proposed algorithm. In [8] and [9], a joint opti- mization framework was proposed in wireless powered MEC systems with different operational paradigms, namely , binary and partial offloading, respectively . In the binary offloading paradigm, the computation task is completely ex ecuted in mobile devices or offloaded into network edge devices for computing. In the partial of floading paradigm, the computation task can be divided into tw o parts, one for local computing and U s e r 1 U s e r 2 U s e r K U s e r k E n e r g y h a r v e s t i n g L o c a l c o m p u t i n g W i r e l e s s pow e r e d l i nk C om put a t i on of f l oa di ng l i nk U s e r k x y z Fig. 1: The system model. the other for offloading. The computation rate was maximized in [8] and the transmit power was minimized in [9] by jointly optimizing the computing frequency and the transmit power . In [6]-[9], the access point or the energy transmitter equipped with an MEC server is deplo yed at the fixed location. It results in a low energy transfer efficienc y due to the severe propagation loss [10], [11]. In order to tackle this issue, the authors designed UA V -enabled wireless powered systems and jointly optimized the resource allocation and trajectory of the U A V [12], [13]. Howe ver , MEC w as not considered in [12] and [13]. Recently , an U A V -enabled MEC system was designed in [14] and the transmit power of users was minimized by jointly optimizing the number of the local computation bits, the offloading computation bits and the downloading bits. Different from [12]-[14], an UA V -enabled wireless powered MEC system is studied in this paper . A power minimization problem is formulated by jointly optimizing the number of the offloading computation bits, the local computation frequencies of users and the UA V , and the trajectory of the U A V . It is challenging to solve the formulated non-con ve x problem due to the existing couple among the optimized v ariables. An alternativ e optimization algorithm is proposed to solve it by using the sequential con ve x optimization (SCA) techniques. Simulation results show that our proposed resource allocation scheme outperforms other benchmark schemes. The remainder of this paper is or ganized as follo ws. The system model is presented in Section II. Section III presents the energy minimization problem. Section IV presents simu- lation results. The paper concludes with Section V . I I . S Y S T E M M O D E L An U A V -enabled wireless powered MEC system is con- sidered in Fig. 1, where an U A V equipped with an MEC server transmits energy to K users and provides MEC services for these users. In this paper, the partial offloading paradigm is applied. Similar to [8] and [9], users can simultaneously perform energy harvesting, local computing and computation offloading. W ithout loss of generality , a three-dimensional (3D) Euclidean coordinate is adopted. Each user is fixed at the ground. The location of the k th ground user is denoted by q k , where q k = [ x k , y k ] , k ∈ K and K = { 1 , 2 , · · · , K } . Boldface lower case letters represent vectors, and x k and y k are the horizontal plane coordinate of the k th ground user . It is assumed that the positions of users are kno wn to the U A V for designing trajectory . A finite time horizon with during T is considered. During the finite time, the UA V flies at a fixed altitude ( H > 0 ). A block f ading channel model is applied. During the finite time, the channel is unchanged. For ease of exposition, the finite time T is discretized into N equal-time slots, denoted by n = 1 , 2 , · · · , N . At the n th slot, it is assumed that the horizontal plane coordinate of the UA V is q u [ n ] = [ x u [ n ] , y u [ n ]] . Similar to [13]-[15], it is assumed that the wireless channel between the UA V and each user is dominated by the LoS channel. Thus, the channel power gain between the UA V and the k th user is denoted by h k [ n ] , gi ven as h k [ n ] = β 0 d − 2 k,n = β 0 H 2 + k q u [ n ] − q k k 2 , k ∈ K , n ∈ N , (1) where β 0 is the channel po wer gain at a reference distance d 0 = 1 m; d k,n is the horizontal plane distance between the U A V and the k th user at the n th slot, n ∈ N , N = { 1 , 2 , · · · , N } , and k·k denotes its Euclidean norm. In order to reach meaningful insights into the design of an UA V -enabled wireless po wered MEC system, similar to [6]-[10], a linear EH model is applied. Thus, the harvested energy at the k th user during n time slots denoted by E k [ n ] , is giv en as E k [ n ] = n X i =1 T η h k [ i ] P u N , (2) where η denotes the energy conservation efficienc y , 0 < η ≤ 1 and P u is the transmit power of the U A V . In this paper , the U A V employs a constant power transmission [12]-[15]. During the n th slot, all users perform energy harvesting, local computing and computation of floading. In order for all users to offload their bits to the U A V for computation, a time division multiple access (TDMA) protocol is applied. The time interval T / N is divided into K time slots with duration λ = T / ( N K ) and K users offload their computation bits to the UA V one by one. Similar to [6]-[9], the recei ved energy and the energy for transmitting the computed results of the UA V are ignored. Let l k [ n ] and f k [ n ] denote the number of the offloading bits and the central processing unit (CPU) frequency (cycles/s) of the k th user at the n th slot, respectiv ely . Thus, the transmit power of the k th user for of floading l k [ n ] computation bits denoted by P k [ n ] , is giv en as P k [ n ] = Γ σ 2  2 l k [ n ] Bλ − 1  h k [ n ] , (3) where B is the communication bandwidth and σ 2 denotes the noise power at the user . In (3) , Γ is a constant related to the gap from the channel capacity o wning to a practical coding and modulation scheme. It is assumed that Γ = 1 in this paper for simplicity . Let f u [ n ] denote the CPU frequency of the U A V at the n th slot. According to [6]-[9], the energy consumed for the local computation at the k th user and that for the offloading computation at the UA V in the n th slot are denoted by E k,l [ n ] and E u,o [ n ] , respectively given as E k,l [ n ] = γ c K λ [ f k [ n ]] 3 , (4a) E u,o [ n ] = γ c K λ [ f u [ n ]] 3 , (4b) where γ c is the ef fective switched capacitance of the CPU. Similar to the works in [14] and [16], the propulsion energy consumption model at the UA V due to the flying in the n th slot denoted by E s [ n ] , is giv en as E s [ n ] = κ k v u [ n ] k 2 , (5a) v u [ n ] = k q u [ n + 1] − q u [ n ] k K λ , (5b) where κ = 0 . 5 W T / N and W is the mass of the UA V . Note that the propulsion energy consumption model employed in this paper only depends on the v elocity . In future work, we will exploit a more general model that considers both the velocity and acceleration of the U A V . I I I . E N E R G Y M I N I M I Z A T I O N D E S I G N A. The Ener gy Minimization Pr oblem F ormulation In the U A V -enabled wireless po wered MEC system, in order to minimize the energy consumed at the U A V while guaranteeing the computation bits of all users, the number of the offloading computation bits and the CPU frequency of the users, the CPU frequency and the trajectory of the UA V are jointly optimized. The energy minimization problem can be formulated as P 1 , given as P 1 : min Ξ N X n =1 κ k v u [ n ] k 2 + T P u + N X n =2 γ c K λ [ f u [ n ]] 3 , (6a) s.t. C 1 : N X n =1 λK f k [ n ] M + N − 1 X n =1 l k [ n ] = R k , k ∈ K , (6b) C 2 : n X i =1 E k,l [ i ] + n X i =1 λP k [ i ] ≤ n X i =1 E k [ i ] , k ∈ K , n ∈ N , (6c) C 3 : n X j =2 f u [ j ] K λ M ≤ K X k =1 n − 1 X i =1 l k [ i ] , k ∈ K , n ∈ N − N , (6d) C 4 : N X j =2 f u [ j ] K λ M = K X k =1 N − 1 X i =1 l k [ i ] , k ∈ K , (6e) C 5 : l k [ N ] = 0 , f u [1] = 0 , k ∈ K , (6f) C 6 : k q u [ n + 1] − q u [ n ] k ≤ V max K λ, n ∈ N , (6g) C 7 : q u [1] = q 0 , q u [ N + 1] = q F , (6h) C 8 : f u [ n ] ≥ 0 , f k [ n ] ≥ 0 , k ∈ K , n ∈ N , (6i) where Ξ denotes the variable set consisting of f u [ n ] , q u [ n ] , l k [ n ] , f k [ n ] ; R k denotes the total number of the computation bits of the k th user; V max is the maximum flying speed of the UA V ; q 0 and q F are the destined initial and final locations of the U A V , respecti vely; M denotes the number of CPU cycles required for computing one bit at the user and the UA V . N − N denotes the set N other than N . The constraint C 1 is the total computation bits required at the k th user; C 2 is the energy causal constraint that the energy consumed for the local computation and offloading computation bits cannot be higher than the harvesting energy; C 3 represents that the number of the computation bits at the U A V in the n th slots cannot be higher than the total number of the offloading computation bits of all users before the n − 1 th slot. Note that the U A V starts to compute the offloading bits at the n th slots only when all users finish offloading the computation bits of the n − 1 th slot; C 4 denotes that all the offloading computation bits of users should be computed; C 5 represents that the U A V does not ex ecute the computation task in the first slot and all users do not offload their computation tasks in the last slot; C 6 is the flying speed constraint and C 7 is the initial and final locations constraint related to the UA V . It is challenging to solve the non-con vex problem P 1 due to the presence of the couple among the optimization variables q u [ n ] , l k [ n ] and f k [ n ] . An alternative algorithm is proposed to solve P 1 in the following subsection. B. Computation Offloading And CPU F requency Optimization It is seen from P 1 that P 1 is conv ex for a given trajectory q u [ n ] . Thus, for a given q u [ n ] , P 1 can be transformed as P 2 , giv en as P 2 : min f u [ n ] ,l k [ n ] ,f k [ n ] N X n =2 γ c K λ [ f u [ n ]] 3 , (7a) s.t. C 1 − C 5 and C 8 . (7b) Since P 2 is con vex, it can be solved by using the Lagrange duality method [17], [18]. By solving P 2 , Theorem 1 can be obtained as follows. Theor em 1: For a given trajectory q u [ n ] , the optimal offloading computation bits and the CPU frequency of the users, and the CPU frequency of the U A V denoted by l k opt [ n ] , f u opt [ n ] and f k opt [ n ] , can be respectiv ely giv en as l k opt [ n ] = B λ log 2            B h k [ n ] " N − 1 P j = n +1 θ j + µ k − θ N # N P j = n ν k,j Γ σ 2 ln 2            , (8a) f u opt [ n ] =            0 , n = 1 s θ N − N − 1 P j = n θ j 3 γ c M , n = 2 , · · · , N − 1 q θ N 3 γ c M , n = N (8b) f k opt [ n ] = v u u u t µ k 3 γ c M K N P j = n ν k,j , k ∈ K , n ∈ N (8c) where µ k ≥ 0 , ν k,n ≥ 0 and θ n ≥ 0 are the dual variables associated with the constraints C 1 , C 2 , C 3 and C 4 , respectiv ely . Pr oof: See Appendix A. Remark 1: Theorem 1 indicates that the CPU frequency of the U A V increases with the time slots since θ N > 0 and θ n ≥ 0 when n = 2 , 3 , · · · , N − 1 . It means that the total number of the of floading computation bits increases with the time slots. Thus, in order to decrease the total energy consumed at the UA V , users need to allocate a high energy for local computation so that the number of the offloading computation can be decreased. It is also seen that the number of the offloading computation bits is increased when the channel condition between the UA V and users is improved. This indicates that the number of the of floading computation bits of users increases with the decrease of the distance between the user and the U A V . Finally , the dual variables can be obtained by using the subgradient algorithm [19]. C. T rajectory Optimization For any giv en number of the offloading computation bits, the CPU frequencies of users and the U A V , the trajectory optimization problem can be formulated as P 3 , given as P 3 : min q u [ n ] N X n =1 κ k v u [ n ] k 2 (9a) s.t. C 2 , C 6 and C 7 . (9b) Due to the constraint C 2 , P 3 is non-con vex. In order to tackle C 2 , the SCA technique is exploited. It can guarantee that the obtained solutions satisfy the Karush-Kuhn-T ucker (KKT) conditions of P 3 . By using the SCA technique, Theorem 2 is giv en as follows. Theor em 2: For any local trajectory q j u [ n ] , n ∈ N at the j th iteration, one has n X i =1 K λη P u β 0 H 2 + k q u [ i ] − q k k 2 ≥ K λη P u β 0 h k [ n ] , (10a) h k [ n ] = n X i =1          H 2 + 2   q j u [ i ] − q k   2 − k q u [ i ] − q k k 2  H 2 +    q j u [ i ] − q k    2  2          , (10b) where the equality holds when q u [ n ] = q j u [ n ] . Pr oof: Let f ( z ) = a b + z , where a and b are positiv e constants, and z ≥ 0 . Since f ( z ) is con ve x with respect to z , the following inequality can be obtain: a b + z ≥ a b + z 0 − a ( b + z 0 ) 2 ( z − z 0 ) , (11) where z 0 is a giv en local point. By using eq. (11) , Theorem 2 is obtained. Using Theorem 2, P 3 can be solved by iterati vely solving the approximate problem P 4 , given as P 4 : min q u [ n ] N X n =1 κ k v u [ n ] k 2 , (12a) s.t. C 6 and C 7 , (12b) n X i =1 E k,l [ i ] + n X i =1 λP k [ i ] ≤ K λη P u β 0 h k [ n ] , k ∈ K , n ∈ N . (12c) It is seen that P 4 is conv ex and can be readily solved by using the software CVX [10]. Based on solving P 2 and P 4 , an alternative optimization algorithm denoted by Algorithm 1 is given to solve P 1 . The details for Algorithm 1 can be found in T able 1. In T able 1, E i u denotes the value of the objectiv e function of P 1 . T ABLE I: The alternativ e optimization algorithm Algorithm 1 : The alternativ e optimization algorithm for P 1 1: Setting: R k , k ∈ K , P u , T , N , V max , q 0 , q F , and the tolerance errors ξ , ξ 1 ; 2: Initialization: The iterativ e number i = 1 and q i u [ n ] ; 3: Repeat 1: calculate l k opt,i [ n ] , f u opt,i [ n ] and f k opt,i [ n ] using eq. (8) for given q i u [ n ] ; update µ k , ν k,n and θ n using the subgradient algorithm; initialize the iterativ e number j = 1 ; Repeat 2: solve P 4 by using CVX for the given l k opt,i [ n ] , f u opt,i [ n ] and f k opt,i [ n ] ; update j = j + 1 , and q j u [ n ] ; if N P n =1    q j u [ n ] − q j u [ n ]    ≤ ξ q i u [ n ] = q j u [ n ] ; break; end end Repeat 2 update the iterativ e number i = i + 1 ; if    E i u − E i − 1 u    ≤ ξ 1 break; end end Repeat 1 4: Obtain solutions: l k opt [ n ] , f u opt [ n ] and f k opt [ n ] and q opt u [ n ] . I V . S I M U L A T I O N R E S U LT S In this section, simulation results are presented to compare the performance obtained by using our proposed design with that achieved by using two benchmark schemes, denoted by Scheme 1 and Scheme 2, respectiv ely . In Scheme 1, the UA V flies straight with a constant speed from the initial position to the final position. In Scheme 2, the U A V flies along the trajec- tory that is a semi-circle with its diameter being k q F − q 0 k . The con verge performance of the proposed algorithm is also ev aluated by simulation results. The simulation settings are based on the works in [9] and [14]. The positions and the total L (Ξ 1 ) = N X n =2 γ c K λ [ f u [ n ]] 3 + K X k =1 µ k " R k − N X n =1 λf k [ n ] M − N − 1 X n =1 l k [ n ] # + K X k =1 N X n =1 ν k,n ( n X i =1 E k,l [ i ] + n X i =1 λP k [ i ] − n X i =1 E k [ i ] ) + N − 1 X n =1 θ n    n X j =2 f u [ j ] K λ M − K X k =1 n − 1 X i =1 l k [ i ]    + θ N    K X k =1 N − 1 X i =1 l k [ i ] − N X j =2 f u [ j ] K λ M    + K X k =1 ρ k l k [ N ] + ϑf u [1] (13) 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 x (m) y (m) UAV’s trajectory being a semi−circle Our optimized UAV’s Trajectory UAV’s trajectory with a constant speed User 1 (0, 0) User 3 (10, 10) User 2 (0, 10) User 4 (10, 0) Fig. 2: The trajectories of the UA V under dif ferent schemes with T = 2 seconds. number of the computation bits of users are set as: q 1 = [0 , 0] , q 2 = [0 , 10] , q 3 = [10 , 10] , q 4 = [10 , 0] , R 1 = 2 Mbits, R 2 = 4 Mbits, R 3 = 6 Mbits, and R 4 = 3 Mbits, respectively . The detail settings are giv en in T able II. T ABLE II: Simulation Parameters Parameters Notation T ypical V alues Numbers of Users K 4 The height of the UA V H 10 m The time length of the UA V flying T 2 sec Numbers of CPU cycles M 10 3 cycles/bit Energy con versation efficiency η 0 . 8 Communication bandwidth B 40 MHz The receiv er noise power σ 2 10 − 9 W The number of time slots N 50 The mass of the UA V W 9 . 65 kg The effecti ve switched capacitance γ c 10 − 28 The channel power gain β 0 − 50 dB The tolerance error ξ , ξ 1 10 − 4 The initial position of the UA V q 0 [0 , 0] The final position of the UA V q F [10 , 0] The maximum speed of the UA V V max 10 m/s The transmit power of the U A V P u 100 dBm Fig. 2 shows the trajectories of the U A V under different schemes. The time length of the U A V flying is set as T = 2 seconds. The trajectories of the U A V under Scheme 1 and Scheme 2 are also presented. As shown in Fig. 2, under our proposed optimal trajectory , the U A V firstly flies smoothly and tends to User 2 and User 3, and then the U A V flies 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3 160 180 200 220 240 260 280 300 The time length of the UAV flying (sec) The total energy consumed at the UAV (Joule) Scheme 1 Scheme 2 Our proposed scheme The energy reduce gain Fig. 3: The total energy consumed at the U A V versus the time length of the UA V flying under different schemes. smoothly with a higher speed to the final position. The reason is that the U A V needs to provide more energy to User 2 and User 3, which has a larger number of computation bits to be offloaded. Moreov er , in order to control the total number of the computation bits of all users offloaded to the UA V , the U A V flies with a higher speed in the end of the flying time so that the harvested energy of users used for offloading the computation bits can be compromised. Fig. 3 shows the total energy consumed at the U A V versus the time length of the UA V flying under different schemes. It is seen that the energy consumed at the UA V by using our proposed scheme is the smallest among those by using the benchmark schemes. This demonstrates that our proposed scheme that jointly optimizes the number of the offloading computation bits, the CPU frequency of users and the U A V , and the trajectory of the U A V can is more ef ficient in terms of the energy minimization of the UA V . It is also seen that the total ener gy consumed at the UA V decreases with the increase of the time length of the U A V flying, irrespectiv e of the adopted scheme. It can be explained by the fact that the total energy consumed at the U A V is dominated by the flying speed and the CPU frequency of the UA V , and the flying speed and the CPU frequency can be decreased when the flying time is increased. Fig. 4 is presented to verify the efficienc y of our proposed alternativ e algorithm. It can be seen that only sev eral number of iterations are required for Algorithm 1 to con verge. 1 2 3 4 5 6 7 8 9 10 210 220 230 240 250 260 270 280 290 The number of iterations The total energy comsumed at the UAV (Joule) T=2 sec T=2.2 sec T=2.4 sec Fig. 4: The total energy consumed at the UA V versus the number of iterations T = 2 , 2 . 2 or 2 . 4 seconds. V . C O N C L U S I O N An UA V -enabled wireless powered MEC system was stud- ied where the U A V provides multiple ground users with com- putation offloading and sustainable operation opportunities. The number of the offloading computation bits and the CPU frequency of users, the CPU frequency and the trajectory of the U A V were jointly optimized in order to minimize the energy consumed at the UA V . An alternativ e algorithm was proposed based on the SCA techniques. Simulation results sho w that our proposed design outperforms other benchmark schemes and that the proposed algorithm only requires se veral number of iterations to conv erge. A P P E N D I X A P RO O F O F T H E O R E M 1 The Lagrangian of P 2 related to the proof is giv en by eq. (13) at the top of the previous page, where ρ k ≥ 0 and ϑ ≥ 0 are the dual variables related to the constraint C 5 ; Ξ 1 is the set consisting of all optimization and dual variables. Thus, the deriv ations of the Lagrangian of P 2 with respect to l k [ n ] and f k [ n ] , can be respectively giv en as ∂ L (Ξ) ∂ l k [ n ] = Γ σ 2  2 l k [ n ] Bλ  ln 2 B h k [ n ] N X j = n ν k,j −   N − 1 X j = n +1 θ j + µ k − θ N   , (14a) ∂ L (Ξ) ∂ f k [ n ] = − µ k λ M + 3 γ c K λ N X j = n ν k,j [ f k [ n ]] 2 . (14b) Let their deriv ations be zero. Thus, eq. (8a) and eq. (8c) are obtained. Let the deriv ation of the Lagrangian of P 2 with respect to f u [ n ] be zero. 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