IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 10, NO. 12, DECEMBER 2011

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A Stochastic Optimization Approach for Joint Relay Assignment and Power Allocation in Orthogonal Amplify-and-Forward Cooperative Wireless Networks Gan Zheng, Member, IEEE, Yangyang Zhang, Chunlin Ji, and Kai-Kit Wong, Senior Member, IEEE

Abstract—This paper addresses the joint relay assignment and power allocation problem for orthogonal multiuser systems using amplify-and-forward (AF) relaying nodes in the downlink. Our aim is to maximize the sum-rate subject to individual and total power constraints on the relays and a relay assignment constraint. In the case of fixed relay selection, the power allocation optimization is convex and an efficient recursive algorithm is proposed to achieve the optimum. The joint optimization of relay selection and power allocation, however, appears to be non-convex and is not known to be tractable. To tackle this, we propose a novel algorithm using Markov chain MonteCarlo with Kullback-Leibler divergence minimization (MCMCKLDM), which is proved to converge to the global optimum almost surely. Results show that the proposed scheme significantly outperforms a greedy approach and achieves near-optimal performance at very low complexity. Index Terms—Amplify-and-forward relay, convex optimization, Kullback-Leibler divergence minimization, Markov chain Monte-Carlo, metropolized independent sampling.

I. I NTRODUCTION ULTIPLE-input multiple-output (MIMO) antenna systems have recently emerged as an effective technique to enhance the performance of wireless communications, e.g., [1–4]. However, its use is tightly constrained by the ability of a mobile device to accommodate multiple antennas. On the other hand, the use of relays is an attracting scheme to extend the range of communications, reduce the transmit power and combat the fading effects. In [5], it was illustrated that singleantenna relays can be used cooperatively to form a distributed virtual MIMO channel. Many relay-based cooperative protocols have been proposed, e.g., amplify-and-forward (AF) [8– 11], decode-and-forward (DF) [12] and many others. Thus far, the use of half-duplex AF relays seems most attractive due to its implementation simplicity without the need of decoding. Though multiple relays can clearly enhance the performance of a point-to-point communication link or a single sourcedestination pair [13, 14], it is revealed that full diversity can be

M

Manuscript received September 24, 2010; revised March 28, 2011 and July 14, 2011; accepted September 9, 2011. The associate editor coordinating the review of this paper and approving it for publication was H. Yousefi’zadeh. G. Zheng is with the Interdisciplinary Centre for Security, Reliability and Trust, The University of Luxembourg, Luxembourg. Y. Zhang and C. Ji are with Kuang-Chi Institute of Advanced Technology, Shenzhen, P. R. China. K. Wong is with the Department of Electronic and Electrical Engineering, University College London, United Kingdom (e-mail: [email protected]). Digital Object Identifier 10.1109/TWC.2011.093011.101685

achieved by selecting only the best relaying terminal [15]. This has motivated the subsequent works on relay selection [16]. However, relay selection for multiuser, point- or multipointto-multipoint, systems is much less known. In particular, to select the best relays jointly for all users is a large-scale integer programming problem and extremely complex to solve. In [17], a utility maximization problem was addressed for the joint optimization of relay selection, cooperative communications, and resource allocation in a cellular network under the ideal assumption of an infinite number of channels. Recently in [18], relay selection and power allocation for AF-operated wireless relay networks were addressed and a semi-distributed algorithm was proposed for multiple source-destination pairs. The method is however heuristic and optimality is not ensured. Most recently, [19] found the optimal solution for relay assignment for multiple source-destination pairs where each pair is assisted by no more than one relay and each relay can only be assigned once. In [20], cooperative orthogonal frequencydivision multiplexing (OFDM) systems were studied where each user can use some subcarriers to relay other users’ data. A suboptimal heuristic algorithm was proposed to optimize the power allocation, subcarrier assignment and relay selection in order to minimize the overall power consumption, under the sum rate constraint over all subcarriers for each user. In this paper and in contrast to prior works, we consider a multiuser downlink system in which the central base station transmitter communicates to many mobile terminals with the aid of AF relays over orthogonal channels. In particular, our focus is on the case where the number of relays is less than the number of users so that each mobile user is assisted by at most one relay while each relay can help many users. This scenario is greatly motivated by, for instance, IEEE 802.16 networks using OFDM access (OFDMA) technologies, which are intended to accommodate a large number of users. Our goal is to maximize the sum-rate of all user terminals by jointly optimizing the power allocation among the relaying terminals and the relay assignment for the users. The power allocation problem for fixed relay selection is convex and can therefore be optimally solved. However, the relay assignment problem is combinatorial and prohibitively complex to solve. To design a near-optimal algorithm with low-complexity, the stochastic optimization method is adopted. The main contributions of this paper are summarized as follows:

c 2011 IEEE 1536-1276/11$25.00 ⃝

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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 10, NO. 12, DECEMBER 2011

hk1

gk1

S

. . .

gkN Fig. 1.

Rk

. . .

Uk1 UkN

1) For a given relay assignment, we show that the power allocation problem over the relaying terminals is convex and devise an efficient algorithm to find the optimum. 2) Moreover, we propose to use Markov chain Monte-Carlo (MCMC) with Kullback-Leibler divergence minimization (KLDM) for jointly optimizing the relay selection and power allocation at the relays. The proposed scheme is guaranteed to converge to the global optimum almost surely and simulation results indicate that our method achieves near-optimal performance at low complexity. The rest of this paper is organized as follows. Section II describes the system model and presents the problem statement. In Section III, the power allocation for fixed relay assignment is first studied and an efficient algorithm is proposed to find the optimum. Section IV then presents MCMC-KLDM for joint relay selection and power allocation. Simulation results are provided in Section V and Section VI concludes the paper. II. S YSTEM M ODEL AND P ROBLEM F ORMULATION Consider a communication link from a central base station, S, to 𝑀 mobile users, labeled as {U𝑚 }, with the help of 𝐾(< 𝑀 ) relays, {R𝑘 }, over 𝑀 orthogonal channels, as depicted in Fig. 1. Each relay operates in the AF manner such that the base station first transmits 𝑀 signals {𝑥𝑚 }𝑀 𝑚=1 , one for each user, via orthogonal channels and then at the second stage, each relay normalizes and forwards the received signals to the corresponding users. As the direct link will not change the problem structure, it is not considered for simplicity. The received signal at user 𝑚 via the 𝑘th relay is written as √

𝑣𝑘,𝑚 ℎ𝑘,𝑚 𝑙𝑘,𝑚 (𝑔𝑘,𝑚 𝑥𝑚 + 𝑛𝑘,𝑚 ) + 𝑢𝑘,𝑚 ,

(1)

with the received signal-to-noise ratio (SNR) given by Γ𝑘,𝑚 =

[√ ] E𝑥,𝑛𝑘,𝑚 ∣ 𝑣𝑘,𝑚 𝑙𝑘,𝑚 (𝑔𝑘,𝑚 𝑥 + 𝑛𝑘,𝑚 )∣2 2 (∣𝑔𝑘,𝑚 ∣2 𝑃𝑚 + 𝑁𝑘,𝑚 ) = 𝑣𝑘,𝑚 . (4) = 𝑣𝑘,𝑚 𝑙𝑘,𝑚

hkN

A relay system communicating to 𝑘𝑁 users via the 𝑘-th relay.

𝑦𝑘,𝑚 =

is the coefficient chosen to limit the average output power of R𝑘 to 𝑣𝑘,𝑚 on channel 𝑚, i.e.,

∣ℎ𝑘,𝑚 𝑙𝑘,𝑚 𝑔𝑘,𝑚 ∣2 𝑣𝑘,𝑚 𝑃𝑚 , ∣ℎ𝑘,𝑚 𝑙𝑘,𝑚 ∣2 𝑁𝑘,𝑚 𝑣𝑘,𝑚 + 𝑈𝑘,𝑚

(2)

where 𝑥𝑚 is the transmitted signal for U𝑚 , with the power of E[∣𝑥𝑚 ∣2 ] = 𝑃𝑚 , 𝑔𝑘,𝑚 denotes the channel from S to R𝑘 on channel 𝑚, 𝑣𝑘,𝑚 denotes the power budget of R𝑘 for U𝑚 with v𝑘 ≜ [𝑣𝑘,1 𝑣𝑘,2 ⋅ ⋅ ⋅ 𝑣𝑘,𝑀 ], ℎ𝑘,𝑚 denotes the channel from R𝑘 to U𝑚 , 𝑢𝑘,𝑚 ∼ 𝒞𝒩 (0, 𝑈𝑘,𝑚 ) (i.e., 𝑢𝑘,𝑚 is complex Gaussian distributed with zero mean and variance of 𝑈𝑘,𝑚 ) denotes the noise at U𝑚 which is assisted by R𝑘 , 𝑛𝑘,𝑚 ∼ 𝒞𝒩 (0, 𝑁𝑘,𝑚 ) is the noise received at R𝑘 on channel 𝑚, 𝑦𝑘,𝑚 is the received signal at U𝑚 contributed by R𝑘 , and 1 𝑙𝑘,𝑚 = √ 2 ∣𝑔𝑘,𝑚 ∣ 𝑃𝑚 + 𝑁𝑘,𝑚

(3)

In our setting, we are interested in the case where 𝐾(< 𝑀 ) relaying terminals are intelligently assigned to 𝑀 users and it is assumed that each user can only get assistance from one relay while each relay should be assigned to help many user terminals for enhancing the overall system capacity. Our aim is to maximize the network sum-rate subject to individual and total power constraints on the relays by jointly optimizing the power allocation and relay assignment, i.e., ⎧ 𝑀 ∑    𝑣𝑘,𝑚 ≤ 𝑄𝑘 ∀𝑘,     𝑚=1    𝑀 𝐾 ∑ ⎨∑ 𝑣𝑘,𝑚 ≤ 𝑄, ℐ s.t. max (5)  {𝑣𝑘,𝑚 ≥0∀𝑘,𝑚}  𝑚=1 𝑘=1    {𝐴𝑘,𝑚 ∈{0,1}∀𝑘,𝑚}  𝐾  ∑    𝐴𝑘,𝑚 = 1 ∀𝑚, ⎩ 𝑘=1

where ℐ≜

𝑀 ∑ 𝑚=1

( log2 1 +

𝐾 ∑

) 𝐴𝑘,𝑚 Γ𝑘,𝑚

(6)

𝑘=1

gives the sum-rate, the first constraint puts the power constraint 𝑄𝑘 on the 𝑘th relay while the second constraint ensures the total relay power to be below 𝑄. The former is due to the distributed nature of the relays while latter limits the overall interference to other co-existed systems. The parameter, 𝐴𝑘,𝑚 , is used to specify if a particular relay is assigned to a user. Specifically, if 𝐴𝑘,𝑚 = 1, this indicates that U𝑚 is served by R𝑘 ; otherwise, 𝐴𝑘,𝑚 = 0. Therefore, the matrix A = [𝐴𝑘,𝑚 ] is referred to as the relay assignment matrix. To find the global optimum, we are required to jointly optimize {v𝑘 }∀𝑘 and A. III. O PTIMAL P OWER A LLOCATION G IVEN A Solving (5) is a challenging task because it is combinatorial and becomes prohibitively complex if either 𝐾 or 𝑀 is large. To proceed, we first study the optimization of power allocation, {v𝑘 }∀𝑘 , if the relay assignment matrix, A, is given. Under this consideration, we define the power vector, which specifies the transmit power at the relays for all the users, as ˜ ≜ [˜ v 𝑣1 , 𝑣˜2 , . . . , 𝑣˜𝑀 ]𝑇 .

(7)

It is noted that originally we have the power allocation values {𝑣𝑘,𝑚 } for 𝑘 = 1, . . . , 𝐾 and 𝑚 = 1, . . . , 𝑀 . Nonetheless, as we have restricted that each user is only served by one relay, there are only 𝑀 non-zero values for {𝑣𝑘,𝑚 }. As such, once ˜ can be introduced to simplify the optimization. A is known, v For a given 𝑚 and 𝑘𝑚 such that [A]𝑘𝑚 ,𝑚 = 1, define ⎧ 2  ⎨ 𝑐𝑚 ≜ ∣ℎ𝑘𝑚 ,𝑚 𝑙𝑘𝑚 ,𝑚 𝑔𝑘𝑚 𝑚 ∣ 𝑃𝑚 , (8) 𝑎𝑚 ≜ ∣ℎ𝑘𝑚 𝑚 𝑙𝑘𝑚 𝑚 ∣2 𝑁𝑘𝑚 𝑚 ,  ⎩ 𝑏𝑚 ≜ 𝑈𝑘𝑚 𝑚 .

ZHENG et al.: A STOCHASTIC OPTIMIZATION APPROACH FOR JOINT RELAY ASSIGNMENT AND POWER ALLOCATION IN ORTHOGONAL AMPLIFY . . . 4093

As a result, (2) can be re-expressed as Γ𝑚 =

B. Both Total and Individual Power Constraints

𝑐𝑚 𝑣˜𝑚 𝑎𝑚 𝑣˜𝑚 + 𝑏𝑚

(9)

and (5) is reduced to max

{˜ 𝑣𝑚 ≥0}

𝑀 ∑

( log2 1 +

𝑚=1

𝑐𝑚 𝑣˜𝑚 𝑎𝑚 𝑣˜𝑚 + 𝑏𝑚

)

s.t. A˜ v ≤ [𝑄1 , 𝑄2 , . . . , 𝑄𝐾 ]𝑇 , 𝑀 ∑

𝑣˜𝑚 ≤ 𝑄.

(10a) (10b) (10c)

𝑚=1

We see that (10) is convex since all the constraints are linear and the objective function to be maximized is concave. Hence, standard interior-point algorithms can be used to obtain the optimal solution. Next, we will tailor-make an efficient recursive algorithm to solve this particular problem. To facilitate such development, we first study (10) with the total power constraint only in Section III-A and then address the general case with both individual and total power constraints in Section III-B.

Now, we show how (10) can be solved. Algorithm 2: 1) Define the set for remaining relays, 𝒮, and initialize it as 𝒮 = {1, 2, . . . , 𝐾}. 2) For a given 𝒮, solve (10) with the total power constraint 𝑄 and no individual power constraints using Algorithm ˜ 1 and let the optimal solution to this problem be 𝒑. 3) Find the set for those relays whose power exceeds their individual power limit to give ˜ > 𝑄𝑘 ∀𝑘}, 𝒮𝑒 = {𝑘∣[A]𝑘,− v

(16)

where [A]𝑘,− denotes the 𝑘th row of A. 4) Update the set for remaining relays as ˜ ≤ 𝑄𝑘 ∀𝑘}. 𝒮 = {𝑘∣[A]𝑘,− v

(17)

5) Solve the following problem for all relays 𝑘 ∈ 𝒮𝑒 : ( ) ∑ 𝑐𝑚 𝑣˜𝑚 max log2 1 + {˜ 𝑣𝑚 ≥0∀𝑚} 𝑎𝑚 𝑣˜𝑚 + 𝑏𝑚 𝑚∈{𝑚∣𝐴 ˜ ˜ 𝑘,𝑚 𝑘,𝑚 ˜ =1} 𝑚∈{𝑚∣𝐴 ˜ =1} ∑ 𝑣˜𝑚 ≤ 𝑄𝑘 . (18) s.t. 𝑚∈{𝑚∣𝐴 ˜ 𝑘,𝑚 ˜ =1}

A. Total Power Constraint Only With the total power constraint only, (10) is reduced to max

{˜ 𝑣𝑚 ≥0}

𝑀 ∑

( log2 1 +

𝑚=1

𝑐𝑚 𝑣˜𝑚 𝑎𝑚 𝑣˜𝑚 + 𝑏𝑚

) s.t.

𝑀 ∑ 𝑚=1

𝑣˜𝑚 ≤ 𝑄. (11)

In this case, the Lagrangian is given by [21] ℒ=−

𝑀 ∑ 𝑚=1

( log2 1 +

) 𝑐𝑚 𝑣˜𝑚 𝑎𝑚 𝑣˜𝑚 + 𝑏𝑚 ) ( 𝑀 𝑀 ∑ ∑ 𝑣˜𝑚 − 𝑄 − 𝜇𝑚 𝑣˜𝑚 , (12) +𝜈 𝑚=1

𝑚=1

in which 𝜈 ≥ 0 and 𝜇𝑚 ≥ 0, ∀𝑚 are Lagrange multipliers. Then, the Karush-Kuhn-Tucker (KKT) conditions are given by ∂ℒ −𝑐𝑚 𝑏𝑚 + 𝜈 − 𝜇𝑚 = 2 ∂˜ 𝑣𝑚 (𝑎𝑚 𝑣˜𝑚 + 𝑏𝑚 ) + 𝑐𝑚 𝑣˜𝑚 (𝑎𝑚 𝑣˜𝑚 + 𝑏𝑚 ) = 0, ∀𝑚, (13a) (13b) 𝑣˜𝑚 𝜇𝑚 = 0, ∀𝑚, 𝑀 ∑

𝑣˜𝑚 = 𝑄.

(13c)

𝑚=1

Combining (13a) and (13b) and solving the equation, we get (14) (see top of next page). Substituting it into (13c) gives (15) on next page and 𝜈 ≥ 0 can be easily solved by root-finding algorithms, such as bi-section search, etc. The algorithm to solve (11) is summarized as follows. Algorithm 1: 1) Solving the root 𝜈 of (15) using a root-find algorithm. 2) The optimal power allocation is given by (14).

Algorithm 1 can be used to solve (18) optimally. 6) Update the total power constraint accordingly as ∑ 𝑄 := 𝑄 − 𝑄𝑘 . (19) 𝑘∈𝒮𝑒

7) Go back to Step 2 until all power allocations are found. Algorithm 2 is very efficient because it is recursive and only Algorithm 1 is used to solve the problems with total power constraints. Now, we prove that Algorithm 2 indeed finds the global optimal solution to (10) as stated below. Theorem 1: Algorithm 2 solves (10) optimally. Proof: See Appendix A. IV. J OINT-O PTIMAL S OLUTION BY MCMC-KLDM Getting the global-optimal solution requires joint optimiza˜ , and the relay assignment tion of the power allocation vector, v matrix, A, under the constraint that each relay can serve more than one users but a user can only obtain∑assistance from only 𝐾 one relay, or 𝐴𝑘,𝑚 ∈ {0, 1}, ∀𝑘, 𝑚, and 𝑘=1 𝐴𝑘,𝑚 = 1, ∀𝑚. As described in Section III previously, we can always find the corresponding optimal power allocation if A is known. As a result, the overall problem reduces to finding the optimal relay assignment matrix, Aopt . In what follows, the most straightforward approach for maximizing (5) is by an exhaustive search (ES), which is prohibitively complex even for not so large 𝐾 or 𝑀 because the complexity grows in the order of 𝒪(𝐾 𝑀 ). To tackle this, we regard (5) as the combinatorial optimization problem: (20) Aopt = arg max ℐ(A), A∈Ω

in which Ω denotes the set of relay assignment matrices that satisfy the assignment constraints. In particular, we propose to solve (20) by a stochastic optimization approach, which we refer to it as MCMC-KLDM. The main idea here is to connect the optimization with a rare event simulation based

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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 10, NO. 12, DECEMBER 2011

⎛ 𝑣˜𝑚 = max ⎝0,

𝑀 ∑

⎛ max ⎝0,

−(2𝑎𝑚 + 𝑐𝑚 )𝑏𝑚 +

4𝑐𝑚 𝑎2𝑚 𝑏𝑚 𝜈

+

4𝑐2𝑚 𝑎𝑚 𝑏𝑚 𝜈

2(𝑎2𝑚 + 𝑎𝑚 𝑐𝑚 )

−(2𝑎𝑚 + 𝑐𝑚 )𝑏𝑚 +

√ 𝑐2𝑚 + 𝑏2𝑚 +

4𝑐𝑚 𝑎2𝑚 𝑏𝑚 𝜈

+

4𝑐2𝑚 𝑎𝑚 𝑏𝑚 𝜈

2(𝑎2𝑚 + 𝑎𝑚 𝑐𝑚 )

𝑚=1

Fig. 2.

√ 𝑐2𝑚 + 𝑏2𝑚 +

⎞ ⎠

(14)

⎞ ⎠=𝑄

(15)

The conversion from A into a.

on a carefully chosen probability density function (pdf). To appropriately represent the feasible solution space by a pdf, the Boltzmann distribution of the objective function ℐ(A) with a properly chosen temperature 𝜏 is used such that [22] { } ℐ(A) 1 𝜋(A) = exp , (21) Γ 𝜏 where Γ=

∑

{ exp

A∈Ω

ℐ(A) 𝜏

} (22)

is a normalization constant and can thus be ignored in the presentation of the method. As a consequence, the maximization of ℐ(A) is equivalent to maximizing 𝜋(A), i.e., Aopt = arg max ℐ(A) = arg max 𝜋(A). A∈Ω

A∈Ω

(23)

Here, the global optimum is regarded as a rare event and finding the optimum can thus be realized by obtaining the pdf for generating the optimal “event” with high probability. In the following, we present the proposed MCMC-KLDM method and discuss how it performs the joint optimization of power allocation and relay assignment in an iterative fashion. A. The MCMC-KLDM Algorithm While MCMC-KLDM is a known technique for stochastic optimization, specific constructions are needed so that it can be applied in our problem. First, in order to facilitate the solution representation, we introduce the vector a from A such that if 𝐴𝑘,𝑚 = 1 for some 𝑘, then 𝑎𝑚 = 𝑘. The transformation process is illustrated in Fig. 2. As a consequence, the probability mass function (pmf) for the 𝑀 -variate random vector, a, with 𝐾 candidate solutions can be defined as 𝜋(a) =

𝑀 ∑ 𝐾 ∏ 𝑚=1 𝑘=1

𝜋𝑘,𝑚 1𝑘 (𝑎𝑚 ), for some {𝜋𝑘,𝑚 }∀𝑘,𝑚 , (24)

such that 𝐾 ∑

𝜋𝑘,𝑚 = 1, ∀𝑚,

(25)

𝑘=1

and the number of candidate solutions, 𝐾, is assumed to be finite. In (24), we have { 1, if 𝑎𝑚 = 𝑘, (26) 1𝑘 (𝑎𝑚 ) ≜ 0, otherwise. The idea is that if we know the pdf, 𝜋(a), which has a proper important sampling focusing more on the increasing value of 𝜋, then this pdf can be used to find a near-optimal solution of a for maximizing 𝜋 (hence giving Aopt ). Note, however, that although for a given vector a, 𝜋(a) can be evaluated by (21), the function 𝜋(⋅) (or 𝝅 ≜ {𝜋𝑘,𝑚 }) is too complicated. In the following, we detail an iterative scheme to learn 𝝅, given an initial proposal pdf 𝑞(a; 𝒑) =

𝑀 ∑ 𝐾 ∏

𝑝𝑘,𝑚 1𝑘 (𝑎𝑚 ),

(27)

𝑚=1 𝑘=1

in which 𝒑 ≜ {𝑝𝑘,𝑚 } is an 𝑀 × 𝐾 matrix whose entry 𝑝𝑘,𝑚 denotes the probability of the 𝑘th relay being selected to help the 𝑚th user. In this process, our aim is to obtain a 𝒑opt that minimizes the Kullback-Leibler divergence 𝒟 [𝜋(a), 𝑞(a; 𝒑)] which is given by (28) (see top of next page) [23]. To solve the minimization of 𝒟 [𝜋(a), 𝑞(a; 𝒑)], which is = 0 ∀𝑘, 𝑚 and thus give rise to convex [24], we set ∂𝑝∂𝒟 𝑘,𝑚 ∫ ℒ(a; 𝑝𝑘,𝑚 ) ≜

𝜋(a) ×

∂ log (𝑞(a; 𝒑)) 𝑑a = 0. ∂𝑝𝑘,𝑚

(29)

The probability mass of each 𝑝𝑘,𝑚 is constrained to sum to 1 ∑𝐾 because 𝑘=1 𝜋𝑘,𝑚 = 1. This can be taken into account by introducing the Lagrange multiplier 𝜆 so that we have (30)

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(

∫ 𝒟[𝜋(a), 𝑞(a; 𝒑)] =

𝜋(a) × log

𝜋(a) 𝑞(a; 𝒑)

)

∫ 𝑑a =

∫ 𝜋(a) × log (𝜋(a)) 𝑑a −

𝜋(a) × log (𝑞(a; 𝒑)) 𝑑a

)] [ (𝐾 ∑ ∂ ℒ(a; 𝑝𝑘,𝑚 ) = 𝜋(a) × 𝑝𝑙,𝑚 − 1 𝑑a log (𝑞(a; 𝒑)) + 𝜆 ∂𝑝𝑘,𝑚 𝑙=1 [ ( 𝑀 𝐾 ) (𝐾 )] ∫ ∏∑ ∑ ∂ = 𝜋(a) × log 𝑝𝑙,𝑚 1𝑙 (𝑎𝑚 ) + 𝜆 𝑝𝑙,𝑚 − 1 𝑑a ∂𝑝𝑘,𝑚 𝑚=1 𝑙=1 𝑙=1 [ ] ∫ 1𝑘 (𝑎𝑚 ) = 𝜋(a) × ∑𝐾 + 𝜆 𝑑a 𝑙=1 𝑝𝑙,𝑚 1𝑙 (𝑎𝑚 )

(28)

∫

(see top of next page). Since ℒ(a; 𝑝𝑘,𝑚 ) = 0 in (29), we have [ ] ∫ 𝑝𝑘,𝑚 1𝑘 (𝑎𝑚 ) 𝜋(a) × ∑𝐾 (31) + 𝜆𝑝𝑘,𝑚 𝑑a = 0. 𝑙=1 𝑝𝑙,𝑚 1𝑙 (𝑎𝑚 ) Adding all (31) for 𝑘 = 1, 2, . . . , 𝐾 together yields ] [𝐾 ∫ 𝐾 ∑ 𝑝𝑘,𝑚 1𝑘 (𝑎𝑚 ) ∑ 𝑝𝑘,𝑚 𝑑a = 0, 𝜋(a) × +𝜆 ∑𝐾 𝑙=1 𝑝𝑙,𝑚 1𝑙 (𝑎𝑚 ) 𝑘=1 𝑘=1 (32) and 𝜆 = −1 is obtained from (32). Thus (31) becomes ] [ ∫ 𝑝𝑘,𝑚 1𝑘 (𝑎𝑚 ) 𝜋(a) × ∑𝐾 − 𝑝𝑘,𝑚 𝑑a = 0 ∀𝑘, 𝑚. (33) 𝑙=1 𝑝𝑙,𝑚 1𝑙 (𝑎𝑚 ) Although it is intractable to obtain a closed-form solution for (33), it can be estimated by an MCMC technique, such as the metropolized independent sampling (MIS) [25] which can be briefly described as follows. First, an initial value a[0] is randomly and arbitrarily chosen. Given the current sample a[0], a candidate sample b is drawn from the proposal distribution 𝑞(a; 𝒑). According to the acceptance probability1 { } 𝜋(b) 𝑞(a[𝑛]) 𝜅 = min 1, , (34) 𝜋(a[𝑛]) 𝑞(b) the new sample will be a[𝑛 + 1] = b if b is accepted, and a[𝑛 + 1] = a[𝑛] otherwise. After 𝑁 iterations, we have a set of samples {a[0], . . . , a[𝑁 ]}, following the distribution 𝜋(a). After sampling from 𝜋(a), we can estimate (33) via Monte Carlo integration [25] [ ] 𝑁 1 ∑ 𝑝𝑘,𝑚 1𝑘 (a𝑚 [𝑛]) − 𝑝𝑘,𝑚 ℋ({a[𝑛]}; 𝑝𝑘,𝑚 ) ≡ ∑𝐾 𝑁 𝑛=1 𝑙=1 𝑝𝑙,𝑚 1𝑙 (a𝑚 [𝑛]) = 0 ∀𝑘, 𝑚,

(35) are the 𝑁 random samples drawn from 𝜋(a) where {a[𝑛]}𝑁 𝑛=1 using MIS with the proposal distribution 𝑞(a; 𝒑). To find the root(s) of (35), we employ the Robbins-Monro algorithm [27] which is a generalized root-finding method when only noisy 1 The

acceptance probability plays a key role in the efficiency for finding the optimal solution that maximizes 𝜋(⋅). In fact, a good estimate on 𝜋(⋅) will be sufficient to obtain the optimal solution for maximizing 𝜋(⋅) if a large number of samples are drawn from the estimate pdf of 𝜋(⋅). However, if the estimate pdf has a stronger tendency in generating samples of greater values of 𝜋(⋅), a much smaller number of samples will be enough to identify the optimal solution. This acceptance probability is designed to serve this purpose.

(30)

measurements of the underlying function are available.2 In our case, we find a root of ℒ(⋅) = 0 based on ℋ(⋅), the noisy measurement of ℒ(⋅). The recursive procedure is written as (𝑡+1)

(𝑡)

(𝑡)

𝑝𝑘,𝑚 = 𝑝𝑘,𝑚 + 𝛼(𝑡+1) ℋ({a[𝑛]}; 𝑝𝑘,𝑚 ) ∀𝑘, 𝑚,

(36)

where 𝑡 indicates the 𝑡th iterate, and {𝛼(𝑡) } denotes a sequence of nonnegative step-sizes, satisfying the conditions: ∑ (𝑡) ∑decreasing (𝑡) 𝛼 = ∞ and (𝛼 )2 < ∞. Under such conditions 𝑡 𝑡 on the step-sizes and some mild condition on the “noise” sequence, the iteration in (36) converges to the optimal solution 𝑝opt almost surely [29, 30], leading to the recursive update: (𝑡+1)

(𝑡)

𝑝𝑘,𝑚 = 𝑝𝑘,𝑚 [ ] (𝑡) 𝑁 1 ∑ 𝑝𝑘,𝑚 1𝑘 (a𝑚 [𝑛]) (𝑡) (𝑡+1) − 𝑝𝑘,𝑚 ∀𝑘, 𝑚. +𝛼 ∑ 𝑁 𝑛=1 𝐾 𝑝(𝑡) 1𝑙 (a𝑚 [𝑛]) 𝑙=1

𝑙,𝑚

(37)

To conclude, we have an updating formula (37) to improve the estimation of the pdf with the hope that this will converge to the optimal pdf 𝒑∗ that reveals the optimal A. The proposed MCMC-KLDM algorithm is given on top of next page. B. Convergence Analysis Different from conventional Markov chain methods, the proposal density used in our proposed algorithm is adaptively updated based on previous samples and the convergence for algorithm of this kind has been a major research issue in recent years [29, 30]. Given the theorem in [30], we have the conditions for convergence of the proposed algorithm. Let {a[𝑛] : 𝑛 ≥ 0} denote a stochastic process on a finite state space evolving according to a collection of transition probabilities: 𝒦𝑛 (a[𝑛], a[𝑛 + 1]) = Prob (a[𝑛 + 1]∣a[𝑛], a[𝑛 − 1], . . . , a[0]) . (38) Note that 𝒦𝑛 (a[𝑛], a[𝑛 + 1]) depends on a[𝑛 − 1], . . . , a[0] although for notational simplicity this is not shown. Writing 𝑝(a[𝑛]) as the proposal pdf of a[𝑛], the following theorem addresses the convergence of our proposed algorithm. 2 To use the Robbins-Monro algorithm, the noise involved has to be Gaussian distributed. In our case, it can be easily seen by the central limit theorem that ℋ (⋅) − ℒ (⋅) is zero-mean Gaussian distributed [28].

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Relay Assignment Algorithm Based on the MCMC–KLDM Method (𝑡)

1 Step 1: Set the iteration counter 𝑡 = 1 and initialize 𝑝𝑘,𝑚 [0] = 𝐾 ∀𝑚, 𝑘; 𝑁 Step 2: Using MIS, generate 𝑁 samples, {a[𝑛]}𝑛=1 , using the proposal 𝑞(a; 𝒑(𝑡) ); (𝑡+1) Step 3: Update the parameter 𝑝𝑘,𝑚 via (37) where 𝛼(𝑡) is the sequence with a decreasing step size and can be chosen to be 1𝑡 to meet the convergence conditions; Step 4: The iteration ends if a convergence criterion is met (e.g., 𝑡 ≥ 𝑇 ). Otherwise, set 𝑡 := 𝑡 + 1 and then go back to Step 2.

35

Theorem 2: In the proposed MCMC-KLDM algorithm, we have sup ∣𝑝(a[𝑛]) − 𝜋(a[𝑛])∣ → 0, as 𝑛 → ∞, (39) which states that if there are enough number of samples then the proposal density will converge to 𝜋(⋅). Proof: It is well known in [29, 30] that (39) holds if the following conditions are satisfied: 1) For every 𝑛, 𝜋(⋅) is the invariant distribution for 𝒦𝑛 (a[𝑛], a[𝑛 + 1]) and ∫ 𝜋(a[𝑛])𝒦𝑛 (a[𝑛], a[𝑛 + 1]) 𝑑a[𝑛] = 𝜋(a[𝑛 + 1]). (40) 2) There exists an 𝜖 > 0 (independent of 𝑛), and a[𝑛 − 1], . . . , a[0], such that 𝒦𝑛 (a[𝑛], a[𝑛 + 1]) ≥ 𝜖𝜋(a[𝑛 + 1]).

(41)

3) ∣𝒦𝑛 (a[𝑛], a[𝑛 + 1]) − 𝒦𝑛+𝑘 (a[𝑛], a[𝑛 + 1]) ∣ ≤ 𝒪(𝑛−𝑟1 ) × 𝒪(𝑘 𝑟2 ) for some 𝑟1 , 𝑟2 > 0. The task remains to show that the proposed MCMC-KLDM algorithm meets all of the above conditions. To proceed, it is easily seen that the first condition is met because MIS is used to generate the candidate solutions [25]. In addition, as the smoothing parameter is set to 𝛼(𝑡) = 1𝑡 , the transition probabilities 𝒦𝑛 (a[𝑛], a[𝑛 + 1]) is always positive and so the second condition is met. To satisfy the third condition, we need to have 𝛼(𝑡) → 0 as 𝑡 → ∞ [30] and because we have chosen 𝛼(𝑡) = 1𝑡 , this condition is also satisfied. V. S IMULATION R ESULTS Computer simulations are conducted to assess the performance of the proposed MCMC-KLDM in Rayleigh flat-fading channels. We have assumed that 𝑔𝑘𝑚 , ℎ𝑘𝑚 ∼ 𝒞𝒩 (0, 1), ∀𝑘, 𝑚, unless otherwise specified, and relays are at about the same distance from the source and the destination. For simplicity, we assume 𝑁𝑘,𝑚 = 𝑈𝑘,𝑚 = 𝑁0 . Results for the performance upper bound given by (42) below and a greedy approach are also provided for comparison. For the performance upper bound, we remove the dependence of the relay assignment matrix A on the objective function, which enlarges the optimization space and relaxes the constraint sets: ( ) 𝑀 𝐾 ∑ ∑ log2 1 + Γ𝑘,𝑚 max ℐ¯ ≜ {𝑣𝑘𝑚 ≥0}

s.t.

⎧      ⎨      ⎩

𝑚=1 𝑀 ∑ 𝑚=1 𝑀 ∑ 𝐾 ∑ 𝑚=1 𝑘=1

𝑘=1

𝑣𝑘𝑚 ≤ 𝑄𝑘 , 𝑣𝑘𝑚 ≤ 𝑄.

(42)

30 Average Capacity (bits/s/Hz)

a[𝑛]

Bound MCMC − KLDM Greedy

25

20 K=4

K=6 15

10

K=8

0

Fig. 3.

5

10

Average capacity versus

15 SNR (dB)

𝑃𝑚 𝑁0

with

20

𝑄 𝑁0

25

30

= 10 dB.

For the greedy method, user 𝑚 simply selects relay 𝑘𝑚 that results in the strongest composite channel, i.e., 𝑘𝑚 = arg max ∣ℎ𝑘,𝑚 𝑔𝑘,𝑚 ∣2 . 𝑘

(43)

After the greedy relay assignment, the proposed Algorithm 2 is applied in order to find the optimal power allocation. In Fig. 3, the average capacity results versus the source SNR 𝑃𝑚 𝑁0 are plotted for a 30-user system with 4, 6 and 8 relays. The total transmit SNR from the relays is set to be 𝑁𝑄0 = 10 1.5𝑄 dB while the individual relay power constraint is 𝑄𝑘 = 𝐾𝑁 . 0 Results show that the performance of the proposed algorithm is very close to the upper bound, indicating the near-optimality, and the capacity gain is about 3 − 5 bps/Hz as compared with the greedy method at high SNRs. Moreover, we can see that the promising performance of the proposed algorithm is independent of the source SNR 𝑃𝑁𝑚0 and the number of relays. In Fig. 4, the average capacity results versus the number of users are plotted for systems with 4, 6 and 8 relays, and with source SNR and total relay SNR 𝑃𝑁𝑚0 = 𝑁𝑄0 = 10 dB. Again, as we can see, the proposed MCMC-KLDM achieves almost the same capacity as that of the upper bound and has about 1 − 2 bps/Hz gain when compared with the greedy method. Results illustrate that the near-optimality of MCMC-KLDM comes irrespective of the numbers of users and relays. To evaluate the impact of asymmetric relay locations on the average capacity, we assume a 2-D geometry model and the distance between the source and destination is normalized to 2. Half relays are fixed in the middle (i.e., distance from the source is 1) and therefore their channel statistics remains the same. The other half relays’ locations 𝑑 vary from 0.2 to 1.8

ZHENG et al.: A STOCHASTIC OPTIMIZATION APPROACH FOR JOINT RELAY ASSIGNMENT AND POWER ALLOCATION IN ORTHOGONAL AMPLIFY . . . 4097

TABLE I C OMPLEXITY COMPARISONS

28

Average Capacity (bits/s/Hz)

26

Bound MCMC − KLDM Greedy

24

(𝐾, 𝑀 )

𝑁

𝑇

22

(4, 40) (6, 40) (8, 40)

25 25 25

35 65 95

20 K=8 18 16 K=4

VI. C ONCLUSION

12 10 10

Fig. 4.

15

20

25 Number of Users

Average capacity versus 𝑀 with

𝑃𝑚 𝑁0

30

=

𝑄 𝑁0

35

40

= 10 dB.

80

Bound MCMC−KLDM Greedy

70

Average Capacity (bits/s/Hz)

440 640 840

875 1625 2375

K=6

14

This paper investigated the joint relay assignment and power allocation for multiuser downlink systems employing AF relays over orthogonal channels. We have been able to develop an efficient iterative algorithm based on MCMCKLDM to achieve the joint optimization, and the convergence conditions of the algorithm have also been discussed. Simulation results have confirmed its ability to achieve nearly optimal performance at much reduced complexity and its considerable performance gain over benchmark methods. A PPENDIX A. P ROOF OF T HEOREM 1 Proof: It suffices to prove that the relays in 𝒮𝑒 of Step 3 are indeed those who should be assigned the maximum individual power. To proceed, we define ( ) ∑ 𝑐𝑚 𝑣˜𝑚 𝑓 (𝑘) = − log2 1 + . (44) 𝑎𝑚 𝑣˜𝑚 + 𝑏𝑚

60

50

𝑚∈{𝑚∣𝐴 ˜ 𝑘,𝑚 ˜ =1}

Then, we can rewrite (11) with total power constraint only as

40

min

30

20 0.2

𝑣 ˜≥0

0.4

0.6

0.8

1 Relay Position

1.2

1.4

1.6

1.8

Fig. 5. Average capacity versus the distance of relays from the source with 𝑄 = 10 dB and 𝑃𝑁𝑚 = 20 dB. 𝑁 0

Total no. of functional calculations MCMC-KLDM ES

0

1 and for those relays, 𝑔𝑘𝑚 ∼ 𝒞𝒩 (0, 𝑑1 ), ℎ𝑘𝑚 ∼ 𝒞𝒩 (0, 2−𝑑 ). The average capacity results versus relays’ locations are shown in Fig. 5 for a 30-user system with 6 relays, and with source SNR, total and individual relay SNRs 𝑃𝑁𝑚0 = 20 dB, 𝑁𝑄0 = 10 1.5𝑄 . It can be seen that the proposed MCMCdB and 𝑄𝑘 = 𝐾𝑁 0 KLDM again achieves nearly optimal capacity and its capacity gain is confirmed when compared with the greedy method.

Table I provides the results for the total number of functional calculations for the proposed MCMC-KLDM and ES.3 It can be observed that the required complexity for MCMCKLDM is significantly lower than that for ES. In particular, it has been demonstrated that using the MCMC-KLDM algorithm, the near-optimal performance can be achieved with less than 1.7 × 10−33 of the computational complexity of the ES method when 𝐾 = 8 and 𝑀 = 40. 3 The complexity is almost negligible since each user simply compares the strength of the composite channel among 𝐾 relays and chooses the best one.

𝐾 ∑ 𝑘=1

𝑓 (𝑘) s.t.

𝐾 ∑

˜ ≤ 𝑄, A𝑘 v

(45)

𝑘=1

where A𝑘 denotes the 𝑘th row of matrix A. Suppose that the ˜ 𝐺 . Without loss of generality, we optimal solution of (45) is v ˜ 𝐺 > 𝑄1 . assume that relay 1 ∈ 𝒮𝑒 , i.e., A1 v The basic idea is to prove that the optimal value of the cost ˜ in the interval function in (10) is non-increasing about A1 v [0, 𝑄1 ]. To do so, we consider the following problem: ⎧ 𝐾 ∑  𝐾 ⎨ ∑ ˜ ≤ 𝑄, A𝑘 v min (46) 𝑓 (𝑘) s.t. 𝑘=1  𝑣 ˜≥0 ⎩ 𝑘=1 ˜ ≤ 𝑄𝑘 for 𝑘 = 2, . . . , 𝐾. A𝑘 v ˜ and Obviously, the optimal cost value is a function about A1 v ˜ is denoted by A1 v ˜𝐹 . suppose that the optimal solution of A1 v ˜ Now, comparing (46) with (45), it can be observed that A𝑘 v ∑𝐾 competes with each other in order to maximize − 𝑘=1 𝑓 (𝑘). ˜ . While (46) has 𝑓 (𝑘) is a non-increasing function about A𝑘 v more power constraints on the other relays than (45), i.e., ˜ can get more resource to A𝑘 𝑣˜ ≤ 𝑄𝑘 ∀𝑘 ∑𝐾= 2, . . . , 𝐾, A1 v contribute to 𝑘=1 𝑓 (𝑘) by resulting in a smaller 𝑓 (1). As a ˜ 𝐹 ≥ A1 v ˜ 𝐺 > 𝑄1 . result, we conclude that A1 v Next, we prove that the objective function (46) is convex ˜ . We study (46) for fixed A1 v ˜ as follows: about A1 v ⎧ 𝐾 ∑  𝐾 ⎨ ∑ ˜, A𝑘 ≤ 𝑄 − 𝐴1 v min 𝑓 (𝑘) s.t. 𝑘=2  𝑣 ˜𝑘 ≥0,𝑘=2,...,𝐾 ⎩ 𝑘=1 A𝑘 𝑣˜ ≤ 𝑄𝑘 for 𝑘 = 2, . . . , 𝐾. (47)

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˜ is not a variable but a parameter here. The Note that A1 v optimal objective value of (46) equals the minimum of that of ˜. (47) with regard to A1 v The Lagrangian of (47) is given by 𝐿=

𝐾 ∑ 𝑘=1

𝑓 (𝑘) +

𝐾 ∑

˜ − 𝑄𝑘 ) 𝛼𝑘 (A𝑘 v

𝑘=2

( +𝜇

𝐾 ∑

) ˜ + A1 v ˜ − 𝑄 , (48) A𝑘 v

𝑘=2

where 𝛼𝑘 , 𝜇 ≥ 0 are Lagrange multipliers. Since the problem (47) is convex, at the optimum, its objective value equals the dual objective max𝜆,𝛼 min𝑣˜ 𝐿, which is the point-wise ˜ and is maximum of a family of affine functions of A1 v ˜ [21]. This further implies therefore a convex function of A1 v that the optimal objective value of (46) is: 1) Monotonically non-decreasing, or 2) Monotonically non-increasing, or ˜ in the inter3) Monotonically non-increasing about A1 v val [0, A1 v𝐹 ] and monotonically non-decreasing in the interval [A1 v𝐹 , 𝑄1 ]. The first case is impossible as the optimal solution A1 v𝐹 > 𝑄1 is assumed nonzero. The latter two cases enable us to conclude that the optimal objective value of (47) is monotonically ˜ in the interval [0, A1 v𝐹 ] and also non-increasing about A1 v ˜ 𝐹 ≥ A1 v ˜ 𝐺 ≥ 𝑄1 , which in the interval [0, 𝑄1 ] because A1 v means that relay 1 should be assigned full power of 𝑄1 . R EFERENCES [1] G. J. Foschini and M. J. Gans, “On limits of wireless communications in a fading environment when using multiple antennas,” Wireless Personal Commun., vol. 6, no. 3, pp. 311–335, Mar. 1998. [2] E. Telatar, “Capacity of multi-antenna Gaussian channels,” Euro. Trans. Telecommun., vol. 10, no. 6, pp. 585–595, 1999. [3] A. Goldsmith, S. A. Jafar, N. Jindal, and S. Vishwanath, “Capacity limits of MIMO channels,” IEEE J. Sel. Areas Commun., vol. 21, no. 5, pp. 684–702, June 2003. [4] S. M. Alamouti, “A simple transmit diversity scheme for wireless communications,” IEEE J. Sel. Areas Commun., vol. 16, pp. 1451–1458, Oct. 1998. [5] J. Laneman, D. Tse, and G. Wornell, “Cooperative diversity in wireless networks: efficient protocols and outage behavior,” IEEE Trans. Inf. Theory, vol. 50, no. 12, pp. 3062–3080, Dec. 2004. [6] J. N. Laneman and G. W. Wornell, “Distributed space-time-coded protocols for exploiting cooperative diversity in wireless networks,” IEEE Trans. Inf. Theory, vol. 49, no. 10, pp. 2415–2425, Oct. 2003. [7] S. Cui, A. J. Goldsmith, and A. Bahai, “Energy-efficiency of MIMO and cooperative MIMO techniques in sensor networks,” IEEE J. Sel. Areas Commun., vol. 22, no. 6, pp. 1089–1098, Aug. 2004. [8] A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversity– part I: system description,” IEEE Trans. Commun., vol. 51, no. 11, pp. 1927–1938, Nov. 2003. [9] A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversity– part II: implementation aspects and performance analysis,” IEEE Trans. Commun., vol. 51, no. 11, pp. 1939–1948, Nov. 2003. [10] S. Wei, D. L. Goeckel, and M. C. Valenti, “Asynchronous cooperative diversity,” IEEE Trans. Wireless Commun., vol. 5, no. 6, pp. 1547–1557, June 2006. [11] A. Adinoyi and H. Yanikomeroglu, “Cooperative relaying in multiantenna fixed relay networks,” IEEE Trans. Wireless Commun., vol. 6, no. 2, pp. 533–544, Feb. 2007. [12] J. Luo, R. S. Blum, L. J. Cimini, L. J. Greenstein, and A. M. Haimovich, “Decode-and-forward cooperative diversity with power allocation in wireless networks,” IEEE Trans. Wireless Commun., vol. 6, no. 3, pp. 793–799, Mar. 2007.

[13] G. Zheng, K. K. Wong, A. Paulraj, and B. Ottersten, “Collaborativerelay beamforming with perfect CSI: optimum and distributed implementation,” IEEE Signal Process. Lett., vol. 16, no. 4, pp. 257–260, Apr. 2009. [14] G. Zheng, K. K. Wong, A. Paulraj, and B. Ottersten, “Robust collaborative-relay beamforming,” IEEE Trans. Signal Process., vol. 57, no. 8, pp. 3130–3143, Aug. 2009. [15] Y. Zhao, R. S. Adve, and T. J. Lim, “Improving amplify-and-forward relay networks: optimal power allocation versus selection,” IEEE Trans. Wireless Commun., vol. 6, no. 8, pp. 3114–3123, Aug. 2007. [16] A. Bletsas, A. Khisti, D. Reed, and A. Lippman, “A simple cooperative diversity method based on network path selection,” IEEE J. Sel. Areas Commun., vol. 24, no. 3, pp. 659–672, Mar. 2006. [17] T. C.-Y. Ng and W. Yu, “Joint optimization of relay strategies and resource allocations in cooperative cellular networks,” IEEE J. Sel. Areas Commun., vol. 25, no. 2, pp. 328–339, Feb. 2007. [18] J. Cai, S. Shen, J. W. Mark, and A. S. Alfa, “Semi-distributed user relaying algorithm for amplify-and-forward wireless relay networks,” IEEE Trans. Wireless Commun., vol. 7, no. 4, pp. 1348–1357, Apr. 2008. [19] Y. Shi, S. Sharma, Y. T. Hou, and S. Kompella, “Optimal relay assignment for cooperative communications,” in Proc. ACM Int. Sym. Mobile Ad Hoc Net. and Comput., May 2008. [20] Z. Han, T. Himsoon, W. P. Siriwongpairat, and K. J. R. Liu, “Resource allocation for multiuser cooperative OFDM networks: who helps whom and how to cooperate,” IEEE Trans. Veh. Technol., vol. 58, no. 6, pp. 2378–2391, June 2009. [21] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge University Press, 2004. [22] P. Laarhoven and E. Aarts, Simulated Annealing: Theory and Applications. Springer, 1987. [23] S. Kullback and R. A. Leibler, “On information and sufficiency,” Ann. Math. Statist., vol. 22, pp. 79–86, 1951. [24] R. Y. Rubinstein and D. P. Kroese, The Cross-Entropy Method: A Unified Approach to Combinatorial Optimization, Monte-Carlo Simulation, and Machine Learning. Springer Verlag, 2004. [25] J. Liu, Monte Carlo Strategies in Scientific Computing. Springer, 2001. [26] R. J. Connor and J. E. Mosimann, “Concepts of independence for proportions with a generalization of the Dirichlet distribution,” J. Amer. Statist. Assoc., vol. 64, no. 325, pp. 194–206, 1969. [27] H. Robbins and S. Monro, “A stochastic approximation method,” Ann. Math. Statist., vol. 22, pp. 400–407, 1951. [28] C. P. Robert and G. Casella, Monte Carlo Statistical Methods. Springer, 2004. [29] J. S. Rosenthal and G. O. Roberts, “Coupling and ergodicity of adaptive MCMC,” J. Applied Prob., vol. 44, pp. 458–475, 2007. [30] D. J. Nott and R. Kohn, “Adaptive sampling for Bayesian variable selection,” Biometrika, vol. 92, No. 4, pp. 747–763, 2005. Gan Zheng received the B. Eng. and the M. Eng. from Tianjin University, Tianjin, China, in 2002 and 2004, respectively, both in Electronic and Information Engineering, and PhD degree in Electrical and Electronic Engineering from The University of Hong Kong (HKU), Hong Kong, in 2008. From December 2007–September 2010, he worked as a Research Associate at University College London, London, UK. Since September 2010, he has been working as a Research Associate at the Interdisciplinary Centre for Security, Reliability and Trust (SnT), University of Luxembourg, Luxembourg. His research interests are in the general area of signal processing for terrestrial wireless communications and multibeam satellite communications, with particular emphasis on applications of optimization techniques, robust and distributed algorithms to multiuser MIMO, cognitive and cooperative system. Yangyang Zhang received the B.S. and M.S. degrees in Electronics and Information Engineering from Northeastern University, Shenyang, China, in 2002 and 2004 respectively, and the Ph.D. degree in Electrical Engineering from the University of Oxford,Oxford, U.K., in 2008. He is currently taking the position of Executive Vice President in KuangChi Institute of Advanced Technology, China. His research interests include MIMO wireless communications and stochastic optimization algorithms. Dr. Zhang has been awarded more than 20 honors. Besides, he also authored and co-authored more than 30 refereed papers.

ZHENG et al.: A STOCHASTIC OPTIMIZATION APPROACH FOR JOINT RELAY ASSIGNMENT AND POWER ALLOCATION IN ORTHOGONAL AMPLIFY . . . 4099

Chunlin Ji received the B.Eng. degree from Northeastern University, Shenyang, China, in 2003, the M.Phil. degree from the University of Cambridge, Cambridge, U.K., in 2006, and the Ph.D. degree in Statistical Science from Duke University, Durham, NC, U.S. in 2009. He was a Junior Research Assistant with the Department of Electronic Engineering, Chinese University of Hong Kong, in 2004, a Research Assistant with the Department of Electronic Engineering, City University of Hong Kong, in 2006, and a Postdoctoral research fellow with the Department of Statistics, Harvard University, in 2010. He is currently the Vice-President of Kuang-Chi Institute of Advanced Technology, Shenzhen, China. His research interests include nonparametric Bayesian modeling, Monte Carlo methods, design of computer experiments, machine learning and statistical signal processing. His collaborations and inter-disciplinary statistical research have spanned areas including metamaterial design, wireless communications, network data analysis, and others. He has published about 30 papers in various journals and conferences.

Kai-Kit Wong (S’99-M’01-SM’08) received the BEng, the MPhil, and the Ph.D. degrees, all in Electrical and Electronic Engineering, from the Hong Kong University of Science and Technology, Hong Kong, in 1996, 1998, and 2001, respectively. He is a Reader in Wireless Communications in the Department of Electronic and Electrical Engineering, University College London, UK. In the past, he took up academic and visiting positions at the University of Hong Kong, the Wireless Communications Research Department of Lucent Technologies, BellLabs, Holmdel, NJ, US, the Smart Antennas Research Group of Stanford University, CA, US, and the University of Hull, UK. Dr Wong is a senior member of IEEE and is on the editorial board of IEEE Wireless Communications Letters, IEEE Communications Letters, IEEE Signal Processing Letters, and KICS Journal of Communications and Networks. His current research interests center around secure wireless communications theory and optimization, performance analysis of single and multiuser MIMO channels, cooperative wireless networks, and cognitive radio.