A distributed algorithm using interference pricing for relay interference channels
 Kien T Truong^{1} and
 Robert W HeathJr^{1}Email author
https://doi.org/10.1186/16876180201326
© Truong and Heath; licensee Springer. 2013
Received: 21 June 2012
Accepted: 11 January 2013
Published: 20 February 2013
Abstract
Relays in cellular systems are sensitive to interference. A good relay design will transmit in a way that avoids excess outofcell interference. This article proposes a twostep algorithm for relay design for the relay interference channel, which models a relaybased cellular system. The transmitters and relays are equipped with multiple antennas while the receivers are equipped with a single antenna. In the first step, we propose to apply existing singlehop strategies to design the transmission parameters of the transmitters. In the second step, we propose to modify the interference pricing approach to design the relays. Interference pricing is used to provide the relays with information on how interference impacts the endtoend achievable rates. A new method is proposed to compute interference prices via an approximation of the endtoend achievable rate to integrate information about the relationship of the parameters in the two hops to alleviate mismatch between the rates on two hops experienced by the direct application of prior algorithms, which are designed specifically for the singlehop interference channel. Simulations show that the proposed algorithm outperforms the other designs, including the naive approach of applying the singlehop interference pricing strategies on two hops.
Keywords
Introduction
The relay interference channel models multiple transmitter–receiver pairs communicating through dedicated relays using the same spectral resource[1–4], as in cellular systems with relays. On the first hop, the transmitters send data to the relays; on the second hop, after some signal processing, the relays forward data to the receivers. Each hop experiences interference, causing resource conflicts, and coupling the achievable rates for the twohop links. In addition, the endtoend achievable rate of a twohop link is limited by rates achieved on each hop. Achieving high endtoend sumrates, therefore, requires strategies that not only mitigate interference[4, 5] but also match the rates on the two hops of the relayaided links. Unfortunately, configuring the relays with limited information about the interference is challenging. Prior work on relay design for cellular systems often neglects interference[6] or ignores matching the rates in the two hops[7–9].
In this article, we maximize the endtoend sumrates from the base stations (i.e., the transmitters) to the mobile stations (i.e., the receivers) through a set of fixed relays. We assume that the transmitters send data to the receivers with the aid of fixed, halfduplex, and decodeandforward (DF) relays via twohop transmissions. The transmitters and relays do not share data. While the transmitters and the relays are equipped with multiple antennas, the receivers have a single antenna. The direct channels from the transmitters to the receivers are neglected due to high pathloss and shadowing attenuation. We assume that the twohop links have a common timesharing value and perfect interuser frame synchronization, i.e., the transmissions in each hop start at the same time and end at the same time. Although this assumption requires some coordination among the transmitters before the actual data transmission, it allows for tractable analysis to obtain insights into the system performance and provides a benchmark for the scenarios with relaxed assumptions on interuser frame synchronization. We also assume that the parameters of the first hop are fixed and focus on the design of transmission parameters at the relays. This is reasonable when the relays are allowed to schedule users[10], as in IEEE 802.16j[11], IEEE 802.16m[12], and 3GPP LTE/LTEAdvanced standards[13]. It is important to note that we consider endtoend achievable rates. Thus, the design challenge is how to take into account information about the fixed firsthop parameters and the timesharing value in the configuration of the second hop while mitigating interference.
Our design problem is to determine transmit precoders at the transmitters and relays to maximize the endtoend sumrates. The singlehop sumrate maximization problem is nonconvex and NPhard, i.e., its globally optimal solution cannot be found efficiently in terms of computational complexity[14]. More complicated, the joint transmit precoder problem for endtoend sumrate maximization is also nonconvex and is expected to be NPhard. This motivates us to develop an algorithm for transmit precoder design that obtains high endtoend sumrates, i.e., to find efficiently suboptimal solutions to the endtoend sumrate maximization problem. Note that prior work often focuses on singlehop sumrate maximization. Then existing algorithms could be applied to maximize independently the sumrates in each hop of the twohop interference channel. This naive approach, however, leads to rate mismatch in the two hops, which reduces the endtoend sumrates. Rate mismatch occurs when some relay links have a dominant first hop while others have a dominant second hop. From a resource allocation perspective, this means it wastes resources to the secondhop dominant relay links while depriving resources from the firsthop dominant relay links.
In this article, we propose a twostep algorithm for designing the transmit precoders at the transmitters and relays. In the first step, we propose to apply directly the existing singlehop strategies for the first hop to design the precoders at the transmitters. The relays then estimate their received signaltointerference plus noise ratio (SINR), which represent the achievable rates on the first hop. In the second step, we use interference pricing[15–17] to develop a distributed relay beamforming algorithm where each relay determines its own transmit precoder without explicit knowledge of the precoders at the other relays and of the channels from the other relays. We propose a new method for computing interference prices at the receivers such that interference prices also include information about the firsthop SINR and the timesharing value. Our approach uses an approximation function of the endtoend achievable rate to take into account the relationship of the parameters on two hops. We describe how the new twohop interference pricing framework is used to develop an asynchronousdistributed pricing algorithm for relay transmit beamforming. In each iteration of the algorithm, one randomly selected relay updates its own precoder based on the knowledge of interference prices corresponding to unintended receivers by solving a nonlinear optimization problem. In general, however, finding the globally optimal solutions to this nonlinear optimization problem may be timeconsuming. To overcome this challenge, we propose a method for solving it approximately, although we are unable to prove analytically the convergence of the modified algorithm. We also present in detail a version of the proposed algorithm, which we refer to as the twohop interference pricing algorithm for relay transmit power control, where we can find closedform expressions for the updates at the relays. This power control algorithm is applicable when the relays have a single antenna. It is also applicable when the relays have multiple antennas but we focus on designing only the norm of the relay precoders. The endtoend sumrate performance of the proposed algorithm is evaluated by Monte Carlo simulations in a multicell cellular network. We observe that the modified algorithm converges in the simulated scenarios. For comparison, we implement a number of relay transmission strategies, including the naive application of the singlehop interference pricing algorithm in[16] to the second hop. The numerical results show that, thanks to the capabilities of interference mitigation and twohop ratemismatch alleviation, the proposed algorithm always outperforms all other algorithms for various system configurations.
Distributed beamforming/precoding has extensively been studied for the singlehop interference channel[16–23];[16, 17, 22] using interference pricing. These singlehop algorithms, including the algorithm in[16], could be applied to maximize independently the sumrates in each hop of the twohop interference channel. This naive approach, however, leads to rate mismatch in the two hops, which reduces the endtoend sumrates. On the contrary, our proposed algorithm alleviates twohop rate mismatch by integrating the firsthop parameters and the timesharing value in relay transmit beamforming design, thus improving the endtoend sumrates. There have been algorithms for distributed relay beamforming[6, 24–26]. Prior work, however, either focuses on another type of relay (i.e., amplifyandforward relays[7, 24, 25, 27–29]) or considers other relay architectures (e.g., the shared relays[6, 30] and the twoway relay[26, 31]). In prior work[6, 24–26], each relay aids multiple pairs of transmitters and receivers at the same time. On the contrary, this study considers oneway DF relays, each of which aids only a single pair of transmitters and receivers. In principle, DF relays can be treated as conventional users on the first hop and acts as base stations on the second hop. Thus, they are readily integrated in a conventional cellular network. Amplifyandforward relays, however, apply the linear transformation to the received signal on the first hop instead of decoding it like a conventional user. Although amplifyandforward relays put less of a signal processing burden on the relays, their integration into conventional cellular networks requires other features like a new pilot structure and different channel estimation methods[32, 33]. IEEE 802.16m considered only DF relays[12] while 3GPP LTE/LTEAdvanced considered both DF and amplifyandforward relays[13].
Power control is a classic technique for interference mitigation in cellular systems (see[34] and references therein). For example, there are several game theorybased power control algorithms for the singlehop interference channel[15, 35–43] and femtocell networks[44, 45]. While Huang et al.[15] uses interference pricing, Stanczak et al.[40, 43] use the framework of adjoint networks that allow for fully distributed implementation. Similar to the case of beamforming, since the prior algorithms are developed specifically for the singlehop interference channel, their direct applications for the second hop cause rate mismatch. The literature on power control algorithms for relayassisted wireless networks, however, is limited[4, 46–48]. An algorithm for power control at the relays and transmitters in interference relay channel is developed in[46], but it aims at minimizing the total transmit power subject to SINR requirements. The prior work in[4, 47] determines allocation of the sum power for each twohop link for its transmitter and relay in a twouser twohop interference channel to maximize the endtoend sumrates. Not only is the problem formulation in[4, 47] different from ours, but also the results cannot be easily extended to the case with more than two users. An algorithm for power control at both the base stations and relays in a multicell network is developed in[48]. The idea is based on the use of pricing factors to reflect the impact of interference, which is in principle similar to that of interference pricing. Nevertheless, pricing factors in[48] are determined numerically rather than by analytical methods. Also, it is unclear how the pricing factors can be extended for the beamforming design.
Our initial results in this article were reported in[49]. Compared with[49], this article presents in more detail the proposed twohop interference pricing, discusses the distributed twohop interference pricing algorithm, and provides more simulations that emphasize the achievable endtoend sumrate performance of the proposed algorithm in comparison with the existing strategies. In addition, we reported some related results on the DF relay broadcast interference channel reported in[50]. Although it also aims at the alleviating twohop ratemismatch while mitigating interference, our prior work in[50] adopts another approach, which is based on a relationship between mean squared error (MSE) values and mutual information. Comparing the two approaches, the idea in[50] has higher channel state information (CSI) requirements and is more complex, but has the advantage of supporting multiuser multipleinput multipleoutput (MIMO) from the relays.
The organization of this article is as follows. First, we introduce the system model of the relay interference channel. Second, we formulate the problem of sum endtoend achievable rate maximization and discusses the challenges in finding its optimal solutions. Third, we present in detail the proposed approach that is based on the interference pricing framework to find highquality suboptimal solutions. Fourth, we present Monte Carlo simulations with a multicell system setting. Finally, we conclude this article and provide suggestions for future research.
Notation
The lowercase and uppercase boldface letters (e.g., h and h) indicate column vectors and matrices, respectively. h ^{∗} and ∥h∥ denote the complex conjugate transpose and the L2 norm of h. I _{ N } stands for the identity matrix of size N × N. We use ν ^{max}(H) to denote the eigenvector corresponding to the maximum eigenvalue of H. a denotes the absolute value of a complex scalar number a. The subscript ()_{{1}} implies the first stage while ()_{{2}} for the second stage. The superscript ()^{(n)} denotes the n th iteration. $\mathbb{E}[\xb7]$ is the statistical expectation operator. For any stacked notation X = (x _{1},…,x _{ K }), we define ${\mathbf{X}}_{k}\triangleq ({\mathbf{x}}_{1},\dots ,{\mathbf{x}}_{k1},{\mathbf{x}}_{k+1},\dots ,{\mathbf{x}}_{K})$ for k ∈ {1,…,K}. We use X and (x _{ k };X _{−k }) interchangeably.
System model
Since halfduplex relays cannot transmit and receive at the same time, the transmission procedure consists of two stages. We assume the transmissions in each stage start at the same time and also end at the same time. This can be justified by some coordination between the transmitters for sharing a predefined common timesharing values and for starting sending data at the same time. The requirement is not that strict because this is typically performed as part of ranging for example in a timedivision multipleaccess (TDMA) system. Such an interuser transmission synchronization assumption is made either explicitly or implicitly in much prior work on interference channel. Moreover, this assumption allows for tractable analysis to obtain insights into the system performance and provides a benchmark for the scenarios with relaxed assumptions on interuser frame synchronization. Using common radio resources, the transmissions in the same stage interfere with each other. For tractable analysis, we assume Gaussian signaling in both stages although it may not be optimal for the relay interference channel. In the first stage, the transmitters send data to their relays. Treating interference signals as additive Gaussian noise, each relay decodes the desired signal, reencodes and retransmits to its receiver in the second stage. Each receiver also treats the interference from unintended relays as additive Gaussian noise when decoding the desired signal from its relay.
Define ${\xi}_{k}\left({\mathbf{F}}_{\mathrm{T}}\right)={2}^{{R}_{1,k}\left({\mathbf{F}}_{\mathrm{T}}\right)}1$. Intuitively, one can think of ξ _{ k }(F _{T}) as an effective SINR at relay k if transmitter k sends a single data stream to relay k.
The design of F _{T} and F _{X} for maximizing R _{sum}(F _{T},F _{X}) should take into account t. Also, note that the units for the achievable rates in (3), (6), (7), and (8) are bps/Hz.
Problem formulation
Interference mitigation is the main challenge in sumrate maximization. Due to interference, there exists coupling among the achievable rates on the same hop. It is the coupling that makes sumrate maximization problems nonconvex and NPhard, even for the singlehop interference channel[52, 53]. The more complicated problem $\left(\mathcal{O}\mathcal{P}\right)$ is expected to be nonconvex and NPhard as well. This means that the globally optimal solutions to $\left(\mathcal{O}\mathcal{P}\right)$ cannot be found efficiently in terms of computational complexity even in a centralized fashion. In addition, due to the definition of the endtoend achievable rates, $\left(\mathcal{O}\mathcal{P}\right)$ has a complicated peruser objective function. In fact, it is challenging to find its stationary points, including their globally and locally optimal solutions[54]. Thus, in this article, we focus on finding suboptimal solutions to $\left(\mathcal{O}\mathcal{P}\right)$ that have high endtoend sumrates.
Note that twohop rate matching is a challenge in solving for suboptimal solutions to $\left(\mathcal{O}\mathcal{P}\right)$. Specifically, for a given t,F _{T} and F _{X}, there may exist a mismatch between the normalized rates on two hops. By definition, a twohop rate mismatch occurs when there exist two twohop links such that one has the dominant firsthop link while the other has the dominant secondhop link. Mathematically, it occurs when there exist $k,m\in {\mathcal{K}}_{\mathrm{X}}$ and k ≠ m, such that t R _{1,k }(F _{T}) > (1−t)R _{2,k }(F _{X}) and t R _{1,m }(F _{T}) < (1−t)R _{2,m }(F _{X}). When a twohop rate mismatch happens, we can always improve the endtoend sumrates by scaling down p _{X,m } and fixing all the other parameters such that the rates on two hops of the m th twohop link are equal to t R _{1,m }(F _{T}). While this power reduction does not change R _{ m }(F _{T},F _{X}), it decreases the interference caused by relay m to unintended receivers on the second hop. This means that the power reduction does not decrease R _{ q }(F _{T},F _{X}) for q≠m, and especially it strictly increases R _{ k }(F _{T},F _{X}). Thus, an efficient system design in terms of endtoend sumrate maximization should not cause any twohop rate mismatch.
A distributed algorithm using interference pricing
In this section, we propose a distributed algorithm for finding highquality suboptimal solutions to $\left(\mathcal{O}\mathcal{P}\right)$. The proposed algorithm consists of two consecutive steps: (i) the first step focuses on designing the transmitters while ignoring the secondhop parameters and (ii) the second step focuses on designing the relays given knowledge of the timesharing value and the firsthop achievable rates resulting from the previous step.
Step 1: transmitter design
In the first step, we focus only on the design of the parameters of the transmitters on the first hop. In particular, we use one of the existing distributed algorithms for the singlehop interference channel to design the firsthop precoders F _{T} and/or transmit power values p _{T}. Due to twohop rate mismatch, an algorithm with a lower firsthop sumrates may achieve higher endtoend sumrates than another algorithm with a higher firsthop sumrates. Thus, in our proposed twostep algorithm, all the existing distributed transmission strategies for the singlehop interference channel are candidates for designing the transmitters’ parameters.
We now briefly describe several strategies for the design of multipleantenna transmitters with increasing requirements for complexity and overhead:

In the first strategy, each transmitter aims at maximizing the desired signal to its associated relay regardless of the interference it may cause to unintended relays[55]. This strategy has the lowest complexity and overhead and is referred to as the maximum ratio transmission (MRT) beamforming. From a game theoretic perspective, the transmitters behave egoistically in this strategy, resulting in no cooperation among them[56]. Each transmitter is required to know only the channel to its associated relay, thus allowing for completely distributed implementation.

In the second strategy, each transmitter behaves altruistically by minimizing the power of interference they cause to unintended relays. The interference on the transmit side is also known as the leakage[21, 57, 58]. Specifically, each transmitter uses multipleantenna techniques for nullifying its leakage signals, similar to zeroforcing (ZF) beamforming on the receive side. The implementation of the strategy requires that each transmitter has the CSI of the channels from itself to all the relays. Thus, it also allows for distributed implementation but with a larger amount of overhead than the MRT strategy.

In the third strategy, each transmitter aims at maximizing the signaltoleakage plus noise ratio (SLNR), which is also known as the virtual SINR[59]. The SLNR for each transmitter is defined as the ratio of the desired signal power at its receiver over the sum power of its leakage signals. The maximization of the SLNR provides some balance between egoism and altruism. Note that, in addition to the CSI of the channel from itself to all the relays, each transmitter needs to know the variance of the noise at its associated receiver.

In the fourth strategy, each transmitter determines its own beamforming vector based on knowledge of how it affects the achievable rates of unintended receivers. This is based on the interference pricing framework, which was developed for the singlehop interference channel in[15–17, 35, 60]. When applied to the first hop, the relays compute interference prices that quantify marginal changes in the achievable rates per unit increase in interference power at the relays[15]. The relays then provide their interference prices to the transmitters via feedback channels. Using the interference prices and the knowledge of the channels from itself to the relays, each transmitter determines its own beamforming vector. Although this strategy allows for distributed implementation, it is iterative and requires more overhead due to the exchange of interference prices.

In the final strategy, each transmitter aims at minimizing its corresponding weighted MSE. The idea is to formulate a weighted MSE minimization problem that has the same stationary points as the sumrate maximization problem but has a betterbehaved objective function based on a relationship between the mutual information and MSE[61]. Thus, we can solve for the stationary points of the weighted MSE minimization problem instead of finding directly those of the sumrate maximization problem. Note that our other results in[50] are based on the same approach as this strategy.
Let ${\stackrel{\u0304}{\xi}}_{k}$ be the resulting received SINR at relay k for $k\in \mathcal{K}$ at the end of the first step. Recall that, by definition, these values represent the achievable rates on the first hop and are used as an input for the relay design in the second step. We assume that relay k knows ${\stackrel{\u0304}{\xi}}_{k}$ perfectly and use it in the relay design. In principle, we can try several candidate distributed strategies in designing the transmitters in the first step and then provide the corresponding values of ${\left\{{\stackrel{\u0304}{\xi}}_{k}\right\}}_{k=1}^{K}$ for the relay design. The relays will be designed with different sets of ${\left\{{\stackrel{\u0304}{\xi}}_{k}\right\}}_{k=1}^{K}$ to select the one with the highest endtoend sumrates. The selected candidate firsthop design strategy will then be informed to the transmitters. Although improving the endtoend sumrates, this increases the requirements for coordination and overhead.
Step 2: relay design
By definition, the endtoend sumrates depend not only on the secondhop achievable rates, but also on t and ${\left\{{\stackrel{\u0304}{\xi}}_{k}\right\}}_{k=1}^{K}$. Thus, in addition to mitigating interference to achieve high secondhop rates, the relay design should match the rates achieved on the second hop with those achieved on the first hop as designed in the previous step. In the second step, given the transmitters’ parameters designed in the first step, we adopt an interference pricing approach to design the relays’ parameters for obtaining high endtoend sumrates. The key idea is to take the advantage of interference prices to exchange information about t and ${\left\{{\stackrel{\u0304}{\xi}}_{k}\right\}}_{k=1}^{K}$ between the relays and receivers for a better relay design. One of our main contributions is to propose a modification of the computation method of interference prices to integrate information about t and ${\left\{{\stackrel{\u0304}{\xi}}_{k}\right\}}_{k=1}^{K}$, which we refer to as the twohop interference pricing framework. We also apply the new framework to develop the corresponding algorithm for solving $\left(\mathcal{B}\mathcal{F}\right)$.
Proposed twohop interference pricing framework
We now present the twohop interference pricing framework for transmit beamforming. Note that it is straightforward to develop the similar twohop interference pricing framework for power control, however, we omit it here to save space. The interference prices are computed based on the Karush–Kuhn–Tucker (KKT) necessary conditions for optimality of the sumrate maximization problem. This computation method is crucial for interference pricingbased algorithms to obtain high sumrates. In the DF relay interference channel, we notice that ${R}_{k}({\stackrel{\u0304}{\xi}}_{k},{\gamma}_{k}({\mathbf{F}}_{\mathrm{X}}\left)\right)$ is not continuously differentiable with respect to γ _{ k }(F _{X}) at the intersection point that makes (1 − t) log2(1 + γ _{ k }(F _{X})) equal to $t\phantom{\rule{0.2em}{0ex}}{log}_{2}(1+{\stackrel{\u0304}{\xi}}_{k})$. It follows that ${R}_{k}({\stackrel{\u0304}{\xi}}_{k},{\gamma}_{k}({\mathbf{F}}_{\mathrm{X}}\left)\right)$ is not continuously differentiable with respect to f _{X,m } at every point for all $m\in \mathcal{K}$. Therefore, it is challenging to use the KKT necessary conditions for $\left(\mathcal{B}\mathcal{F}\right)$ to find directly even its locally optimal solutions to[62]. To overcome this, we propose to use an approximate function of the endtoend achievable rate, which we refer to as R _{soft,k }(γ _{ k }(F _{X})) and is a function of ${\left\{{\stackrel{\u0304}{\xi}}_{k}\right\}}_{k=1}^{K}$, t, and γ _{ k }(F _{X}).
The quantity C _{ k }(R _{soft,k }(γ _{ k }(F _{X}))) is referred to as the coefficient of relative risk aversion of the utility function R _{soft,k }(γ _{ k }(F _{X}))[63]. The larger value of C _{ k }(R _{soft,k }(γ _{ k }(F _{X}))) indicates that R _{soft,k }(γ _{ k }(F _{X})) is “more concave”[35]. If these conditions are satisfied, then it is possible to develop an algorithm with asynchronous beamforming vector updates such that ${\sum}_{k=1}^{K}{R}_{\text{soft},k}\left({\gamma}_{k}\right({\mathbf{F}}_{\mathrm{X}}\left)\right)$ is nondecreasing after each iteration provided that the interference prices are current (i.e., they are updated after each iteration)[16, 35].
Because 0 < γ _{ k }(F _{X}) < (1 + γ _{ k }(F _{X})) and 0 < γ _{ k }(2 + γ _{ k }(F _{X})) < (1 + γ _{ k }(F _{X}))^{2}, we have C _{ k }(R _{ip1,k }(γ _{ k }(F _{X}))), C _{ k }(R _{ip2,k }(γ _{ k }(F _{X}))) ∈ (0,2]. Both R _{ip1,k }(γ _{ k }(F _{X})) and R _{ip2,k }(γ _{ k }(F _{X})) are functions of γ _{ k }(F _{X}), t and ${\left\{{\stackrel{\u0304}{\xi}}_{k}\right\}}_{k=1}^{K}$. If ${\gamma}_{k}\left({\mathbf{F}}_{\mathrm{X}}\right)>{\varphi}_{k}({\stackrel{\u0304}{\xi}}_{k},t\phantom{\rule{0.2em}{0ex}})$, then they are larger than ${R}_{k}({\stackrel{\u0304}{\xi}}_{k},{\gamma}_{k}({\mathbf{F}}_{\mathrm{X}}\left)\right)$. The gaps between these approximate functions and the exact function increase with γ _{ k }(F _{X}) and are upperbounded by (1−t)/ ln2. It is still unclear how to evaluate the quality of the approximate endtoend functions in terms of exact endtoend sumrate maximization. Roughly speaking, we expect that the one with a smaller gap with the exact function, which is R _{ip2,k }(γ _{ k }(F _{X})) in this case, outperforms the other. Although this prediction is confirmed by our numerical results, we do not have a formal proof. Note that the two approximate functions can be used to develop distributed twohop interference pricing algorithms for solving $\left(\mathcal{B}\mathcal{F}\right)$ and $\left(\mathcal{P}\mathcal{C}\right)$. The next sections present the details of how such algorithms are developed using R _{ip1,k }(γ _{ k }(F _{X})). The algorithms corresponding to the use of R _{ip2,k }(γ _{ k }(F _{X},p _{X})) can be developed in the same way.
Distributed algorithm development
This section describes a distributed twohop interference pricing algorithm for solving $(\mathcal{B}\mathcal{F}\mathcal{I}\mathcal{P})$.
To compute θ _{BF,ip1,k }(F _{X}), receiver k itself estimates γ _{ k }(F _{X}) and A _{ k }(F _{X}). In addition, it obtains ${\varphi}_{k}({\stackrel{\u0304}{\xi}}_{k},t\phantom{\rule{0.2em}{0ex}})$ from relay k via a feedforward channel. We need θ _{BF,ip1,k }(F _{X}) and γ _{ k }(F _{X}) to compute β _{BF,ip1,k }(F _{X}).
According to Proposition 3 in[35], the resulting algorithm based on R _{ip1,k }(γ _{ k }(F _{X})) is guaranteed to converge to a stationary point of $(\mathcal{B}\mathcal{F}\mathcal{I}\mathcal{P})$.
We can show that the solution is ${\mathbf{f}}_{\mathrm{X},k}^{\left(n\right)}={\nu}^{\text{max}}\left({\mathbf{Y}}_{\text{BF},k}\right({\mathbf{F}}_{\mathrm{X}}^{(n1)}\left)\right)$, the eigenvector corresponding to the maximum nonnegative eigenvalue of ${\mathbf{Y}}_{\text{BF},k}\left({\mathbf{F}}_{\mathrm{X}}^{(n1)}\right)$. If ${\mathbf{Y}}_{\text{BF},k}\left({\mathbf{F}}_{\mathrm{X}}^{(n1)}\right)$ has no nonnegative eigenvalue, then the relay does not update its beamforming.
Algorithm. Distributed twohop interference pricing algorithm for relay beamforming design

Initialization: each relay k feeds forward ${\varphi}_{k}({\stackrel{\u0304}{\xi}}_{k},t\phantom{\rule{0.2em}{0ex}})$ to receiver k and selects a random beamformer ${\mathbf{f}}_{\mathrm{X},k}^{\left(0\right)}$.

Iteration n(n ≥ 1): the following updates are performed

Interference price: each receiver k computes ${\theta}_{\text{BF},\text{ip}1,k}\left({\mathbf{F}}_{\mathrm{X}}^{(n1)}\right)$ and broadcasts to all the relays.

Desired signal reward: each receiver k computes ${\beta}_{\text{BF},\text{ip}1,k}\left({\mathbf{F}}_{\mathrm{X}}^{(n1)}\right)$ and ${\gamma}_{k}\left({\mathbf{F}}_{\mathrm{X}}^{(n1)}\right)$. Then it feeds back the information to relay k.

Beamforming vector: a relay k is randomly selected to update ${\mathbf{f}}_{\mathrm{X},k}^{\left(n\right)}={\nu}^{\text{max}}\left({\mathbf{Y}}_{\text{BF},k}\right({\mathbf{F}}_{\mathrm{X}}^{(n1)}\left)\right)$.

Although we have not been able to prove analytically the convergence of the modified algorithm, we observed from our numerical results that it converges in all the cases considered.
For updating its beamforming vector, each relay k needs to know the following information: ${\left\{{\mathbf{h}}_{m,k}\right\}}_{m=1}^{K}$, θ _{BF,ip1,m } for m ≠ k, and β _{BF,ip1,k }. While this information can be made available at the relay via feedback, thus allowing for distributed implementation, a large amount of overhead may incur. In addition, in this article, we assume that the relays have obtained this information perfectly. The inaccuracy of this information, however, may affect significantly the achievable sumrates. Methods for reducing overhead and investigating the impacts of the uncertainty of this information on the sumrate performance are interesting topics for future research.
Note that our proposed algorithm is different from the prior algorithm in[16]. Prior work in[16] is able to maximize the sumrates on a single hop, i.e., either the first hop or the second hop. It ignores ${\left\{{\gamma}_{k}^{\left\{1\right\}}\right\}}_{k=1}^{K}$ and t and hence results in twohop rate mismatch. On the contrary, our approach is able to take into account ${\left\{{\stackrel{\u0304}{\xi}}_{k}\right\}}_{k=1}^{K}$ and t to alleviate twohop rate mismatch to obtain higher endtoend sumrates. In addition, the relays need to feedforward information to their associated receivers.
Recall that when N _{X} > 1, then $(\mathcal{B}\mathcal{F}{\mathcal{N}\mathcal{P}}_{k})$ is a nonlinear optimization problem, of which finding the globally optimal solutions may be timeconsuming. Although the modified algorithm can approximately solve $(\mathcal{B}\mathcal{F}{\mathcal{N}\mathcal{P}}_{k})$, we are unable to prove analytically the convergence of the modified algorithm. This challenge may not appear when we focus on updating only the actual transmit power p _{X}, for example, when the relays have a single antenna or when we update only the norm of the relay precoders but not their shape. For notational convenience, we denote ${\mathbf{F}}_{\mathrm{X}}=\sqrt{{p}_{\mathrm{X}}}{\stackrel{\u0304}{\mathbf{F}}}_{\mathrm{X}}$, where ${\stackrel{\u0304}{\mathbf{F}}}_{\mathrm{X}}$ with $\parallel {\stackrel{\u0304}{\mathbf{F}}}_{\mathrm{X}}{\parallel}_{F}^{2}=1$ represents the shape of the transmit precoders at the relays. Given fixed beamforming vectors ${\stackrel{\u0304}{\mathbf{F}}}_{\mathrm{X}}$, we define the effective channel from relay k to receiver m as ${\stackrel{\u0304}{h}}_{m,k}\triangleq {\mathbf{h}}_{m,k}^{\ast}{\stackrel{\u0304}{\mathbf{f}}}_{\mathrm{X},k}\in \mathbb{C}$ for $k,m\in \mathcal{K}$. We can always formulate an interference pricingbased relay power control problem to determine p _{X} given the knowledge of effective channel ${\stackrel{\u0304}{h}}_{m,k}$, which is referred to as $(\mathcal{P}\mathcal{C}\mathcal{I}\mathcal{P})$ and is a special case of $(\mathcal{B}\mathcal{F}\mathcal{I}\mathcal{P})$. Note that in the relay beamforming design problem, each relay needs to determine simultaneously several complex numbers to update its beamforming vectors. In the relay power control algorithm, each relay needs to determine only a positive real number for its transmit power value. This may simplify the implementation of the resulting algorithm. Specifically, by relaxing the power range constraints and using similar steps for finding highquality solutions for $(\mathcal{B}\mathcal{F}\mathcal{I}\mathcal{P})$, we can develop a twohop interference pricing power control algorithm for determining p _{X}. This algorithm is iterative and asynchronous where in each iteration only one relay is allowed to update the transmit power. Let k be the index of the selected relay for transmit power update in iteration n≥1. One important characteristic of this algorithm is that the candidate value for power update in iteration n is provided in closed form as
Incorporating back the power range constraints, we propose that relay k updates its transmit power as ${p}_{\mathrm{X},k}^{\left(n\right)}={\stackrel{\u0304}{p}}_{\mathrm{X},k}^{\left(n\right)}$ only if $0\le {\stackrel{\u0304}{p}}_{\mathrm{X},k}^{\left(n\right)}\le {p}_{\mathrm{X},k}^{\text{max}}$. According to Proposition 2 in[35], the iterative and asynchronous interference pricing algorithm for power control is guaranteed to converge to a stationary point of $(\mathcal{P}\mathcal{C}\mathcal{I}\mathcal{P})$. Note that this algorithm does not maximize directly the exact endtoend sumrates. Thus, its convergence in terms of exact endtoend sumrate maximization is not guaranteed. In our simulations, however, this algorithm converges in all the considered scenarios.
Discussion
In this section, we provide some remarks on the order of optimization, distributed implementation, and complexity analysis of the algorithm.
Order of optimization
There is another optimization order where the relays are first designed to maximize the achievable sumrates on the second hop. Then interference pricing is used to take into account the timesharing and secondhop rates in the design the transmitters to approximately maximize endtoend achievable rates. Depending on channel realizations on two hops, one order of the optimization outperforms the other in terms of endtoend sumrate maximization and vice versa. We prefer our order of optimization due to overhead considerations. Specifically, as the relays themselves estimate the received SINR on the first hop, our proposed order of optimization only requires the receivers estimate and send back the secondhop SINR to the relays to perform twohop rate matching. The other order of optimization, however, requires the relays and receivers to send back the SINR on two hops to the transmitters to perform twohop ratematching.
Distributed implementation
The implementation of the proposed algorithm requires that the transmissions on each hop start at the same time and end at the same time. The relays are informed about the predetermined common time sharing value t. The transmitters need to agree with each other on the transmission strategy on the first hop as listed in Step 1. Each relay k itself estimates the received SINR on the first hop ${\stackrel{\u0304}{\xi}}_{k}$ and computes ${\varphi}_{k}({\stackrel{\u0304}{\xi}}_{k},t\phantom{\rule{0.2em}{0ex}})$. Each relay k feeds forward ${\varphi}_{k}({\stackrel{\u0304}{\xi}}_{k},t\phantom{\rule{0.2em}{0ex}})$ to receiver k and selects a random beam former ${\mathbf{f}}_{\mathrm{X},k}^{\left(0\right)}$. In each iteration n≥1, the following updates are performed in the predetermined order: (i) each receiver k computes ${\theta}_{\text{BF},\text{ip}1,k}\left({\mathbf{F}}_{\mathrm{X}}^{(n1)}\right)$ and broadcasts it to all the relays, (ii) each receiver k computes ${\beta}_{\text{BF},\text{ip}1,k}\left({\mathbf{F}}_{\mathrm{X}}^{(n1)}\right)$ and ${\gamma}_{k}\left({\mathbf{F}}_{\mathrm{X}}^{(n1)}\right)$ and then sends them back to relay k, and (iii) a relay k _{0} is randomly selected to update its beamforming vector ${\mathbf{f}}_{\mathrm{X},{k}_{0}}^{\left(n\right)}={\nu}^{\text{max}}\left({\mathbf{Y}}_{\text{BF},{k}_{0}}\right({\mathbf{F}}_{\mathrm{X}}^{(n1)}\left)\right)$. Note that this implementation requires feedback and feedforward mechanisms between the relays and receivers. It also requires a little coordination among the relays to determine which relay is selected to update its beamforming vector in each iteration.
Complexity analysis
For simplicity in the complexity analysis of the algorithm, we assume that N _{ k } = N _{X} for $k\in \mathcal{K}$. Note that the initialization of the algorithm requires the estimation of the received SINR on the first hop ${\stackrel{\u0304}{\xi}}_{k}$ and then the computation of ${\varphi}_{k}({\stackrel{\u0304}{\xi}}_{k},t\phantom{\rule{0.2em}{0ex}})$ based on (15). Moreover, this step is performed only once at the beginning. Thus, in the complexity analysis, we ignore the initialization and focus on the iterations. Recall that in each iteration, only one relay is allowed to update its beamformer. We now provide a rough complexity analysis of the individual steps in each iteration. First, in the interference price update step, we need to compute ${I}_{k}\left({\mathbf{F}}_{\mathrm{X}}\right)={\sum}_{q\ne k}{\mathbf{f}}_{\mathrm{X},q}^{\ast}{\mathbf{h}}_{k,q}{}^{2}$ and ${A}_{k}\left({\mathbf{F}}_{\mathrm{X}}\right)=\parallel {\mathbf{f}}_{\mathrm{X},k}^{\ast}{\mathbf{h}}_{k,k}{\parallel}^{2}$ for each $k\in \mathcal{K}$, which allow us to compute γ _{ k }(F _{X}) according to (7) and then θ _{BF,ip1,k }(F _{X}) according to (25). Thus, this step yields a complexity of $\mathcal{O}\left({K}^{2}{N}_{\mathrm{X}}\right)$. Second, in the desired signal reward update step, we need to compute β _{BF,ip1,k }(F _{X}) from θ _{BF,ip1,k }(F _{X}) and γ _{ k }(F _{X}) according to (26). This step can be ignored from the complexity analysis. Third, in the beamforming vector update step, we need to compute ${\mathbf{Y}}_{\text{BF},k}\left({\mathbf{F}}_{\mathrm{X}}^{(n1)}\right)$ according to (30) only for the selected relay k. This computation has the complexity of $\mathcal{O}\left(K{N}_{\mathrm{X}}^{2}\right)$. We also need to perform an eigenvalue decomposition of ${\mathbf{Y}}_{\text{BF},k}({\mathbf{F}}_{\mathrm{X}}^{(n1)}\in {\mathbb{C}}^{{N}_{\mathrm{X}}\times {N}_{\mathrm{X}}}$, which yields a complexity of $\mathcal{O}\left({N}_{\mathrm{X}}^{3}\right)$. Thus, this step has a complexity of $\mathcal{O}\left(K{N}_{\mathrm{X}}^{2}\right)+\mathcal{O}\left({N}_{\mathrm{X}}^{3}\right)$. In short, the periteration complexity of the algorithm is $\mathcal{O}\left({K}^{2}{N}_{\mathrm{X}}\right)+\mathcal{O}\left(K{N}_{\mathrm{X}}^{2}\right)+\mathcal{O}\left({N}_{\mathrm{X}}^{3}\right)$.
Numerical results
For comparison, we implement several baseline relay beamforming algorithms. While the ZFlike beamforming minimizes the sum of interference power each relay cause to the other receivers of interest, the MRT beamforming maximizes the desired signal power each relay sends to its associated receiver. We also implement the beamforming algorithm resulting from the direct application of singlehop interference pricing beamforming in[16] to the secondhop, which we refer to as the secondhop interference pricing beamforming.
Conclusions
We proposed an algorithm for designing the transmission parameters at the transmitters and relays in the DF relay interference channel to maximize endtoend sumrates. The algorithm copes with both interference and mismatch between the rates on two hops, two main challenges in designing the DF relay systems. Our key contribution is the twohop interference pricing framework that allows for the integration of information about the firsthop performance and timesharing value in the computation of interference prices on the second hop for design the relays. The proposed algorithm allows for distributed implementation, making it suitable for practical systems. Simulations showed that the proposed algorithm obtains higher endtoend sumrates than the existing strategies, including the naïve approach of applying independently the singlehop interference pricing algorithms on two hops.
Declarations
Acknowledgements
This study was supported by a grant from Huawei Technologies, Inc.
Authors’ Affiliations
References
 Simeone O, Somekh O, BarNess Y, Poor HV, Shamai S: Capacity of linear twohop mesh networks with rate splitting, decodeandforward relaying and cooperation. In Proceedings of the Allerton Conference on Communication, Control, Computing. Monticello, IL; 2007.Google Scholar
 Thejaswi PSC, Bennatan A, Zhang J, Calderbank R, Cochran D: Rateachievability strategies for twohop interference flows. In Proceedings of the Allerton Conference on Communication Control Computing. Montecillo, IL; 2008:14321439.Google Scholar
 Cao Y, Chen B: Capacity bounds for twohop interference networks. In Proceedings of the Allerton Conference on Communication, Control, Computing. Montecillo, IL; 2009:272279.Google Scholar
 Shi Y, Wang J, Letaief K, Mallik R: A gametheoretic approach for distributed power control in interference relay channels. IEEE Trans. Wirel. Commun 2009, 8(6):31513161.View ArticleGoogle Scholar
 Ahmed AMA, Marsland I: Cochannel interference cancellation in multihop relay networks. In Proceedings of the IEEE International Conference on Communication. Beijing, China; 2008:6267.Google Scholar
 Peters SW, Panah AY, Truong KT, Heath RWJr: Relay architectures for 3GPP LTEAdvanced. EURASIP J. Wirel. Commun. Netw 2009, 2009(618787):114.Google Scholar
 Ning H, Ling C, Leung KK: Relayaided interference alignment: feasibility conditions and algorithm. In Proceedings of the IEEE International Symposium on Information Theory. Austin, TX; 2010:390394.Google Scholar
 Nourani B, Motahari SA, Khandani AK: Relayaided interference alignment for the quasistatic interference channel. In Proceedings of the IEEE International Symposium on Information Theory. Austin, TX; 2010:405409.Google Scholar
 Gou T, Jafar SA, Wang C, Jeon SW, Chung SY: Aligned interference neutralization and degrees of freedom of the 2x2x2 interference channel. IEEE Trans. Inf. Theory 2012, 58(7):43814395.MathSciNetView ArticleGoogle Scholar
 Yilmaz E, Boccardi F, Alexiou A: Distributed and centralized architectures for relayaided cellular systems. In Proceedings of the IEEE Vehicular Technology Conference. Barcelona, Spain; 2009:15.Google Scholar
 IEEE 802.16j: Standard for local and metropolitan area networks—Part 16: air interface for broadband wireless access systems—amendment 1: Multiple relay specificationGoogle Scholar
 IEEE 802.16m Broadband Wireless Access Working Group: IEEE 802.16m system requirementGoogle Scholar
 3GPP TR 36.814: Further enhancements for EUTRA physical layer aspectsGoogle Scholar
 Luo ZQ, Zhang S: Dynamic spectrum management: complexity and duality. IEEE J. Sel. Topics Signal Process 2008, 2: 5773.View ArticleGoogle Scholar
 Huang J, Berry RA, Honig ML: Distributed interference compensation for wireless networks. IEEE J. Sel. Areas Commun 2006, 24(5):10741084.View ArticleGoogle Scholar
 Shi C, Berry RA, Honig ML: Distributed interference pricing with MISO channels. In Proceedings of the Allerton Conference on Communication, Control, Computing. Monticello, IL; 2008:539546.Google Scholar
 Shi C, Berry RA, Honig ML: Local interference pricing for distributed beamforming in MIMO networks. In Proceedings of the IEEE Military Communications Conference. Boston, MA; 2009:16.Google Scholar
 RashidFarrokhi F, Liu KJR, Tassiulas L: Transmit beamforming and power control for cellular wireless systems. IEEE J. Sel. Areas Commun 1998, 16(8):14371450. 10.1109/49.730452View ArticleGoogle Scholar
 Scutari G, Palomar D, Barbarossa S: Competitive design of multiuser MIMO systems based on game theory: a unified view. IEEE J. Sel. Areas Commun 2008, 26(7):10891103.MathSciNetView ArticleGoogle Scholar
 Larsson E, Jorswieck E: Competition versus cooperation on the MISO interference channel. IEEE J. Sel. Areas Commun 2008, 26(7):10591069.View ArticleGoogle Scholar
 Peters SW, Heath RWJr: Cooperative algorithms for MIMO interference channels. IEEE Trans. Veh. Technol 2011, 60: 206218.View ArticleGoogle Scholar
 Schmidt D, Shi C, Berry RA, Honig ML, Utschick W: Distributed resource allocation schemes. IEEE Signal Process. Mag 2009, 26(5):5363.View ArticleGoogle Scholar
 Zakhour R, Ho ZKM, Gesbert D: Distributed beamforming coordination in multicell MIMO channels. In Proceedings of the IEEE Vehicular Technology Conference. Barcelona, Spain; 2009:15.Google Scholar
 Chen H, Gershman AB, Shahbazpanahi S: Distributed peertopeer beamforming for multiuser relay networks. In Proceedings of the IEEE International Conference on Acoustics, Speech Signal Processing. Taiwan: Taipei; 2009:22652268.Google Scholar
 Nguyen DHN, Nguyen HH, Pham TT: Distributed beamforming in multiuser multirelay networks with guaranteed QoS. In Proceedings of the IEEE Global Telecommunication Conference. Honolulu, HI; 2009:16.Google Scholar
 Joung J, Sayed AH: Multiuser twoway amplifyandforward relay processing and power control methods for beamforming systems. IEEE Trans. Signal Process 2010, 58(3):18331846.MathSciNetView ArticleGoogle Scholar
 Jeon SW, Chung SY, Jafar SA: Degrees of freedom of multisource relay networks. In Proceedings of the Allerton Conference on Communication, Control, Computing. Monticello, IL; 2009:388393.Google Scholar
 Truong KT, Heath RWJr: Interference alignment for multipleantenna amplifyandforward relay interference channel. In Proceedings of the IEEE Asilomar Conference on Signals, Systems Computers. Pacific Grove, CA, USA; 2011:15.Google Scholar
 Truong KT, Sartori P, Heath RWJr: Cooperative algorithms for MIMO amplifyandforward relay networks. IEEE Trans. Signal Process 2013, 61(5):12721287. (to appear) [http://arxiv.org/abs/1112.4553] [] (to appear)MathSciNetView ArticleGoogle Scholar
 Sahin O, Bala E, Yang R, Pietraski P: Relaying with distributed interference alignment in multinode networks. In Proceedings of the IEEE Personal Indoor Mobile Radio Communication. Istanbul, Turkey; 2010:543547.View ArticleGoogle Scholar
 HavaryNassab V, Shahbazpanahi S, Grami A: Optimal distributed beamforming for twoway relay networks. IEEE Trans. Signal Process 2010, 58(3):12381250.MathSciNetView ArticleGoogle Scholar
 Patel CS, Stuber GL: Channel estimation for amplify and forward relay based cooperation diversity systems. IEEE Trans. Wirel. Commun 2007, 6(6):23482356.View ArticleGoogle Scholar
 Gao F, Cui T, Nallanathan A: On channel estimation and optimal training design for amplify and forward relay networks. IEEE Trans. Wirel. Commun 2008, 7(5):19071916.View ArticleGoogle Scholar
 Chiang M, Hande P, Lian T, Tan CW: Power Control in Wireless Cellular Communications. Foundations and Trends in Communications and Information Theory. Norwell, MA: Now Publishers Inc; 2008.Google Scholar
 Shi C, Berry RA, Honig ML: Monotonic convergence of distributed interference pricing in wireless networks. In Proceedings of the IEEE International Symposium on Information Theory. Seoul, Korea; 2009:16191623.Google Scholar
 Foschini GJ, Miljanic Z: A simple distributed autonomous power control algorithm and its convergence. IEEE Trans. Veh. Technol 1993, 42(4):641646. 10.1109/25.260747View ArticleGoogle Scholar
 Yates RD: A framework for uplink power control in cellular radio systems. IEEE J. Sel. Areas. Commun 1995, 13(7):13411347. 10.1109/49.414651MathSciNetView ArticleGoogle Scholar
 Saraydar CU, Mandayam NB, Goodman DJ: Efficient power control via pricing in wireless data networks. IEEE Trans. Commun 2002, 50(2):291303. 10.1109/26.983324View ArticleGoogle Scholar
 Sung CW, Wong WS: A noncooperative power control game for multirate CDMA data networks. IEEE Trans. Wirel. Commun 2003, 2: 186194. 10.1109/TWC.2002.806394View ArticleGoogle Scholar
 Stanczak S, Wiczanowski M, Boche H: Distributed utilitybased power control: objectives and algorithms. IEEE Trans. Signal Process 2007, 55(10):50585068.MathSciNetView ArticleMATHGoogle Scholar
 Su Y, van Der Schaar M: A new perspective on multiuser power control games in interference channels. IEEE Trans. Wirel. Commun 2009, 8(6):29102919.View ArticleGoogle Scholar
 Qian LP, Zhang YJ, Huang J: MAPEL: achieving global optimality for a nonconvex wireless power control problem. IEEE Trans. Wirel. Commun 2009, 8(3):15531563.View ArticleGoogle Scholar
 Stanczak S, Wiczanowski M, Boche H: Fundamentals of Resource Allocation in Wireless Networks. New York: Springer; 2009.MATHGoogle Scholar
 Chandrasekhar V, Andrews JG, Muharemovic T, Shen Z, Gatherer A: Power control in twotier femtocell networks. IEEE Trans. Wirel. Commun 2009, 8(8):43164328.View ArticleGoogle Scholar
 Jo HC, Mun C, Moon J, Yook JG: Interference mitigation using uplink power control for twotier femtocell networks. IEEE Trans. Wirel. Commun 2009, 8(10):49064910.View ArticleGoogle Scholar
 Zhou S, Xiao H, Niu Z: Distributed power control for interferencelimited cooperative relay networks. In Proceedings of the IEEE International Conference on Communication. Dresden, Germany; 2009:16.Google Scholar
 Shi Y, Mallik RK, Letaief KB: Power control for relayassisted wireless systems with general relaying. In Proceedings of the IEEE International Conference on Communication. Cape Town, South Africa; 2010:15.Google Scholar
 Xiao L, Cuthbert L: Multicell noncooperative power allocation game in relay based OFDMA systems. In Proceedings of the IEEE Vehicular Technical Conference. Barcelona, Spain; 2009:15.Google Scholar
 Truong K, Heath RWJr: Relay beamforming using interference pricing for the twohop interference channel. In Proceedings of the IEEE Global Telecommunication Conference. Houston, TX; 2011:15.Google Scholar
 Truong KT, Heath RWJr: Joint transmit precoding for the relay interference broadcast channel. IEEE Trans. Veh. Technol 2013, 62(3):12011215. available at (to appear) [http://arxiv.org/abs/1206.4275] available at [] (to appear)View ArticleGoogle Scholar
 Chakrabarti A, Sabharwal A, Aazhang B: Sensitivity of achievable rates for halfduplex relay channel. In Proceedings of the IEEE Signal Processing Advanced Wireless Communication. New York City, NY; 2005:970974.Google Scholar
 Liu YF, Dai YH, Luo ZQ: On the complexity of optimal coordinated downlink beamforming. In Proceedings of the IEEE International Conference on Acoustics, Speech Signal Processing. Dallas, TX; 2010.Google Scholar
 Jorswieck EA, Larsson EG, Danev D: Complete characterization of the Pareto boundary for the MISO interference channel. IEEE Trans. Signal Process 2008, 56(10):52925296.MathSciNetView ArticleGoogle Scholar
 Ko K, Lin CL: Minimax and Applications chap. On the Complexity of MinMax Optimization Problems and Their Approximation. Kluwer; 1995.Google Scholar
 Lo TKY: Maximum ratio transmission. IEEE Trans. Commun 1999, 47(10):14581461. 10.1109/26.795811View ArticleGoogle Scholar
 Chung ST, Kim SJ, Lee J, Cioffi JM: A gametheoretic approach to power allocation in frequencyselective Gaussian interference channels. In Proceedings of the IEEE International Symposium on Information Theory. Yokohama, Japan; 2003:316316.Google Scholar
 Gomadam K, Cadambe VR, Jafar SA: Approaching the capacity of wireless networks through distributed interference alignment. In Proceedings of the IEEE Global Telecommunication Conference. LA: New Orleans; 2008:16.Google Scholar
 Peters SW, Heath RWJr: Interference alignment via alternating minimization. In Proceedings of the IEEE International Conference on Acoustics, Speech Signal Processing. Taipei, Taiwan; 2009:24452448.Google Scholar
 Sadek M, Tarighat A, Sayed AH: A leakagebased precoding scheme for downlink multiuser MIMO channels. IEEE Trans. Wirel. Commun 2007, 6(5):17111721.View ArticleGoogle Scholar
 Kim SJ, Giannakis GB: Optimal resource allocation for MIMO ad hoc cognitive radio networks. IEEE Trans. Inf. Theory 2011, 57(5):31173131.MathSciNetView ArticleGoogle Scholar
 Shi Q, Razaviyayn M, Luo ZQ, He C: An iteratively weighted MMSE approach to distributed sumutility maximization for a MIMO interfering broadcast channel. IEEE Trans. Signal Process 2011, 59(9):43314340.MathSciNetView ArticleGoogle Scholar
 Bertsekas DP: Nonlinear Programming. Belmont, MA: Athena Scientific; 1999.MATHGoogle Scholar
 MasColell A, Whinston MD, Green JR: Microeconomic Theory. Oxford, UK: Oxford University Press; 1995.MATHGoogle Scholar
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