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56
A Tutorial on Decomposition Methods for Network Utility Maximization
 IEEE J. SEL. AREAS COMMUN
, 2006
"... A systematic understanding of the decomposability structures in network utility maximization is key to both resource allocation and functionality allocation. It helps us obtain the most appropriate distributed algorithm for a given network resource allocation problem, and quantifies the comparison ..."
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Cited by 185 (4 self)
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A systematic understanding of the decomposability structures in network utility maximization is key to both resource allocation and functionality allocation. It helps us obtain the most appropriate distributed algorithm for a given network resource allocation problem, and quantifies the comparison across architectural alternatives of modularized network design. Decomposition theory naturally provides the mathematical language to build an analytic foundation for the design of modularized and distributed control of networks. In this tutorial paper, we first review the basics of convexity, Lagrange duality, distributed subgradient method, Jacobi and Gauss–Seidel iterations, and implication of different time scales of variable updates. Then, we introduce primal, dual, indirect, partial, and hierarchical decompositions, focusing on network utility maximization problem formulations and the meanings of primal and dual decompositions in terms of network architectures. Finally, we present recent examples on: systematic search for alternative decompositions; decoupling techniques for coupled objective functions; and decoupling techniques for coupled constraint sets that are not readily decomposable.
Theory and applications of Robust Optimization
, 2007
"... In this paper we survey the primary research, both theoretical and applied, in the field of Robust Optimization (RO). Our focus will be on the computational attractiveness of RO approaches, as well as the modeling power and broad applicability of the methodology. In addition to surveying the most pr ..."
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Cited by 110 (16 self)
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In this paper we survey the primary research, both theoretical and applied, in the field of Robust Optimization (RO). Our focus will be on the computational attractiveness of RO approaches, as well as the modeling power and broad applicability of the methodology. In addition to surveying the most prominent theoretical results of RO over the past decade, we will also present some recent results linking RO to adaptable models for multistage decisionmaking problems. Finally, we will highlight successful applications of RO across a wide spectrum of domains, including, but not limited to, finance, statistics, learning, and engineering.
Optimal Beamforming for TwoWay MultiAntenna Relay Channel with Analogue Network Coding
, 2009
"... This paper studies the wireless twoway relay channel (TWRC), where two source nodes, S1 and S2, exchange information through an assisting relay node, R. It is assumed that R receives the sum signal from S1 and S2 in one timeslot, and then amplifies and forwards the received signal to both S1 and S ..."
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Cited by 90 (6 self)
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This paper studies the wireless twoway relay channel (TWRC), where two source nodes, S1 and S2, exchange information through an assisting relay node, R. It is assumed that R receives the sum signal from S1 and S2 in one timeslot, and then amplifies and forwards the received signal to both S1 and S2 in the next timeslot. By applying the principle of analogue network coding (ANC), each of S1 and S2 cancels the socalled “selfinterference ” in the received signal from R and then decodes the desired message. Assuming that S1 and S2 are each equipped with a single antenna and R with multiantennas, this paper analyzes the capacity region of the ANCbased TWRC with linear processing (beamforming) at R. The capacity region contains all the achievable bidirectional ratepairs of S1 and S2 under the given transmit power constraints at S1, S2, and R. We present the optimal relay beamforming structure as well as an efficient algorithm to compute the optimal beamforming matrix based on convex optimization techniques. Lowcomplexity suboptimal relay beamforming schemes are also presented, and their achievable rates are compared against the capacity with the optimal scheme.
Dynamic resource allocation in cognitive radio networks
 IEEE Signal Process. Mag
, 2010
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MultiUser MISO interference channels with SingleUser detection: Optimality of beamforming and the achievable rate region
 IEEE TRANS. INFORM. TH
, 2009
"... For a multiuser interference channel with multiantenna transmitters and singleantenna receivers, by restricting each transmitter to Gaussian input and each receiver to a singleuser detector, computing the largest achievable rate region amounts to solving a family of nonconvex optimization probl ..."
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Cited by 29 (1 self)
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For a multiuser interference channel with multiantenna transmitters and singleantenna receivers, by restricting each transmitter to Gaussian input and each receiver to a singleuser detector, computing the largest achievable rate region amounts to solving a family of nonconvex optimization problems. Recognizing the intrinsic connection between the signal power at the intended receiver and the interference power at the unintended receiver, the original family of nonconvex optimization problems is converted into a new family of convex optimization problems. It is shown that, for such interference channels with each receiver implementing singleuser detection, transmitter beamforming can achieve all boundary points of the achievable rate region.
Fast algorithms and performance bounds for sum rate maximization in wireless networks
 in Proceedings of IEEE INFOCOM
, 2009
"... Abstract — Sum rate maximization by power control is an important, challenging, and extensively studied problem in wireless networks. It is a nonconvex optimization problem and achieves a rate region that is in general nonconvex. We derive approximation ratios to the sum rate objective by studying t ..."
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Cited by 28 (10 self)
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Abstract — Sum rate maximization by power control is an important, challenging, and extensively studied problem in wireless networks. It is a nonconvex optimization problem and achieves a rate region that is in general nonconvex. We derive approximation ratios to the sum rate objective by studying the solutions to two related problems, sum rate maximization using an SIR approximation and maxmin weighted SIR optimization. We also show that these two problems can be solved very efficiently, using much faster algorithms than the existing ones in the literature. Furthermore, using a new parameterization of the sum rate maximization problem, we obtain a characterization of the power controlled rate region and its convexity property in various asymptotic regimes. Engineering implications are discussed for IEEE 802.11 networks. Index Terms — Duality, Distributed algorithm, Power control, Weighted sum rate maximization, Nonnegative matrices and applications,
On the Proximity Factors of Lattice ReductionAided Decoding
"... Lattice reductionaided decoding enables significant complexity saving and nearoptimum performance in multiinput multioutput (MIMO) communications. However, its remarkable performance largely remains a mystery to date. In this paper, a first step is taken towards a quantitative understanding of ..."
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Cited by 22 (7 self)
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Lattice reductionaided decoding enables significant complexity saving and nearoptimum performance in multiinput multioutput (MIMO) communications. However, its remarkable performance largely remains a mystery to date. In this paper, a first step is taken towards a quantitative understanding of its performance limit. To this aim, the proximity factors are defined to measure the worstcase gap to maximumlikelihood (ML) decoding in terms of the signaltonoise ratio (SNR) for given error rate. The proximity factors are derived analytically and found to be bounded above by a function of the dimension of the lattice alone. As a direct consequence, it follows that lattice reductionaided decoding can always achieve full receive diversity of MIMO fading channels. The study is then extended to the dualbasis reduction. It is found that in some cases reducing the dual can result in smaller proximity factors than reducing the primal basis. The theoretic bounds on the proximity factors are further compared with numerical results.
ECOS: An SOCP solver for embedded systems
 in European Control Converence
, 2013
"... Abstract — In this paper, we describe the embedded conic solver (ECOS), an interiorpoint solver for secondorder cone programming (SOCP) designed specifically for embedded applications. ECOS is written in low footprint, singlethreaded, libraryfree ANSIC and so runs on most embedded platforms. Th ..."
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Cited by 11 (3 self)
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Abstract — In this paper, we describe the embedded conic solver (ECOS), an interiorpoint solver for secondorder cone programming (SOCP) designed specifically for embedded applications. ECOS is written in low footprint, singlethreaded, libraryfree ANSIC and so runs on most embedded platforms. The main interiorpoint algorithm is a standard primaldual Mehrotra predictorcorrector method with NesterovTodd scaling and selfdual embedding, with search directions found via a symmetric indefinite KKT system, chosen to allow stable factorization with a fixed pivoting order. The indefinite system is solved using Davis ’ SparseLDL package, which we modify by adding dynamic regularization and iterative refinement for stability and reliability, as is done in the CVXGEN code generation system, allowing us to avoid all numerical pivoting; the elimination ordering is found entirely symbolically. This keeps the solver simple, only 750 lines of code, with virtually no variation in run time. For small problems, ECOS is faster than most existing SOCP solvers; it is still competitive for mediumsized problems up to tens of thousands of variables. I.
Robust power allocation for energyefficient locationaware networks
 IEEE/ACM Trans. Netw
"... Abstract—In wireless locationaware networks, mobile nodes (agents) typically obtain their positions using the range measurements to the nodes with known positions. Transmit power allocation not only affects network lifetime and throughput, but also determines localization accuracy. In this paper, ..."
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Cited by 11 (4 self)
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Abstract—In wireless locationaware networks, mobile nodes (agents) typically obtain their positions using the range measurements to the nodes with known positions. Transmit power allocation not only affects network lifetime and throughput, but also determines localization accuracy. In this paper, we present an optimization framework for robust power allocation in network localization with imperfect knowledge of network parameters. In particular, we formulate power allocation problems to minimize localization errors for a given power budget and show that such formulations can be solved via conic programming. Moreover, we design a distributed power allocation algorithm that allows parallel computation among agents. The simulation results show that the proposed schemes significantly outperform uniform power allocation, and the robust schemes outperform their nonrobust counterparts when the network parameters are subject to uncertainty. Index Terms—Localization, resource allocation, robust optimization, secondorder conic programming (SOCP), semidefinite programming (SDP), wireless networks. I.
A Tractable Method for Robust Downlink Beamforming in Wireless Communications
"... Abstract—In downlink beamforming in a multipleinput multipleoutput (MIMO) wireless communication system, we design beamformers that minimize the power subject to guaranteeing given signaltointerference noise ratio (SINR) threshold levels for the users, assuming that the channel responses between ..."
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Cited by 10 (0 self)
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Abstract—In downlink beamforming in a multipleinput multipleoutput (MIMO) wireless communication system, we design beamformers that minimize the power subject to guaranteeing given signaltointerference noise ratio (SINR) threshold levels for the users, assuming that the channel responses between the base station and the users are known exactly. In robust downlink beamforming, we take into account uncertainties in the channel vectors, by designing beamformers that minimize the power subject to guaranteeing given SINR threshold levels over the given set of possible channel vectors. When the uncertainties in channel vectors are described by complex uncertainty ellipsoids, we show that the associated worstcase robust beamforming problem can be solved efficiently using an iterative method. The method uses an alternating sequence of optimization and worstcase analysis steps, where at each step we solve a convex optimization problem using efficient interiorpoint methods. Typically, the method provides a fairly robust beamformer design within 5–10 iterations. The robust downlink beamforming method is demonstrated with a numerical example. I.