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A Tutorial on Geometric Programming
"... A geometric program (GP) is a type of mathematical optimization problem characterized by objective and constraint functions that have a special form. Recently developed solution methods can solve even largescale GPs extremely efficiently and reliably; at the same time a number of practical problems ..."
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Cited by 123 (11 self)
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A geometric program (GP) is a type of mathematical optimization problem characterized by objective and constraint functions that have a special form. Recently developed solution methods can solve even largescale GPs extremely efficiently and reliably; at the same time a number of practical problems, particularly in circuit design, have been found to be equivalent to (or well approximated by) GPs. Putting these two together, we get effective solutions for the practical problems. The basic approach in GP modeling is to attempt to express a practical problem, such as an engineering analysis or design problem, in GP format. In the best case, this formulation is exact; when this isn’t possible, we settle for an approximate formulation. This tutorial paper collects together in one place the basic background material needed to do GP modeling. We start with the basic definitions and facts, and some methods used to transform problems into GP format. We show how to recognize functions and problems compatible with GP, and how to approximate functions or data in a form compatible with GP (when this is possible). We give some simple and representative examples, and also describe some common extensions of GP, along with methods for solving (or approximately solving) them.
Channel Access Algorithms with Active Link Protection for Wireless Communication Networks with Power Control
 IEEE/ACM Trans. on Networking
, 2000
"... A distributed powercontrol algorithm with active link protection (DPC/ALP) is studied in this paper. It maintains the quality of service of operational (active) links above given thresholds at all times (link quality protection). As network congestion builds up, established links sustain their qual ..."
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Cited by 120 (8 self)
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A distributed powercontrol algorithm with active link protection (DPC/ALP) is studied in this paper. It maintains the quality of service of operational (active) links above given thresholds at all times (link quality protection). As network congestion builds up, established links sustain their quality, while incoming ones may be blocked and rejected. A suite of admission control algorithms, based on the DPC/ALP one, is also studied. They are distributed/autonomous and operate using local interference measurements.
Auctionbased spectrum sharing
 WiOpt '04
, 2004
"... Abstract. We study auctionbased mechanisms for sharing spectrum among a group of users, subject to a constraint on the interference temperature at collocated receivers. The users access the channel using spread spectrum signaling and thus generate interference with each other. Each user receives a ..."
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Cited by 119 (18 self)
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Abstract. We study auctionbased mechanisms for sharing spectrum among a group of users, subject to a constraint on the interference temperature at collocated receivers. The users access the channel using spread spectrum signaling and thus generate interference with each other. Each user receives a utility that is a function of the received signaltointerference plus noise ratio. We propose two auction mechanisms for allocating the received power. The first is an SINRbased auction, which, when combined with logarithmic utilities, leads to a weighted maxmin fair SINR allocation. The second is a powerbased auction that maximizes the total utility when the bandwidth is large enough. Both auction mechanisms achieve social optimality in a large system limit where bandwidth, power and the number of users are increased in a fixed proportion. We also give sufficient conditions for global convergence of a distributed updating algorithm and discuss the convergence speed. 1
A utilitybased powercontrol scheme in wireless cellular systems
 IEEE/ACM TRANS. ON NETWORKING
, 2003
"... Distributed powercontrol algorithms for systems with hard signaltointerference ratio (SIR) constraints may diverge when infeasibility arises. In this paper, we present a powercontrol framework called utilitybased power control (UBPC) by reformulating the problem using a softened SIR requirement ..."
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Cited by 105 (3 self)
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Distributed powercontrol algorithms for systems with hard signaltointerference ratio (SIR) constraints may diverge when infeasibility arises. In this paper, we present a powercontrol framework called utilitybased power control (UBPC) by reformulating the problem using a softened SIR requirement (utility) and adding a penalty on power consumption (cost). Under this framework, the goal is to maximize the net utility, defined as utility minus cost. Although UBPC is still noncooperative and distributed in nature, some degree of cooperation emerges: a user will automatically decrease its target SIR (and may even turn off transmission) when it senses that traffic congestion is building up. This framework enables us to improve system convergence and to satisfy heterogeneous service requirements (such as delay and bit error rate) for integrated networks with both voice users and data users. Fairness, adaptiveness, and a high degree of flexibility can be achieved by properly tuning parameters in UBPC.
Optimal Sequences, Power Control, and User Capacity of Synchronous CDMA Systems with Linear MMSE Multiuser Receivers
 IEEE TRANS. INFORM. THEORY
, 1999
"... There has been intense effort in the past decade to develop multiuser receiver structures which mitigate interference between users in spreadspectrum systems. While much of this research is performed at the physical layer, the appropriate power control and choice of signature sequences in conjuncti ..."
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Cited by 102 (5 self)
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There has been intense effort in the past decade to develop multiuser receiver structures which mitigate interference between users in spreadspectrum systems. While much of this research is performed at the physical layer, the appropriate power control and choice of signature sequences in conjunction with multiuser receivers and the resulting network user capacity is not well understood. In this paper we will focus on a single cell and consider both the uplink and downlink scenarios and assume a synchronous CDMA (SCDMA) system. We characterize the user capacity of a single cell with the optimal linear receiver (MMSE receiver). The user capacity of the system is the maximum number of users per unit processing gain admissible in the system such that each user has its qualityofservice (QoS) requirement (expressed in terms of its desired signaltointerference ratio) met. Our characterization allows us to describe the user capacity through a simple effective bandwidth characterization: Users are allowed in the system if and only if the sum of their effective bandwidths is less than the processing gain of the system. The effective bandwidth of each user is a simple monotonic function of its QoS requirement. We identify the optimal signature sequences and power control strategies so that the users meet their QoS requirement. The optimality is in the sense of minimizing the sum of allocated powers. It turns out that with this optimal allocation of signature sequences and powers, the linear MMSE receiver is just the corresponding matched filter for each user. We also characterize the effect of transmit power constraints on the user capacity.
Radio Link Admission Algorithms for Wireless Networks with Power Control and Active Link Quality Protection
 IN PROC. IEEE INFOCOM
, 1994
"... In this paper we present a distributed power control scheme, which maintains the SIRs of operational (active) links above their required thresholds at all time (link quality protection), while new users are being admitted; furthermore, when new users cannot be successfully admitted, existing ones do ..."
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Cited by 96 (5 self)
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In this paper we present a distributed power control scheme, which maintains the SIRs of operational (active) links above their required thresholds at all time (link quality protection), while new users are being admitted; furthermore, when new users cannot be successfully admitted, existing ones do not suffer fluctuations of their SIRs below their required thresholds values. We also present two admission /rejection control algorithms, which exercise voluntary dropout of links inadmissible to the network, so as to reduce interference and possibly facilitate the admission of other links.
Optimal linear precoding strategies for wideband noncooperative systems based on game theory – Part II: Algorithms
 IEEE Trans. Signal Process
, 2008
"... In this twoparts paper we propose a decentralized strategy, based on a gametheoretic formulation, to find out the optimal precoding/multiplexing matrices for a multipointtomultipoint communication system composed of a set of wideband links sharing the same physical resources, i.e., time and band ..."
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Cited by 86 (10 self)
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In this twoparts paper we propose a decentralized strategy, based on a gametheoretic formulation, to find out the optimal precoding/multiplexing matrices for a multipointtomultipoint communication system composed of a set of wideband links sharing the same physical resources, i.e., time and bandwidth. We assume, as optimality criterion, the achievement of a Nash equilibrium and consider two alternative optimization problems: 1) the competitive maximization of mutual information on each link, given constraints on the transmit power and on the spectral mask imposed by the radio spectrum regulatory bodies; and 2) the competitive maximization of the transmission rate, using finite order constellations, under the same constraints as above, plus a constraint on the average error probability. In Part I of the paper, we start by showing that the solution set of both noncooperative games is always nonempty and contains only pure strategies. Then, we prove that the optimal precoding/multiplexing scheme for both games leads to a channel diagonalizing structure, so that both matrixvalued problems can be recast in a simpler unified vector power control game, with no performance penalty. Thus, we study this simpler game and derive sufficient conditions ensuring the uniqueness of the Nash equilibrium. Interestingly, although derived under stronger constraints,
Power Control and Capacity of Spread Spectrum Wireless Networks
 Automatica
, 1999
"... Transmit power control is a central technique for resource allocation and interference management in spreadspectrum wireless networks. With the increasing popularity of spreadspectrum as a multiple access technique, there has been significant research in the area in recent years. While power contr ..."
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Cited by 77 (5 self)
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Transmit power control is a central technique for resource allocation and interference management in spreadspectrum wireless networks. With the increasing popularity of spreadspectrum as a multiple access technique, there has been significant research in the area in recent years. While power control has been considered traditionally as a means to counteract the harmful effect of channel fading, the more general emerging view is that it is a flexible mechanism to provide QualityofService to individual users. In this paper, we will review the main threads of ideas and results in the recent development of this area, with a bias towards issues that have been the focus of our own research. For different receivers of varying complexity, we study both questions about optimal power control as well as the problem of characterizing the resulting network capacity. Although spreadspectrum communications has been traditionally viewed as a physicallayer subject, we argue that by suitable abstr...
Power Control in TwoTier Femtocell Networks
, 2008
"... Two tier cellular networks, comprising of a central macrocell underlaid with short range femtocell hotspots offer an economical way to improve cellular capacity. With shared spectrum and lack of coordination between tiers, crosstier interference limits overall capacity. To quantify nearfar effects ..."
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Cited by 74 (6 self)
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Two tier cellular networks, comprising of a central macrocell underlaid with short range femtocell hotspots offer an economical way to improve cellular capacity. With shared spectrum and lack of coordination between tiers, crosstier interference limits overall capacity. To quantify nearfar effects with universal frequency reuse, this paper derives a fundamental relation providing the largest feasible macrocell SignaltoInterferencePlusNoise Ratio (SINR), given any set of feasible femtocell SINRs. A distributed utilitybased SINR adaptation at femtocells is proposed in order to alleviate crosstier interference at the macrocell from overlaid femtocell infrastructure. The FoschiniMiljanic (FM) algorithm is a special case of the adaptation. Each femtocell maximizes its individual utility consisting of a SINR based reward less an incurred cost (interference to the macrocell). Numerical results show greater than 30 % improvement in mean femtocell SINRs relative to FM. In the event that crosstier interference prevents a macrouser from obtaining its SINR target, an algorithm is proposed that adaptively curtails transmission powers of the strongest femtocell interferers. The algorithm ensures that a macrouser achieves its SINR target even with 100 femtocells/cellsite, and requires a worst case SINR reduction of only 16 % at femtocells. These results motivate design of power control schemes requiring minimal network overhead in twotier networks with shared spectrum.