MAPEL: Achieving Global Optimality for a Non-convex Wireless Power Control Problem

MAPEL: Achieving Global Optimality for a Non-convex Wireless Power Control Problem Liping Qian, Student Member, IEEE, Ying Jun (Angela) Zhang, Member, IEEE, and Jianwei Huang, Member, IEEE Department of Information Engineering, The Chinese University of Hong Kong Shatin, New Territories, Hong Kong {lpqian6, yjzhang, jwhuang}@ie.cuhk.edu.hk Abstract Achieving weighted t hroughput maxim ization (WTM ) throug h power c ontrol has been a l ong standin g open problem in interference-limited wireless network s. The complicated coup lin g between the mutual interferences of links gives rise to a non-convex optimization pro blem. Previous work h as consider ed the WT M problem in the high signal to interference -and-noise rat io (SINR ) regime, w here the probl em can be approxim ated and transformed into a c onvex optimization prob lem through proper chan ge of variables. In the genera l SINR regime, however, the approx imation and transform ation approac h does not work . This pa per prop oses an al gorithm , MAPEL, whic h globally converges t o a global optim al soluti on of the WTM proble m in the gene ral SINR regime. T he MAPEL alg orithm is designed base d on three key obse rvations of t he WTM pro blem: (1) t he objecti ve function i s monotoni cally increasi ng in SINR , (2) the objective funct ion can be t ransform ed into a pro duct of exp onentiated l inear fractio n functi ons, and (3) the feasible set of the equivale nt transform ed problem is alway s “normal” al though not necessarily conve x. The MA PLE algorithm finds the desired opt imal powe r control sol ution by co nstructing a se ries of poly blocks that approxim ate the feasible SINR region in i ncreasing precision. Furtherm ore, by tu ning the a pproximati on factor i n MAPEL, we could e ngineer a desirable tradeoff b etween optimality and convergence time. MAPEL provides an important benchmark for performance evaluation of other heu ristic algori thms target ing the sam e problem . With the help of MAPEL, we e valuate the performance of several respective algorithms through extensiv e simulations. Index Terms Wireless Ad Hoc Networks, Power Control, Global Optimization. I. I NTRODUCTION Due to the broadcast natu re of wireless comm unications, simu ltaneous transmissions in the same channel interfere with each other and limit the wireless network performance. One im portant interference mitigation technique is transm it-power control at the phys ical layer. This technique has been well studied and implemented in the context of wireles s cellular communications (see a rec ent survey in [1]). The research in this area can be di vided into two main threads. The first thread is concerned with achieving fixed sign al to interference-plus-noise ratio (SINR) targets with m inimum transmission power (e.g., [2]-[9]). This form ulation is m otivated by traditional voice communications, where an SINR highe r than the threshold is not useful in term s of user perceived Quality of Service (QoS). The second thread is concerned with join t SINR allocation and power control. This formulation is motivated by data communication applications, where higher SINR means higher data rate and better QoS. Such jo int optim ization becomes mo re important as data applications will be dom inant in next gene ration wireless networks (e.g., 4G all IP-based communication systems). The joint SINR allocation and power control problem is more difficult to solve than the fixed SINR target case. This is because we need to op timize over th e entire feasible SINR region, which is typically non-convex due to co mplicated interference coupli ng between links. One important instance of this joint optimization is weighted throughput maxim ization (WTM), where the objective function to be maximized is . Here is user i’s weight and is user i ’s achievable date rate (bps/Hz). Re searchers have spent significant am ount of efforts on studying this WTM problem in the past. For example, the authors in [11] considered the high SINR regime where the SINR of each link is mu ch larg er than 0dB, in which case the individual data rate can be approximated by . Under such approximation, the WTM problem can be transformed into a convex one in the form of geometric programming (GP) by proper change of variables, and thus can be solved efficiently in a centralized fashion. A different approach was considered in [8], where the authors showed that the feasible SINR region is convex in the logarithm of SINR. This also explains why the approximati on and convexification in [11] is suitable under the high SINR regime. Unfortunately, the high-SINR assumption is not valid in general for practical wireless ad-hoc networks when nearby links heavily interfere with each other. As a result, standard GP often yields a solution that is far from op timum due to possible str ong interferences between links nearby. Compared with GP, the work [ 12] does not require hi gh-SINR assumption. In particular, the authors in [12] first transforme d the WTM problem into an equivalent signomial programming (SP), which is provably NP hard to solve. Then the authors adopted a success ive 2 log ( 1 SIN R ) ii i w + ∑ i w 2 log ( 1 SINR ) i + 2 log (SINR ) i convex programming method, SP Condensation (SPC ) algorithm to solve SP. Similar to m any algorithms used to solve non-convex optimization, the SPC algorithm only guarantees local optimal solutions. An improper initialization m ay consider ably degrade the system throughput. To date, achieving a global optimal solution o f the WTM problem still is an open problem. In this paper, we propose a MAPEL (MLFP-bAsed PowEr aLlocation) algorithm, which is the first algorithm in the literatur e that can achieve the global optimal solution of the WTM problem in the general SINR regime. There are three key observati ons th at enable MAPEL to eff iciently solve the non-convex optimization problem. First, the objec tive function of W TM is monotonically increasing in SINR. This means the optimal solution is achie ved at the boundary of th e feasible SINR region. Second, the objective function of WTM can be transformed into a product of exponentiated linear fractional functions, wh ich can be further form ulated into a multiplicative linear fractional programming (MLFP) problem with nice computational f eatures. Last, the feasible set of the equivalent transformed problem, a lthough m ay be not convex, is always normal 1 . This, together with monotonicity, allows us to construct a sequence of polyblocks to approximate that SINR region boundary with increasing level of accuracy. Given an arbitrary sm all and finite error tolerance level, MAPEL is guaranteed to find one global optimal so lution of the W MT problem within finite amount of time. A flexible tradeoff between performance and convergence time can be achieved by tuning the approximation factor. The main benefit of MAPEL is to provide a be nchmark for all algorithm s that are designed to tackle the WMT problem, whether it is existing or to be proposed, cen tralized or distributed, optimal or heuristic. In this paper, we show how such benchmark is useful in elevating the performance of two state-of-art centralized and distributed algorithms ([12], [16]) in this area. Finally, we note that some wo rk has been done on the problem of maximizing the m inimum achievable SINR of each link in wireless networks [4], [10]. This is motivated partially by fair allocation among various users in the network. All existing algorithms for solving this problem are centralized. Interestingly, our MAPEL algorithm can be easily adapted to solve the same max-m in optimization problem in a different and also centralized manner. We will b riefly discuss this extension as well. The remainder of this paper is organized as follo ws. System m odel is disc ussed in Section II. In Section III, we transform the throughput-maximization power cont rol problem into a MLFP problem. Some properties of the feasible region in MLFP problem are also discussed. The MAPEL algorithm is proposed and analyzed in Section IV. A brief discussion on the extensi on to the max-min SINR 1 Various math preliminar ies and de finitions are g iven in Section V. problem is also provided. In Section V, we ev aluate the perform ance of MAPEL through several simulations. With the benchm ark established by MAPEL, we evaluate the performance of two existing algorithms in Section VI. Th e paper is concluded in Section VII. Throughout the paper , vectors are de noted in bold small letter , e.g . , , with its th component . Matrices are denoted by bold capitalized letters, e.g . , , with z i i z Z ij Z denoting the {, th component. Sets are denoted by Euler le tters, e.g . , . } ij A II. S YSTEM M ODEL W e consider a wireless ad hoc network with a set of {1 } M = ,,  M distinct links 2 . Each link consists of a transmitter node and a receiver node i T i R . The channel gain between node and node i T j R is denoted by , which is determined by various factors such as path loss, shadowing and fading effects. The com plete channel matrix is denoted by ij G [ ij G ] = G . Let i p denote the transmitting power of link (i.e., from node ), and denote the receiving noise on link (i.e., measured at node i i T i n i i R ). The received signal to in terferen ce-plus-noise ratio (SINR) of link is i () ii i i ji j i ji Gp Gp n γ ≠ = , + ∑ p (1) and the data rate calculated based on the Shannon capacity formula is 2 log ( 1 ( )) i γ + p 3 . T o simplify notations, we use and ( ) i pi =, ∀ ∈ M p ( ) ( ( ) ) i i γ = ,∀ ∈ γ pp M to represent the transm ission power vector and achieved SI NR vector of all links. W e want to find the optimal power allocation ∗ p that maximizes the weighted sum throughput subject to individual data rate constraints. Mathem atically we want to solve the follo wing optimization problem 2 1 2m i n ma ximiz e log ( 1 ( )) subject to log ( 1 ( )) 0 M ii i ii max ii w ri pP i γ γ = , + + ≥, ∀ ∈ ≤≤ , ∀ ∈ . ∑ p p p M M , (P1) Here is the minim um data ra te requirement of link i (including the special case of min 0 i r , ≥ 2 For example,this could represent a network snaps hot under a part icular schedule of transmissions determined by an underlying routing and MAC protocol. 3 T o better model the achiev able rates in a practical s ystem, we can re-norm alize i γ by i β γ , where [0 1 ] β ∈, represents the system’ s “gap” from capacity . Such modification, however , does not change the analysis in this pap er . min 0 i r , = , i.e., no rate constraint), and is the priority weight of link i . Without loss of generality, the weights are normalized so that 0 i w > 1 1 M i i w = = ∑ i w . Notice that if ’s are too large, there may not exist a fea sible solution. min i r , ,min ,m i n 2 i r i γ F o r a u s e r i , its received SINR valu e needs to be at least 1 = − in order to satisfy its minimum rate requ irement. Consider the fo llowing matrix B , ,min 0, . , ij i ii G ij G ij B ij γ = ⎧ ⎪ = ⎨ ≠ ⎪ ⎩ According to Theorem 2.1 in [1], if the maximum eigenvalue of is larger than 1, then there is n o feasible solution to Problem P1. Othe rwise, we can find a power allocation as follows, B ˆ p 1 ˆ () − =− pI B u , where is the I M M × identity matrix and is a u 1 M × vector with elements ,min ii ii n i G u γ = . By Theorem 2.2 in [1], Problem P1 is feasible if and only if the components of satisfies for all i ,. Therefore, the procedure of checking the feasibility of Problem P1 is as follows: ˆ p ˆ 0 max ii pP ≤≤ Procedure 1 Check the feasibility of ’s min i r , 1: T ransform minimum data constraints into minimum SINR constraints th rough for all i . ,min ,m in () 2 1 i r i γ =− p 2: Compute the maximum eigenvalue of m atrix and check if it is smaller than 1 . If not, ’ s are infeasible. Otherwise, go to step 3. B min i r , 3: Compute the power allocation and check if it satisfies for all i . If so, ’ s are feasible. Otherwise, ’ s are infeasible. 1 ˆ () − =− pI B u ˆ 0 max ii pP ≤≤ min i r , min i r , It has been shown that (e.g., [1 1], [12], [16], [17]) Problem P1 is a non-convex optimization problem in terms of the transm it power p . Thus, it is dif ficult to find a global optimal solution ef ficiently even in a centra lized fashion. In Sect ion III, we w ill show Problem P1 can be transfor med into a Multiplicative Linear Fractional Program ming (MLFP) problem, which can then be solved ef ficiently by the MAPEL algor ithm presented in Section IV . III. P OWE R C ONTROL AS M ULTIPLICATIVE L INEAR F RACTIONAL P ROGRAMMING (MLFP) In this section, we first introduce the definitio n of Generalized Linear Fraction al Programming, and show that Problem P1 can be form ulated as a special c ase of the GLFP (which we refer to as MLFP). W e further discuss several key properties of the new formulation that are critical f or developing the MAPEL algorithm. Definition 1 GLFP: [20] An optimization problem belongs to the class of Generalized Linear Fractional Programming (GLFP) if it can be repr esented by one of the following two form ulations: 1 1 () () max imi ze () () var i able s M M ff gg ⎛⎞ Φ, , ⎜⎟ ⎝⎠ ∈  D xx xx x (2) or 1 1 () () min i mi ze () () va riable s , M M ff gg ⎛⎞ Φ, , ⎜ ⎝ ∈  D ⎟ ⎠ x x x x x (3) where the domain is a nonempty polytope D 45 in N R , functions are linear af fine on 11 N MM ff g g ,, , ,, : →  RR N R , and function is incr easing on M Φ: → RR M + R . By the properties of the l ogarithm function, we can rewrite P roblem P1 as follows: 1 () maxi mize () variabl es , i w M i i i f g = ⎛⎞ ⎜⎟ ⎝⎠ ∈ ∏ P p p p (P2) with the feasible set min () {0 2 } () i r max i ii i f pP i g , =| ≤ ≤ , ≥ , ∀ ∈ P p p p , M (4) which is a nonempty polytope in M R . Here () ii i i j i j ji i f Gp G p n ≠ = + ∑ p + j i n + and for all . It is c lear that the objective function of Problem P2 is a product of exponentiated linear fractional functions, and the function is an increasing () ij i ji gG p ≠ = ∑ p i 1 () ( ) i M w i i z = Φ= ∏ z 4 Polytope means the generalization to any dimension of polygon in two dimensions , polyhedron in three dimensions, and polychoron in four dimensions. 5 denotes -dim real domain and denotes -dim non-negative domain. N R N + R N N function on M + R . That is, for any two vectors 1 z and 2 z such that 1 V 2 z z 6 , we have 1 () ( ) Φ≥ Φ 2 z z . Therefore, Problem P2 is a special case of G LFP, which we refer to as Multiplicative Linear Fractional Programm ing (MLFP) due to the multiplicative n ature of the objective function. We further note that and in Problem P2 are always strictly positive due to the existence of positive noise power . Based on this, we can further rewrite Problem P2 as ( ) i f p ( ) i g p i n (P3) 1 maxi mize ( ) ( ) va riable s i M w i i z = Φ= ∈, ∏ G z z where the feasible set () {0 } () i i i f zi g =| ≤ ≤ , ∀ ∈ , ∈ p zp p GM . P (6) Since () Φ z is an increasing function in z , the optimal solution to Problem P3, denoted by ∗ z , must occur at places where () () i i f i g z = p p for all . If we can find a power allocatio n i ∗ p corresponding to the optimal solution ∗ z such that () () i i f i g z ∗ ∗ ∗ = p p for all i , then such is clearly the optimal solution to Problem P2. Finding such ∗ p ∗ p requires solving M linear equations with () () 0 ii i zg f ∗∗ ∗ −= pp M variables 1 M pp ∗ ∗ , ,  . As the coefficients of and () i f ∗ p () i g ∗ p consist of random channel gains ’s, we can show with prob ability 1 that the ij G M equations are linearly independent, implying there is a unique solution ∗ p . Hence Problems P1, P2 and P3 are all equivalent with each other. We will focus on how to solve Problem P3 efficiently in the rest of the paper. Before attempting to solve Problem P3, it is critical to understand several important properties of the feasible set in (5). The following definition w ill be useful in later discuss ions. G Definition 2 (Box) : Given any vector M + ∈ v R , the hyper rectangle [0 ] { 0 } , =| UU vx x v is a box with vertex 7 v . According to this definition, the feasible set can be characterized as a union of infinite num ber of boxes with vertices of a ll boxes belonging to the set G () {} () i i i f ci g | =, ∀ ∈ , ∈ p cp p MP . Each element in this set is d etermined by a power vec tor that is feasible in Problem P1 ( and Problem p 6 In this paper , means is component-wise smaller than , means is component-wis e lar ger than , means is component-wise sma ller than or equal to , and means is component-wise la rger than or equal to . ab ≺ a b ab  a b ab U a b ab V a b 7 In this paper , is a 1 0 M × vector with every element being 0, and is a 1 1 M × vector with every element being 1 P2). Definition 3 (Normal) : An infinite set M + ⊂ FR is normal if f or any element , the set . ∈ v F [0 ] ,⊂ v F Proposition 1: The intersection and union of a fa mily of normal sets are normal sets. Remark 1: Since the feasible set of Problem P3 is the union of in finite number of boxes, it is a normal set. G Fig. 1 illustrates one possible example of the shape of in a 2-link network. Note that is in general a non-convex set. However , this paper show s that convexity of th e feasible set is not important in obtaining the global optimal solution. It is the monotonicity of the objective function in the reformulated problem P3 that facilitates ef ficient calculation of the global optim al solution. G G Before leaving this section, note that () () i i f g p p is lower bounded by for . Consequently , the optimal solution min 2 i r , ∈ p P ∗ z to P3, which occurs only at places where () () i i f i g z = p p for all i , is also lower bounded by . In other words, the optimal solution min 2 i r , ∗ z must reside in the set , where . ∩Θ G min {2 } i r i zi , Θ= | ∀ ∈ z M V Fig. 1. Shapes of and for a two-link network G L IV. T HE MAPEL A LGORITHM In this section, we propose a novel algorithm, MAPEL, to solve Problem P3 based on the special characteristics of MLFP. Som e mathematical prelim inaries will be introduced f irst before we present the algorithm. A. Related Mathematical Preliminaries Definition 4 (Polyblock): Given any finite set M + ⊂ TR with elements , the union of all the boxes is a polyblock with vertex set . v [0 ] i , v T Definition 5 (Proper): An element is proper if there does not exist such that ∈ v T ∈ v  T ≠ vv  and . In other words, a proper elem ent is Pareto optim al. If every element is proper, then the set is a proper set. vv  V ∈ v T T Proposition 2: I f : is an increasing function of , then the maximum of () Φ v N + → R + R v () Φ v over a polyblock occurs at one prop er vertex of this polyblock. Proof: Let be a global optimal solution of ∗ v () Φ v over a polyblock . If is not a proper vertex of , then for som e proper vertex S ∗ v S ∗ v  U v ∗ ≠  vv . Since due to the increasing property of , it follows that is also a global optimal solution of , which is a contradiction to being a global optimal solution. Hence, Proposition 2 follows immediately. ■ ( ) ( ) ∗ Φ≤ Φ  vv () Φ v v  () Φ v ∗ v Definition 6 (Projection) : Given any nonempty normal set M + ⊂ FR and any 8 \{ 0 } M + ∈ v R , is a projection of on if () π v F v F ( ) π λ = v F v with max{ } λ αα = |∈ v F . In other words, is the un ique point where the half line from through meets the upper boundary of . () π v F 0 v F Definition 7 (Upper boundary): A point M + ∈ y R is an upper boundary point of a bounded normal set if while 9 F ∈ y F \ M K + ⊂ y RF . The set of upper boundary points of is the upper boundary of . F F We illustrate the above concepts in Fig. 2(a). In Fig. 2(a), the rectangles 10 1 0 ac v and represent boxes and , respectively. and are the respective vertices of thes e two boxes. The area cons isting of rectangles and represents polyblock with proper vertex set 2 0 bd v 1 [0 ] , v 2 [0 ] , v 1 v 2 v 1 0 ac v 2 0 bd v 1 [0 ] [0 ] =, ∪ , v S 2 v } 12 { = , vv T . If we choose any point , it is clear that the rectangle belongs to polyblock , i.e., 3 ∈ v S 3 0 ef v S 3 [0 ] , ⊂ v S . Hence, polyblock is normal. Being the only intersec tion of the halfline from through and the upper boundary of , is a projection of on . Moreover, if S 0 4 v S 4 () π v S 4 v S () Φ v is an increasing function on , S } 8 In this paper , denotes the set \ AB {a n d |∈ ∉ . xx x AB 9 { } ' ' a nd ' M K + =∈ ≠ y yy y y  R y 10 The rectangle is denoted using four letters in its four vertices. then for all 12 () m a x { ( ) ( ) } Φ≤ Φ , Φ vv v ∈ v S . In other words, the maximum of the increasing function occurs only at either or , a proper vertex of () Φ v 1 v 2 v () Φ v . Now let’ s use the above concepts to illustrate how we can construct a series of polyblocks th at approximate a set with increasing level of accuracy . F Proposition 3 : L e t M + ⊂ SR be a polyblock with proper vertex set . Also let be a nonempty normal closed set that is contained in , i.e., T F S M + ⊂⊂ FS R . For a given vertex i ∈ v T , let be the set obtained from by replacing the vertex with ' T T i v M new vertices, . Here the new vertex , where 1 ( ii M ,, vv  ) ) ) (( ij i i j j i j v π , =− − vv v e F j e is the j th unit vector of M + R 11 , is the ij v , j th element of the o ld vertex , and is the i v () ji π v F j th element of the projection . Note that some of the new vertices () i π v F 1 ( ii M ) , , vv  m ight not be proper . If we further remove all improper elem ents from set and obtain a new set , then the polyblock with vertex set ' T ∗ T ∗ S ∗ T satisfies . In this way , we have constructed a smaller polyblock that still contains . ∗ ⊂⊂ FS S ∗ S F The detailed proof of Proposition 2 is omitted due to space lim itation, and interested readers are referred to the Proposition 3 in [20]. W e use Fig. 2(b) to illustrate the above procedure. As shown in Fig. 2(b), given and where , we can obtain a polyblock F S 2 + ⊂⊂ FS R ∗ S with proper vertex set 11 2 {} ∗ = , vv T satisfying . is obtained by replacing in with , , and then deleting the improper elem ent from . ∗ ⊂⊂ FS S 11 2 { ∗ =, vv T } } ) ) v π , =− − vv v e T 12 i =, 1 v 12 { =, vv T 111 1 (( ii i i 12 v 11 12 2 '{ } =, , vv v T 11 In this paper , the j th unit vector of M + R , , denotes the vector whose every element is equal to zero except the j e j th element being 1. Fig. 2. Illustration about related mathematical pr eliminaries for MAPEL algorithm B. The MAPEL Algorithm The MAPEL algorith m works as follows . W e first construct a polyblock that contains the feasible set of Problem P3, . Let denote the proper vertex set of . By Proposition 2, the maximum of the objective function of Problem P3 (i.e., ) over set occurs at some proper vertex 1 S G 1 T 1 S 1 () ( ) i M w i i z = Φ= ∏ z 1 S 1 z of , i.e., . If 1 S 1 ∈ z T 1 1 z happens to reside in as well, then it solves Problem P3 and G 1 ∗ = z z . Otherwise, based on Proposition 3 we can construct a smaller polyblock that still contains but excludes 2 ⊂ SS 1 G 1 z . This is achieved by cons tructing the vertex set by first replacing 2 T 1 z in with 1 T M new vertices 11 1 () M , , z z  , where 11 1 1 (( jj j z π , =− − ) ) j z zz T e , and then removing improper vertices. W e can repeat this procedure un til an optimal solution is found. This leads to a sequence of polyblocks containing : G 12 ⊃ ⊃⊃  SS G . Obviously , 12 () ( ) ( ) ∗ Φ≥ Φ ≥ ≥ Φ z zz  , where ( ) i Φ z is the optimal vertex that maximizes () Φ z over set . The algorithm term inates at the th iteration if i S k k ∈ z G . For practical im plementation, we say when k ∈ z G max { ( ( )) } ki i k ki zz π δ ,, − ≤ z G , where 0 δ > is a small positive number representing the error tolerance level. W e can further expedite the above process by selecting k z from a smaller set , where . This will not af fect the optimality of the algorithm since the optim al solution k ∩Θ T min {2 i r i zi , Θ= | ∀ ∈ z M V } ∗ z is lower bounded by . min 2 r A critical step in constructing new polyblocks an d checking the termination criterion is calculating the projection () k π z G . This is, however , by no means tr ivial, since the upper boundary of is not explicitly known. In particular , G () kk k π λ = z z G is obtained by solving the following max-min problem for k λ : 1 1 () max { } ma x{ mi n } () () max mi n () i kk iM ki i i iM ki i f zg f zg λα α α α ≤≤ , ≤≤ ∈ , =| ∈ =| ≤ , ∈ =. p p zp p p p P GP (6) This is again a generalized linear fractional prog ramm ing problem by Definition 1. W e solve this problem using the Dinkelbach-type algorithm in [ 21] with slight modifications. The details are shown in Algorithm 1 12 . Algorithm 1 Max-Min Projection Al gorithm (for finding () k π z G ) 1: Initialization : Choose and let (0) [0 ] max ∈, pP 0 j = . 2: rep e a t 3: Given , solve () j p () () () () () 1 min j i j ki i f j k zg iM λ , ≤≤ = p p . 4: Given () j k λ , solve . (1 ) ( ) 1 arg max min ( ( ) ( )) jj ik k i i iM fz g λ + , ≤≤ ∈ =− p pp P p 5: . 1 jj =+ 6: until (1 ) max min ( ( ) ( ) ) 0 j ik k i i i fz g λ − , ∈ − ≤ p pp P . 7: The projection is (1 ) () Gj kk πλ − = k z z . Definition 8 (Q-super linear convergence): [21] A sequence {1 2 } j sj , =, , ∈  R with limit s ∞ converges Q-super (quotient super) linearly if 12 In fact, each Step 4 of Algorithm 1 is a linear programming in the convex power do main. 1 lim 0 j j j ss ss +∞ →∞ ∞ − = . − (7) Theorem 1: Since and are linear affine functions on for all and there is a unique optimal solution to (6), the sequence ( ) i f p () ki i zg , p p i () {1 2 j k j λ } , =, ,  converges Q-super linearly to the optimal solution. Proof : Immediate from Theorem 8.7 in [21]. Having introduced the basic operations, we now formally present the MAPEL algorithm as follows. Algorithm 2 The MAPEL Algorithm 1: Initialization : Check the feasibility of m inimum data rate require ments ’s based on Procedure 1. If ’s are infeasible, term inate the algorithm. Otherwise, choose the approximation factor min i r , min i r , 0 δ > , and let 1 k = . 2: repeat 3: If , construct the initial polyblo ck with vertex set 1 k = 1 S 1 {} = b T , where the th element of vector is i b () max 1 () max ii i i i ii fG P b gn ∈ == + , ∀ ∈ p p p P M i . ( 8 ) It is clear that polyb lock is a box 1 S [0 ] , b containing . If , construct a smaller polyblock with vertex set by replacing G 1 k > k G k T 1 k − z in with 1 k − T M new vertices , where 11 1 () kk −− ,, zz  M ) ) 11 1 1 (( kj k k j j k j z π −− − , − =− − z zz G e , and removing improper vertices. 4: Find k z that maximizes the objective f unction of Problem P3 over set , i.e., k ∩Θ T a r g m a x { ( ) } kk = Φ| ∈ ∩ Θ zz z T . ( 9 ) 5: Find () k π z G based on Algorithm 1. 6: . 1 kk =+ 7: until 11 1 max{ ( ( )) } ki i k ki i zz π δ −, − −, −≤ z G . 8: Compute the optimal power allocation ∗ p (i.e., optim al solution to Problem P1) by solving () 1 () () i i f ik g π − = p p z G for all i C. Global Convergence Theorem 2: The MAPEL algorithm globally converges to a global optimal solution of Problem P3. Proof: The MAPEL algorithm generates a sequence { } k z for 12 k = ,,  . Each component is calculated as (9) for each newly construc ted polyblock. W e can find a subsequence {} n k z within the sequence { } k z such that 11 1 1 1 11 1 ( ( )) ( ( )) nn n n n n ki i i k k k i i k zz π + ,, =− − , , = − − n i π , z zz e z z z  G e G (10) where . denotes the th element of vector 12 1 n kk k << < < <  nn ki z , n i n k z , where is the only position in which n i 1 n k + z differs from n k z . This subsequence can be t hought as the “off-springs” of vertex 1 z through a series of projections , and they are not necessarily adjacent since th ere might be projections of other vertices in be tween. It can be shown that there is at leas t one such subsequence that has infinite length. With a slight abuse of notation, let {1 } n k n , ∀≥ z denote such one subsequence. Since () n k π n k z z U G , (10) implies that . Hence, min 1 1 2 n kk  r zz z VV V V V 1 lim 0 nn kk n + →∞ −→ zz . From (10) we know that n k z and 1 n k + z only differ in the ’s position, thus n i 11 () 0 w h e n n n nn n n nn n n k k ki k i ki i k zz z n π ++ ,, , −= − = − → → zz z G ∞ . n k (11) Since () nn kk πλ = z z G and , (11) implies that min 2 n k r z V lim 1 n k n λ →∞ = . That is, lim ( ) n k n π →∞ → n k . z z G (12) Eqn. (12) implies that the subsequence {} n k z conver ges to the boundary of the feasible region . Since it is a maximizer over the set , it is thus also the g lobal optimum of Problem P3. Note that the MAPEL algorithm terminates once the optimal so lution to Problem P3 is found. Therefore, the convergence of the subsequence G n k S {} n k z guarantees the convergence of the algorithm to the global optimal solution . ■ D. Trade-off between Performance and Convergence Time The converg ence time of MAPEL is in finite if the approximation factor 0 δ = . However , it can be easily shown that MAPEL always terminates with finite steps when 0 δ > [20]. Next, we analyze the influence of the approximation factor δ on the performance. Definition 9( optimal ε − solution): Given an 0 ε ≥ , we say that a vector is an ∈ y G optimal ε − solution of Problem P3 if ( ) ( 1 ) ( ) ε ∗ Φ≤ + Φ zy . Theorem 3: The solution obtained by MAPEL (i f the algorithm converges) is an optimal ε − solution with 1 δ δ ε − ≤ . Proof : MAPEL terminates when () max ki i k ki z z i π δ , , − ≤ z G . Consequently, together with , 1 1 M i i w = = ∑ ( ) ( 1 )( ( ) )( )( kk δπ ∗ Φ − ≤Φ ≤Φ ≤Φ ) k z zz G z leading to () ( () ) () kk k π δ Φ− Φ ≤ . Φ zz z G Note that () ( ) k ∗ Φ≤ Φ z z implies () ( () ) () k k π δ ∗ Φ− Φ ≤ . Φ zz z G Consequently, () ( () ) () (( ) ) (( ) ) 1 kk kk π δ δ π πδ ∗ Φ− Φ Φ ≤ ≤, ΦΦ zz z zz G GG − which leads to the following inequality that p roves Theorem 3: () ( () ) 1 1 k z δ π δ ∗ ⎛⎞ Φ≤ Φ + ⎜⎟ − ⎝⎠ z G . ■ Remark 3: We note that 1 δ δ δ − ≈ when 1 δ  .F u r t h e r m o r e , 1 δ δ − is generally a conservative estimate of ε . In practice, we ofte n obtain a error that is much sm aller than δ . An advantage of the MAPEL algorithm is that we can trade off performance for conver gence time by tuning δ . The smaller δ , the longer the algorithm runs and the more accurate the optimal solution is. E. Extension to Max-min SINR Power Control As discussed in the Introduction, some previous work on power control aim ed at maximizing the minimum SINR of all links. Mathematic ally, they tried to solve the fo llowing problem max min ( ) max mi n ii i i ii P ji j i ji Gp Gp n γ ∈∈ ≠ = + ∑ pp p P (13) Obviously , this is a generalized lin ear fractional programming defined in (2). In fact, this formulation is similar to the one in describ ed (6). Hence, the Dinkelbach-type algorith m (Algorithm 1) that is adopted to solve (6) can be easily extended to solve the max-min SINR problem in (13). V. P ERFORMANCE E VALUATION OF MAPEL We illustrate the effectiveness of the MAPEL algorithm through several numerical exam ples. Example 1 (Performance and convergence time trad eoff through the approximation factor δ ): We consider a four-link network where the links are randomly placed in a 10m -by-10m area. The resultant channel gain matrix is (14) 1 0 4310 0 0002 0 2605 0 0039 0 0002 0 3018 0 0008 0 0054 0 0129 0 0005 0 4266 0 1007 0 0011 0 0031 0 0099 0 0634 .... ⎡⎤ ⎢⎥ .... ⎢ = ⎢ .... ⎢⎥ .. .. ⎣⎦ G ⎥ . ⎥ Assume that max P =[0.7 0.8 0.9 1.0]mW, 0 1 i n µ = . W for all link , and the priority weigh ts = i w 111 1 663 3 [ ] . Also we do not consider minimum data rate constraints in this example. 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 4.647 4.648 4.649 4.65 4.651 4.652 4.653 4.654 4.655 4.656 4.657 Approximation Factor δ Obtained Weighted Sum Throughput (bps/Hz) 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Iterations Number of Iterations Weighted Sum Throughput Fig. 3. Obtained weighted sum throughput and number of iterations for diff erent approximation factor δ In Fig. 3, we plot the optimal weighted-sum throughput obtained by MAP EL, together with the needed number of iterations versus δ . It is not surprising to see that the algor ithm perform ance improves with a decreasing value δ , which has been predicted by Theorem 3. On the other hand, the total number of iterations increases when δ decreases, and the change is drastic when δ is close to 0. Moreover, the al gorithm performance is not sensitive to the value of δ . For example, when 01 δ =. , we achieve a weighted-sum throughput of 4.655bps/Hz that is only 0.025% away from the exact optimum. Th is illustrates that the performance bound obtained in Theorem 3 is quite loose, and actual performance can be m uch better. It is also c lear that param eter δ provides a tuning knob for achieving desired trade-off betw een algorithm perform ance and computational complexity. Example 2 (Global optimal power alloca tion): MAPEL enables us to ea sily characterize the glo bal optimal solution 13 of the WTM problem for an arbitrary wire less network. This is not possible before without exhaustive search. W e consider a dif ferent 4-link network in Fig. 4 as a simple illus trating example. The length of each link is 4m, while the distances between to i T j R for , denoted by , are proportional to . The four links have diff erent cha nnel gains due to dif ferent fading states: , , , i ≠ j ij l d 11 1 G = 22 07 5 G =. 33 05 0 G =. 44 02 5 G = . . The priority weight of each link is equal. Meanwhile, , 4 ij ij Gl − = max P =[0.7 0.8 0.9 1.0]mW , 0 1 i n µ = . W for all . In Fig. 5, the optimal transmit power of each link is plotted against the topology param eter . It can be seen that when the links are very close to each other , only th e link with the lar gest cha nnel gain (i.e., Link 1) is active with maximum transm it power , while all the other links keep silent. When increases, a quantum jump in i d 1 max P d 2 p from 0 to is observed. As further in creases, Link 3 sta rts to transm it, followed by Link 4. In this particular example, prio rity is always given to the lin k with a larg er channel gain. Although the result may not be general, this toy example illustrates the possib ility of using MAPEL as a tool to investig ate the characteristics of global opt im al solutions to power control problems. 2 max P d d d d d 1 R 1 T 2 R 2 T 3 T 3 R 4 T 4 R Fig. 4. The relationship between op timal transmit power and distance d 13 MAPEL will only find one of the possible many global optimal solutions, depending on the choice of initial conditions. 2 3 4 5 6 7 8 9 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Distance d (meter) Optimal Transmit Power (mW) Link 1 Link 2 Link 3 Link 4 Link 1 Link 2 Link 3 Link 4 Fig. 5. A network topology with four links VI. P ROVIDING B ENCHMARK FO R E XISTING P OWER C ONT ROL A LGORITHMS A key application of MAPEL is to provide performance benchmark for other centralized or distributed algorithms that have been (or to be proposed) to solve W TM problem. W i th MAPEL, we are able to give quantitative measu rements of th ese algorithm s’ performances (e.g., the chances of achieving global optimal solution and the gap of sub-optim ality) under a wide range of network scenarios (e.g, dif ferent netw ork densities and topologies). A. Review of Existing Power Control Algorithms As we mentioned in Introduction, the current existing power cont rol algorithms are essentially divided into two categorie s: centralized and distributed. Here we will review one “representativ e” algorithm from each category that repres ents the state-of-art in this ar ea. Notice that the focus here is to show how MAPEL can be used to provide effec tive benchmark f or the algorithms that tack le the same problem (i.e., Problem P1). Readers can ch oose your favorite algorithm to conduct the study . 1) Centralized algorithm: Signomial Progra mming Condensation (SPC ) Algorithm [12]: SPC Algorithm is one of the best ex isting centralized algor ithms for solving Problem P1. It utilizes the fact that Problem P1 can be rewritten as m ini mizing a ratio between two posynomials (i.e., a SP): 1 () min imi ze () var i ab l es i i w M i w i i g f = ∈ ∏ p p p P . ( 1 5 ) The key idea of SPC Algorithm is to im prove the solution of Problem (15) through successive approximations until a KKT point is reached. During each step, th e SP is approximated by a GP , which can be solved ef ficiently using a centralized interior point method. 2) Asynchronous Distributed Pricing (ADP) Algorithm [16 ]: ADP Algorithm is a distributed algorithm that can be used to solve Problem P1 w ithout minimum data rate constraints. In ADP , each link announces a price that reflects its sensitivity to the received interference, and updates its own transmit power based on the prices announced by other links. The price and power values need to be updated iteratively and asynchronousl y until a convergent point is f ound. T o implement the updates, each link only needs to acquire limited inform a tion from the network. W e observe that ADP algorithm conver ges very fast in our numerical expe rim ents, mainly because no stepsize is used in the updates. Its theore tical conver gence to the gl obal optimal point, however , is difficult to prove in general. B. Performance Study of SPC Algorithm and ADP Algorithm In this subsection, we evaluate the performan ce of both algorithms th rough several exam ples by utilizing the benchmark provided by MAPEL. Example 3 (Probability of achieving global optimal solutio n): MAP EL always guarantees global optimality, while the SPC algorith m and the ADP al gorithm f ail to do so. Using the same 4-link network given in Example 1 (topology ), we simulate three algorithms based on 500 different random initializations and show the results in Fig. 6 and Fig. 7, re spectively. Then we change the topology to with channel matrix illustra ted in (16), and simulate both algorithm s again in Fig. 8 and Fig. 9, respectively. 1 G 2 G (16) 2 0 1476 0 0105 0 0018 0 0402 0 0034 0 1784 0 0013 0 2472 0 0014 0 0017 0 3164 0 0046 0 0048 0 4526 0 0012 0 6290 ... . ⎡⎤ ⎢⎥ .... ⎢ = ⎢ .... ⎢⎥ .... ⎣⎦ G ⎥ . ⎥ Other system param e ters are the same as in Ex am ple 1. The figures show that MAPEL always converges to the global optimal solu tion, regardless of the initial power allocation. On the other hand, the SPC algorithm and the ADP algorithm are trapped in local optimal solution s from time to time. For example, Fig. 6 and Fig. 7 show that SPC and ADP algorithms obtain the global optim al solution 70.8% and 62.6% of the time, resp ectively. However, Fig. 8 and Fig. 9 show that in a different topology SPC and ADP algorithms obt ain the global optimal solution 96% and 93.6% of the tim e, respectively. 0 50 100 150 200 250 300 350 400 450 500 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 Experiment Index Obtained Weighted Sum Throughput (bps/Hz) SPC Algorithm MAPEL Algorithm Fig. 6. Maximal weighted sum throughput achieved by MAPEL algorithm as well as SPC algorithm for 500 different initial feasible power allocations in network 1 G 0 50 100 150 200 250 300 350 400 450 500 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 Experiment Index Obtained Weighted Sum Throughput (bps/Hz) ADP Algorithm MAPEL Algorithm Fig. 7. Maximal weighted sum throughput achieved by MAPEL algorithm as well as ADP algorithm for 500 different initial feasible power allocations in network 1 G 0 50 100 150 200 250 300 350 400 450 500 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5 5.1 Experiment Index Obtained Weighted Sum Throughput (bps/Hz) SPC Algorithm MAPEL Algorithm Fig. 8. Maximal weighted sum throughput achieved b y MAPEL algorithm as well as SPC algorithm for 500 diff erent initial feasible power allocations in network 2 G 0 50 100 150 200 250 300 350 400 450 500 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5 5.1 Experiment Index Obtained Weighted Sum Throughput (bps/Hz) ADP Algorithm MAPEL Algorithm Fig. 9. Maximal weighted sum throughput achieved b y MAPEL algorithm as well as ADP algorithm for 500 differ ent initial feasible power allocations in network 2 G Example 4 (Average algorithm performance without m inimu m data rate constraints): In Fig. 10, we compare the average performance of the SPC algor ithm , the ADP algorithm, and the GP algorithm with MAPEL under different ne twork densities. Compared w ith SPC and ARP, GP [12] approximates and solves the W TM problem based on high-SINR assumptions. For each fixed total number of links , we place the links randomly in a 10m -by-10m area. The length of each link is uniformly distributed within [1m, 2m]. The prior ity weight of each lin k is equal. Meanwhile, we have =1mW, n max i P 0 1 i n µ =. W, and initial power allocation is fixed at . We vary the to tal number of links from 1 to 10. Each point is obtained by averaging over 500 different topologies of the same link density. On average, the perfor mance loss of SPC compared with respect to the global optimality is about 2%, thus is quite small. Notice that the performan ce loss of each particular realization might be sm aller (e.g., 0% when reac hing the g lobal optim ality) or larger (when trapped in a local optimal). The average perform ance degr adation of the ADP algorithm is about 10%, which implies that ADP is trapped in local optim um more often than SPC. Noticeab ly, the gap between SPC (or ADP) and the global optimum is not know n before this work, as there does not exist previous algorithm that can guarantee the global op tim al solution. This is in fact one of the key contributions of this paper. In addition, Fig. 10 shows that GP works reasonably well when the network density is low, where all (or m ost) links ar e active and some of them are indeed in the high SINR regime. However, the gap from the global op timum is m uch bigger wh en the network density becomes high, where many links need to be silent in order to avoid heavy interferences to their neighbors. 2 max / P n 1 2 3 4 5 6 7 8 9 10 8 10 12 14 16 18 20 22 n (Number of Links) Average Sum Throughput (bps/Hz) MAPEL Algorithm SPC Algorithm ADP Algorithm GP Algorithm Fig. 10. Average sum throughput of different algorithms Table I gives more detailed statistics about the perform ances of two al gorithms. As shown in Table I, SPC achieves the global optimality with a proba bility that is always larger than 65% with the number of links up to 10. In contrast, the probability of ADP achieving th e global optimality can b e very low, e.g., only 0.6% in 10-link networks. It suggests that the init ial power allocation of 2 max P is a good initial point for SPC, but may not for ADP . On the other hand, we find that SPC has a high-mean and low-variance average perform ance co mpared to the global optim ality, which implies that SPC can achieve close-to -optimal perform an ce with the initial power allocation of 2 max P for most topologies. However, ADP has a low-mean and high-variance average performance, which implies that ADP maintains a large degradation for som e topologies. Example 5 (Average algorithm performance with minim u m data rate constraints): We consider a series of 4-link networks with minimum data rate constraints on each link. The four links are randomly placed within a 10m x 10m area, and the le ngth of each link is unifo rmly distribu ted within the interval [1m, 2m ]. max P =[0.7 0.8 0.9 1.0]mW, 0 1 i n µ = . W for all . Meanwhile, the priority weight of each link is equal. In Fig. 11, the perf orm ance of MAPEL, GP, a nd SPC is plotted against the data rate constraint of each link. Each point f or sum throughput on the curves is an average over 500 different topologies. We elimin ate the topologies that are not feasible. Since ADP algorithm performs poorly in this case, we do not show its performance here. i It is not surprising to s ee that the sum throughputs of all algorithms drop as the data rate constraints become more stringent. One in teresting observation is that the gap between GP and MAPEL becomes smaller when the data ra te constraints are high. This is due to the fact that links are forced to operate in the high SINR regime when a high data rate is to b e ensured. The high SINR assumption made by GP becomes mo re reasonable in this case. TABLE I O PTIMALITY OF SPC A LGORITHM AS WELL AS ADP A LGORITHM SPC Algorithm ADP Algorithm Number Of Links Probability of achieving global optimality A verage performance Coefficient of variation Probability of achieving global optimality A verage performance Coefficient of variation 2 69.8% 96.9% 7.22% 50.6% 89.6% 17.8% 4 80.4% 98.7% 3.91% 25.0% 94.3% 8.79% 6 77.2% 98.9% 3.13% 6.0% 93.4% 7.89% 8 69.4% 98.8% 2.58% 1.4% 92.7% 7.42% 10 65.6% 98.7% 2.81% 0.6% 92.1% 8.18% 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 13.5 14 14.5 15 15.5 16 Data Rate Constraint (bps/Hz) Average Sum Througput (bps/Hz) MAPEL Algorithm SPC Algorithm GP Algorithm Fig. 11. Average sum throughput of different algorithms versus the data rate cons tr aint in 4-link networks VII. C ONCLUSIONS AND D ISCUSSIONS In this paper, we proposed the MAPEL algorithm that solves the open problem of weighted throughput maximization in general interference- limited wireless networks. The MAPEL algorithm is guaranteed to globally converge to an optimal solution despite the nonc onvexity of the problem. The key idea behind the algorithm is to reform ulate the WTM problem into an MLFP and then construct a sequence of shrinking polyblocks that eventually closely approximate the upper boundary of the feasible regi on around the global optimum. We have also established the tradeoff relationship between performance and convergence time of the MAPEL algorithm. Although a centralized algorithm, MAPEL provide s an important benchmark for performance evaluation of existing and newly proposed power co ntrol heuristics in this area. For example, by comparing with MAPEL through extensive simulations , we have gained deep er understanding of two state-of-the-art centralized a nd distributed power control al gorithms: SPC algorithm and A DP algorithm. Simulations show that both algorithms achieve close-to-optimal average perform a nce in the general SINR regime. This paper helps to pave the way for furthe r study of power control problems with various objectives and constraints. An interesting future research direction is to study power control that maximizes general utility func tions, including both concave a nd non-concave functions. Optimal power control in time-varying channels is a nother challenging topic for future research. The MAPEL algorithm presented in this paper is not the only way to ef ficiently obtain the global optimal solution. V ariants of the algorithm can be developed to expedite the convergence and reduce the computational com plexity . For example, it can be proved that the projection of a vertex of on m ust contain at least one element equal to . Such characteristics could be used to design a faster projection algorithm to re place Algorithm 1. Anothe r possibility is to exploit the shape of the feasible region . S G max i P G R EFERENCES [1] M. Chiang, P. Hand e, T. Lan a nd C. W. Tee, “Power Control Wirele ss Cellular Networks,” to appear in Found ations and Trends in Networking , 2008 [2] J. Zander, “P erformance of optim um transmitter power control in cellular radio s ystems,” in IEEE Trans. Veh. Technol. , vol. 41, no. 1, pp. 57-62, February 1992. [3] G. J. Foschini and Z. Mi ljanic, “A simple distributed au tonomous pow er con trol algorithm a nd its convergence, ” in IEEE Trans. Veh. Technol. , vol. 42 , no. 6, pp. 641-646, November 1993. [4] S. A. Grandhi and J. Za nder, “Constrained Power Control in Cellular Radio S ystems,” Proc. IEEE VTC , Sto ckholm, Sweden, June 1994. [5] S. A. Grandhi, R. Vijayan, D. J. Goodman, and J. Zander, “Centralized Po wer Control in Ce llular Radio Systems,” IEEE Trans. Veh. Technol. , vol. 42 , no. 6, pp. 466-468, November 1993. [6] R. D. Yates, “ A Framework for Uplink Power Control in Cellular Radio Systems,” IEEE J. Sel. Areas Commun. , vol. 13, no. 7, pp. 1341-1347, September 1995. [7] N. Bambos, S. C. Chen, and G. J. Pottie, “ Channel access algorithms with activ e link protection for wireless communication networks with power control,” in IEEE/ACM Trans. Netw. , vol. 8 , no. 5, pp. 583-597, October 2000. [8] C. W. Sung, “Log-Convexity Property of the Feas ible SIR Region in Power-Controlled Cellular Systems,” IEEE Commun.Letters , vol. 6, no. 6, pp 248-249, June 2002. [9] H. Boche and S. Stanczak,“Convexity of some feasible QoS regions and as ymptotic behavior of the minimum total power in CDMA systems, IEEE Trans. Commun. , vol. 52, no. 12, pp. 2190 - 2197, December 2004. [10] M. Schubert and H. Boche, “Solution of the Multi-user Do wn link Beamforming Problem with Individu al Sir Constraints,” IEEE Trans. Veh. Technol. , vol. 53, no. 1, pp . 18-28, January 2004. [11] D. Julian, M. Chiang, D. O. Neill, and S. Boyd, “Qos and fairness constrained c onvex op timization of resource alloc ation f or wireless cellular and ad hoc networks,” Proc. IEEE INFOCOM , vol. 2, pp . 477-486, June 2002. [12] M. Chiang, C. W. Tan, D. Palomar, D. O’Neill, and D. Julian, “Power contro l by geometric progr amming,” IEEE Trans. Commun. , vol. l, no. 7, pp. 2640-2651, July 2007. [13] M. Chiang and J. Bell, “Balancing supply and demand of wireless bandwidth: Joint rate allocation and power control,” Proc. IEEE INFOCOM , Hong Kong, China, March 20 04. [14] D. ONeill, D. Julian, and S . Boyd, “A daptiv e management of network resources,” Proc. IEEE VTC , Orlando, FL, October 20 03. [15] M. Xiao, N. B. Shroff, and E. Chong, “A utility-b ased power-control scheme in wireless cellul ar systems,” IEEE/ACM Trans. Networking , vol. 11, no. 2, pp. 210 - 221, April 20 03. [16] J. Huang, R. A. Berry, and M. L. Honig, “Dis tributed Int erference Compensation for W ireless Networks,” IEEE J. S el. Areas Commun. , vol. 24, no. 5, pp. 107 4 - 1084, May 2006. [17] P. Hande, S. Rangan, and M. Chiang, “Distributed uplink power control f or optimal sir assignment in cellular data networks ,” Proc. IEEE INFOCOM , April 2006. [18] M. Chiang, “Balancing trans port and physical layers in wireless multihop networks: Jointl y op timal congestion control and power control,” IEEE J. Sel. Areas Commun. , vol. 23, no. 1 , pp. 104-116, January 2005. [19] X. Lin, Ness B. Shroff and R. Srikant, “A Tuto rial on C ross-Layer Optimiza tion in Wireless Networks,” IEEE J. Sel. Areas Commun. , vol. 24, no. 8, pp. 145 2-1463, August 2006. [20] N. T. H. Phuong and H. Tuy, “A Unif ied Monotonic Approach to Gen eralized Linear Fractiona l Programming,” Journal of Global Optimization, Kluwer Academic Publishe rs, pp. 229-259, 2003. [21] J. B. G. Frenk and S. Sch aible, “Fractional Programming,” Handbook of Generalized Convexity and Generalized Monotonicity, pp. 335-386, 2006. [22] S. Boyd and L. Vandenberg he, “ Convex Optimization ,” Cam bridge, U.K.: Camb ridge Univ. Press, 2004.