A cutting plane optimization algorithm for intra-cell link adaptation problem

A cutting plane optimization algorithm for intra-cell link adaptation problem
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  2005 IEEE 16th International Symposium on Personal,Indoor and Mobile Radio Communications   utting Plane Optimization Algorithm for Intra-cell Link Adaptation Problem Alireza Babaei(1) rezababaci aec.,i   Mansoor Isvand Yousefil2 tnansoor( ee..shari fledL Bahman Abolhassani(t) abolhassania iust c ir Naser Sadati sadati caishari fedu  1 : College of Electrical Engineering, Iran University of Science and Technology, Tehran, Iran, FAX: (98-21)745-4055  2 : Department of Electrical Engineering, Sharif University of Technology, Tehran, Iran Abstract- Link adaptation of a wireless cellular network is of much attention due to the desire of maximizing both the a erage lin throughput and coerage reliability, which have conflicting effect on each other. In this paper, the intra-cell lin adaptation problem is formulated as a non-differentiable constrainedoptimization problem to maximize average link throughput while guaranteeing the best possible coverage reliability. To achieve this, a cutting plane optimization algorithm is employed. The performance of our method:adaptive modulation/coding with power management  AMCWPM) using proposed algorithm is compared with that of an adaptive modulation/coding  AMC) with no power management. Simulation results show that the performance ofour method improves bothaverage link throughputand coverage reliability by at least 10 while in each case the other parameter  i.e. respectively, coverage reliability and average link throughput is hold as much as that of the  M with no powermanagement. Keywords- Adaptive modulation/coding, power distribution management, decomposition techniques, non-differentiable optimization, convex analysis. I. INTRODUCTION Future wireless mobile communication networks are expected to provide extremelyhigh data rate applications,especially in the down-link. Themain obstacle to satisfying this expectation is the mobile wireless channels, which is quite harsh.   mobile channel usually consists of multiplepaths whose parameters (attenuations, timedelays and phases) are highly time-variant. Providing reliable communication links over these channels necessitates thatthe system is designed forthe worstchannel conditions, which is resource wasting. To avoid this, system designers have to applyadaptivemethods. Of these methods is adaptation of the modulation/coding with or without power control in the base station transmitter, basedon link conditions. The performance of adaptive transmitters has been studied for both single-user and multi-user environments. In [1], the Shannon capacity of a fading channel is obtained fora single user, in the presence ofchannel side information (CSI) at both thereceiver and transmitter. As well, joint adaptation of transmitted power and rate is proposed using idealcase of infinite number of coding rates. In [2], a sequel of [1], a variable-rate variable-power MQ M modulation is proposed to remove the impracticality of [1]. Moreover, it is shown that there is a constant gap between the Shannon capacity and the throughput provided by the method proposed in [2]. References [3] and [4], considering multi- user case, propose new iterative methods to maximize aggregate throuighput of a channel, which results in maximizing aggregate throughput of the network. In [5], considering practical conditions such as SINRmin, SINRmax and finite number of modulation/coding rates, performance of different intra-cell link adaptation strategies is studied. It is shown that water-filling algorithm is optimal and provides the greatest amountof average link throughput, at the expense of loosing coverage reliability  i.e., reduction in the number ofmobiles that can be supported). To compensate this coverage reliability,in[5-6], different link adaptation methods are analyzed to provide equations for the network capacity. As well, a hybrid of both  M and power control is also proposedand its performance is addressed. Yet, the performanceof optimal intra-celllink adaptation, considering the practical constraints, has not been investigated. Optimal intra-cell link adaptation with practical constraints is a hard optimization problem. In this paper, we propose a new method (adaptive modulation/coding with powermanagement  AMCWPM)) for maximization of both the average link throtughput and coverage reliability, which have conflicting effect oneach other. We formulate the problem with practicalconstraints asa constrained non-convex non-smooth optimization problem. A decomposition technique is used to solve the srcinal large-scale problem: replacing it by a sequenceof reduced-dimensional local problems that are coordinated by a master program. A cutting planeoptimization algorithm is proposed to overcome the non-smoothness arising in the master program. Since locations of mobiles are random, the constraints of the problem sometimes cannot be met, and therefore no feasible solution can be found. To overcome this, we are obliged to relax some of theconstraints corresponding to links having sm ll SINR values. Performance average link throughput andcoverage reliability) of the proposed method  AMCWPM) is 978-3-8007-2909-8/05/ 20.00 ©2005 IEEE 1 895  2005IEEE 16th International Symposium on Personal, Indoor and Mobile Radio Communications compared with that of the AMC with no power management, through computer simulations. The remainderof this paper is organized as follows. Section 1I describes mathematical model of the problem. Section III briefly describes the exploited decomposition technique and thecutting plane algorithm. The proposed method is evaluated through computer simulations in Section IV, and conclusions are presented in Section V. II. MATHEMATICAL MODEL OFPROBLEM In this section, the mathematical model of the problem, used forthe maximizationof average link throughput, is presented. Themodel of throughput is a modification of Shannon capacity for AWGN channels. In this direction,the capacity is divided by the bandwidth, and a link degradation parameter a, where0<a<1, is used in the model, to consider any practical link degradation from ideal Shannon case. It is notable that the model used for throughput inthis paper, is the one employed in[5-6]. The link throughput t is given by t = log2  1   or.SINR),  1) where SINR is the signal to noise and interference ratio. Optimal intra-celllink adaptation problem is the bestdistribution of power among the N downlinks of a specific cell, whichmaximizes the average link throughput and meanwhile satisfies theconstraints. It is desirable to giveservice to all links ina cell. This requires that the SINR of each link to be larger than a threshold value. There are K modulations/coding rates available at the transmitter  i.e. ri,...-,rK, where rl<r2<.. .<rK). Each modulation/coding rate r- i 1,2, ,K, corresponds to a SINR value denoted by y- i 1,2, ,K. Moreover, the base station  BS initially obtains a primary profile of each link SINR  i.e. [SINRk,...,SINRN]), by transmitting an equal amount of power  i.e. P/N in each of N active links and receiving a feedback from each mobile. The modulation/coding ratein each link is chosen to be r if the measured SINR of the link   <SINR< y±i The power distribution should result in SINR values that lie in the interval [Yi 7K]. That is because, links with SINR values less than y, will be considered out of service, and links with SINR values more than YK are just wasting power;because such links are assigned the modulation coding rate rK, no matter howmuch their SINR values are more than YK. The base stationdistributes its composite transmitted power P among   links such that to maximize total downlink throughput. For this purpose, as was mentioned in previousparagraph, the base station initially distributes the total transmitted power ofP equally among N active links to obtain a profile ofeach link SINR. Then, it changes the transmitted power assigned to link i from P/N to  P/N) +xi, where xi is the power variation according to the measured SINR of link i. This modification, in the transmitted power, causes the SINR of link i changes from SINRk to SINR Il+xiN/P). The objective function T  i.e. total throughput) to be maximized is given by Max T = E fIlog2Il+ a.SINRtp   i=   2) where 0 x<rl r1 r x<r2 f x) = r2 r2 < x < r3 rKl rK1 x rK r,K x 2rK Solution must satisfy the following constraints: N 1) Zx,=0. i=1 This guarantees that the total transmitted power remains equal toP. 2 x> x, where x=[xl x2 ... xN]T is the vector of power corrections applied in each link, and x =[x*l x ... XN]T is the minimum threshold for power correction vector to assure that the SINR of link i after adaptation remains above 71, and its i h   l- SINRi p element is = S SINRi N 3 x< x,   ** K -SINRi p where x   with itch element as x =i SINR. N the maximum threshold for power correctionvector to guarantee that the SINR, after adaptation, remains below YK. 4) x> e where e=[1IlI...,]T. N This constraint is to guarantee that thetransmitted power from each link is nonnegative. The objectivefunction, given by  2), is discontinuous since function f is discontinuous. Therefore, this is a constrained non-convex non-smooth optimization problem, which is hardly tractable. Using a decomposition technique, we will split this large-scale program to a set of simpler subproblems coordinated by a master program. Then, we propose a simple iterative algorithm to solve the problem efficiently. III. SOLUTION METHOD As mentioned, our optimization problem is a large-scale problem with a non-smooth objective function. This problem is quite difficult to be solved by classical algorithms. So,other algorithms appropriate for our optimization problem must be employed. One of these algorithms is cutting plane, which is described below. 978-3-8007-2909-8/05J 20.00 ©2005 IEEE 1 896  2005 IEEE 16th International Symposium on Personal,Indoor and Mobile RadioCommunications To achieve a better solution, in the cutting plane algorithm, a generalized concept of derivative for objectivefunction is employed. Rather than using a gradient of the objective function at a given x, the vector containing thevariables that must be found, a certain set will be associated with x. This set is calledthe generalized Clarke sub- differential at x, which is defined as follows [7], af x) := conv{ lim Vf  x ) Vx ; Vf  x ) exist }, where conv stands for convex hull. Each elementof x is called a sub-gradient. Forconvex functions f is minimized at i if and only if 0 e af x) [7]. Cutting plane based algorithms are first proposed by Kelly [8], and they are to minimize a constrained convex non- smooth objective function. These algorithms rely on using information from pastiterations to find a polyhedral approximation of convex functions, and using this approximation to find a search direction for the new iteration. In fact, it will transform the initial problem to a set of linear programs with a number of affine constraints, which increase as the algorithm progresses. Keeping track of past information allows defining a model for the objective function. Denoting an arbitrary element of subdifferential of f at xi by s , cutting plane method uses k values off so far obtained, which are given by fi := f x ), s := s xi); i = I.k, to construct the following piecewise linear model for f: fk  Y) = max {fi+ Y-x   s }  3) i=il..k The minimization of the objective function model Ik on a convex compact set S to be determined in advance, gives a new vector of variables for iteration k+1  i.e., gives xk+1 ). A variant of this algorithm appropriate for our problem is employedwhose steps are given below. A. ProposedAlgorithm  cutting planes) Let tol . 0 be a givenstopping tolerance and let S X 0 be a compact convex set containing a minimum point of f Choose x E S and set k = 1   Define A- STEP 1  Implement stopping test): k ik computeda := f x  -f  x ).If 6s, < tol, stop. k k- I STEP 2  Find candidatesearch direction):findthe search direction d by solving the following linear program  LP) min d,r r.2/+ x -x ) s +d s, I k  4) xk +dESand re RSTEP 3  skipping Line-search): set tk = 1. k+I k k STEP 4  Loop): define x = X +tkd Change k to k+land go to 1. The key idea is to incorporate inequality constraints in the reference set X. With this setting, our maximization problem changes to the minimization of the opposite sign of the objectivefunction, given by  2), that is N N xeMX - jf  log 2  1 + aSINR i   aSINR i -xi N s.t. E xi = O i=1 where  5) N i-A =l P SINRmi SINR SINRmi IR _xE R max - min   )<X< min - SINR N SINRiSINRi in this setting the Lagrangianof  3) has a simple separable structure[7] N p N L x, A) =- af Iog2  1   SINR, + axSINRi -xi )) +  Z xi)  6) i=1 N i=- with A E R. According to [9], solving  5) is equivalent to solving inf sup L x, A Interchanging order of inf and x A sup  under no assumptions) we have dualproblem sup 0 A) AieR where O A)   inf L x,A) xeX  7)  8) is the associate dual function. Note that, L x, A), X and thus0 A) have separable structure. Price decomposing technique is used to solvelargescale problem  5), replacing it by N one dimensional local problems linked by a master program, 0 A)f=  n [ log  ai + bi xi)) + Axi J}  9) Typically, the master program will choose a multiplier A  price) sent to the   ocal solvers. Once thesolutions x of  9) are obtained for all i, the master program evaluates0 A) and computes a new A using a cutting plane scheme.Note that both master and local programs are non-smooth. Each local problem can be solved by searching for all possible states of function f, which is veryeasy to solve and may be performed analytically. Using these local solutions, the value of dual function o A) is found by  9), and according to proposition 2.2.2 in chapter XIIof [7], an arbitrary subgradient is now available: - E  9 2 10 i=l Once the value of the dual function o A) and an arbitrary element of its subdifferential determined, a black-box optimization technique such as cutting plane methodmaybe 978-3-8007-2909-8/05/ 20.00 ©2005 IEEE   1897  2005IEEE 16th International Symposium on Personal,Indoor and Mobile Radio Communicationsadopted to solvedual problem (7) andcompute a new iterate k+I. Thisprice  2k+ is then given to the slave(local) programs and this procedure is repeated until the convergence criterion is met. It can be shown [7] that the algorithm converges linearly only if number of affine functionsdefining the model (3) goes toinfinity. Note that, although our optimization problem, given by (5), is non-convex, after solving slave programs the dual function in (8) would be affine (and therefore concave) in 2, and therefore, using a convex optimizationtechnique makes no problem. However the aforementioned dual approach is ultimatelysuccessful if an initial solution can be recovered from a dual solution (7). The difference between optimalobjective values in(5) and (7) is calledthe duality gap and measures how good the dualization scheme is. Complete recovery corresponds to a zero duality gap. This occurs if and only if (5) has convex data and a constraint is satisfied. In our problem, although the duality gap is nonzero, the numerical resolution of (7) generates a sequence {   k converging to 2*, with sub-gradients hopefully close to zero for large k (see (10)).  hen by (10) there will be primal points xk := x for which s Ak is close to zero. Theorem 2.1.1 of [7] then implies that those x kare very good approximation of a primal optimum. A serious drawback of thecutting plane method is the infinite accumulation of affine function defining the model  3 . More precisely, as k grows, the LP (4) has more andmore constraints and many of them are similar. So in general the LP will become extremely difficult to solve. As our formulation shows, incorporating complete decoupled constraints in thereference set X and thenpass them to the local problems allows Lagrangian (6) to be one dimensional inA . Inthis way LP (4)exits vector notation and will be very easy to solve even manually so that the aforementioned tailing- off effect will not manifest itself. IV. SIMULATION RESULTS Inthis section, the performance of adaptivemodulation/coding  AMC), with no power management, as well as the performance of our method with power management  AMCWPM), are compared through computer simulations. We have considered a cellular hexagonal system with frequency reuse of 1. Fading effects are assumed to be removed by a sufficient number of diversity branches at the transmitter or receiver. A distance dependent path loss model, using a path loss exponent is employed for our propagationmodel. The system alsosuffers fromlognormal shadowing with a standarddeviation of 10 dB. The value of link degradation parameter is set to 0.398, which corresponds to 4 dB deviation from Shannon capacity limit. The supported modulation/coding rates along their corresponding SINR intervals havebeen listedin table I. TABLE I SUPPORTED MOD/COD RATES AND THEIR CORRESPONDING SINR INTERVALS mod/cod rate initial SINR  dB) final SINR  dB) 0.375 -1.27 2.33 0.75 2.33 6.62 1.5 6.62 12.45 3 12.45 17.35 4.5 17.35 22 6 22 The transmitted power of the base station is considered to be 100   and the number of links in each cell is set to be 100. It is considered that the BS has a primary profile of each link s SINR by transmitting an equal amount of power  i.e. 1W) in each link. In order to include the effect of the randomnessof users locations, we have averaged the parameter of interest over a number of different users locations in our simulations. We havefound the coverage reliability and average link throughput of adaptive modulation/coding  AMC) through simulation and have listed them in table II. In this case, the vector of power correction  i.e. x) equals 0. TABLE II COVERAGE RELIABILTY AND AVERAGE LTNK THROUGHPUT OF AMC n 2.5 3 3.5 4 4.55 Average link throughput 0.93 1.33 1.71 2.092.432.74 (bit/s/Hz) reliability (0) 45 53 60 66 70 74 Employing the primary profile, the BS uses the optimization method discussed in the pervious section, to findthe optimal vector x which leads to maximum throughput. Since locations of users are random, the resulting SINR values may lead to an empty feasiblespace. In this case, someof the weak links  i.e. links with low SINR values)should be considered out of service sequentially. By doing this the coverage reliability reduces from ideal coverage of 100 . The power of theselinks is allocated to the remaining active links, resulting to a newproblem withreduced number of dimensions. In order to compare the performance of the  M and  M WPM for each path loss exponent n, we set the same coverage reliabilityfor both link adaptation methods. For this purpose, using table II, a proper number of the weakest linksare considered out of service to have a coverage reliability which is at least as good as AMC. Then our algorithm maximizes the average link throughput within the remaining links. Figure 1 illustrates the improvement in average link throughput of the  M WPM over that of the  M when the coverage reliability ofboth is equal for each n). As it can be seen, the use of power management has led to a significant improvement in the average link throughput, while the coverage reliability is at least as much asthat of the AMC. Coverage reliability of  M WPM and  M with no power control is compared in Figure 2. Foreach path loss 978-3-8007-2909-8/05/ 20.00 ©2005 IEEE 1 898  2005IEEE 16th International Symposium on Personal, Indoor and Mobile Radio Communications exponent n, the same average link throughput is considered for both link adaptation methods. To provide the same average link throughput for AMCWPC, some of the weakest links should be considered out of service. It can be seen that while both of AMC and AMCWPM provide the same average link throughput, the coverage reliability of AMCWPM is at least 10% more than that of the AMC with no power control. V. CONCLUSIONS In this paper, we formulated the intra-cell link adaptation problem considering all practicallimitations as a non-convex non-smooth constrainedoptimization problem.Exploiting the problem structure, a price decomposition techniquealong a cutting plane optimization algorithm was used to solvetheresulting problem.Simulation results showed more than 10% improvement in the coverage reliability and 0.5 bits/S/Hz improvement in the average link throughput obtained from our proposed methodcompared to those of the AMC with no power control.IV. REFERENCES [1] A. J. Goldsmith and P. Varaiya,  Capacity of fading channelswith channel side information, IEEE Trans. Inform Theory, vol.43, Nov. 1997. [2] A. J. Goldsmithand S. G Chua,  Variable-rate variable-power MQAM for fading channels, IEEE Trans. Commun., vol.45, Oct.1997. [3] X. Qiu andK. Chawla,  On the performance of adaptive modulation in cellular systems, IEEE Trans. Commtin., vol.47,June. 1999. [4] A. Babaeiand B. Abolhassani,  A new iterative method forjoint powerand modulation adaptation in cellular systems, Proceedingsof WOCN2005, Dubai, UAE, March 2005. [5] K.L. Baum, T.A. Kostas, P.J. Sartori, and B.K. Classon,  Performance Characteristics of Cellular Systems with Different Link Adaptation Strategies, IEEE Trans. Veh. Technol, Vol. 52, NOV 2003. [6] B. Abolhassani and A. Babaei,  Capacity enhancement of wireless networks using adaptivemodulation/coding, Proceedingsof IEEEGCC- 2004 Bahrain, Nov. 2004. [7] J.-B. Hiriart-Urruty and C. Lemarechal, Convex Analysis and Minimization Algorithms(volumes I and II Springer-Verlag, Berlin, 1993. [8] J.E. Kelly,  The Cutting Plane Method for Solving Convex Problems, Journal of SIAM, 1960. [9] S.Boyd, L.Vandenbergh, Convex Optimization, Cambridge universitypress, 2004. z ;Z I- C :3 0 z Iz  2 Z a r AMCWPM   24 2-   _1-   2.   , ot ,3   _____ ______L___S__ _   ____ / , ,,,-   - X   __ . - .,/ /,, .. ' ,/ ,, _, _ -- X- - --- - r-- -- --1- ,/ _____ ______e______I______   PATHLOSS E PONENT  n) Fig. 1. Average link throughput of AMC and AMCWPM at the same coverage reliability I AMC   AMCWPM _I / I _ I- m lY co 4   UC   2 e PATH LOSS E PONENT  n) Fig. 2. Coverage reliability of AMC and AMCWPM at the same average link throughput 978-3-8007-2909-8/05/ 20.00 ©2005 IEEE  ^ 1899
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