Optimal resource allocation in wireless communication and networking

Optimal design of wireless systems in the presence of fading involves the instantaneous allocation of resources such as power and frequency with the ultimate goal of maximizing long term system properties such as ergodic capacities and average power consumptions. This yields a distinctive problem structure where long term average variables are determined by the expectation of a not necessarily concave functional of the resource allocation functions. Despite their lack of concavity it can be proven that these problems have null duality gap under mild conditions permitting their solution in the dual domain. This affords a significant reduction in complexity due to the simpler structure of the dual function. The article discusses the problem simplifications that arise by working in the dual domain and reviews algorithms that can determine optimal operating points with relatively lightweight computations. Throughout the article concepts are illustrated with the optimal design of a frequency division broadcast channel. Introduction Operating variables of a wireless system can be separated in two types. Resource allocation variables p(h) determine instantaneous allocation of resources like frequencies and transmitted powers as a function of the fading coefficient h. Average variables x capture system’s performance over a large period of time and are related to instantaneous resource allocations via ergodic averages. A generic representation of the relationship between instantaneous and average variables is x ≤ E [f1 (h, p(h))] , (1) where f1(h, p(h)) is a vector function that maps channel h and resource allocation p(h) to instantaneous performance f1(h, p(h)). The system’s design goal is to select resource allocations p(h) to maximize ergodic variables x in some sense. An example of a relationship having the form in (1) is a code division multiple access channel in which case h denotes the vector of channel coefficients, p(h) the instantaneous transmitted power, f1(h, p(h)) the instantaneous communication rate determined by the signal to interference plus noise ratio, and x the ergodic rates determined by the expectation of the instantaneous rates. The design Correspondence: [email protected] Department of Electrical and Systems Engineering, University of Pennsylvania, 200 S. 33rd St., Philadelphia, PA, 19096, USA goal is to allocate instantaneous power p(h) subject to a power constraint so as to maximize a utility of the ergodic rate vector x. This interplay of instantaneous actions to optimize long term performance is pervasive in wireless systems. A brief list of examples includes optimization of orthogonal frequency division multiplexing [1], beamforming [2,3], cognitive radio [4,5], random access [6,7], communication with imperfect channel state information (CSI) [8,9], and various flavors of wireless network optimization [10-18]. In many cases of interest the functions f1(h, p(h)) are nonconcave and as a consequence finding the resource allocation distribution p∗(h) that maximizes x requires solution of a nonconvex optimization problem. This is further complicated by the fact that since fading channels h take on a continuum of values there is an infinite number of p∗(h) variables to be determined. A simple escape to this problem is to allow for time sharing in order to make the range of E [f1(h, p(h))] convex and permit solution in the dual domain without loss of optimality. While the nonconcave function f1(h, p(h)) still complicates matters, working in the dual domain makes solution, if not necessarily simple, at least substantially simpler. However, time sharing is not easy to implement in fading channels. In this article, we review a general methodology that can be used to solve optimal resource allocation problems in wireless communications and net© 2012 Ribeiro; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Ribeiro EURASIP Journal onWireless Communications andNetworking 2012, 2012:272 Page 2 of 19 http://jwcn.eurasipjournals.com/content/2012/1/272 working without resorting to time sharing [19,20]. The fundamental observation is that the range of E [f1(h, p(h))] is convex if the probability distribution of the channel h contains no points of positive probability (Section “Duality in wireless systems optimization”). This observation can be leveraged to show lack of duality gap of general optimal resource allocation problems (Theorem 1) making primal and dual problems equivalent. The dual problem is simpler to solve and its solution can be used to recover primal variables (Section “Recovery of optimal primal variables”) with reduced computational complexity due to the inherently separable structure of the problem Lagrangians (Section “Separability”). We emphasize that this reduction in complexity, as in the case of time sharing, just means that the problem becomes simpler to solve. In many cases it also becomes simple to solve, but this is not necessarily the case. We also discuss a stochastic optimization algorithm to determine optimal dual variables that can operate without knowledge of the channel probability distribution (Section “Dual descent algorithms”). This algorithm is known to almost surely converge to optimal operating points in an ergodic sense (Theorem 5). Throughout the article concepts are illustrated with the optimal design of a frequency division broadcast channel (Section “Frequency division broadcast channel” in “Optimal wireless system design”, Section “Frequency division broadcast channel” in “Recovery of optimal primal variables”, and Section “Frequency division broadcast channel” in “Dual descent algorithms”). One of the best known resource allocation problems in wireless communications concerns the distribution of power on a block fading channel using capacityachieving codes. The solution to this problem is easy to derive and is well known to reduce to waterfilling across the fading gain, e.g., [21, p. 245]. Since this article can be considered as an attempt to generalize this solution methodology to general wireless communication and networking problems it is instructive to close this introduction by reviewing the derivation of the waterfilling solution. This is pursued in the following section. Power allocation in a point-to-point channel Consider a transmitter having access to perfect CSI h that it uses to select a transmitted power p(h) to convey information to a receiver. Using a capacity achieving code the instantaneous channel rate for fading realization h is r(h) = log(1 + hp(h)/N0) where N0 denotes the noise power at the receiver end. A common goal is to maximize the average rate r := E[ r(h)] with respect to the probability distribution mh(h) of the channel gain h—which is an accurate approximation of the long term average rate—subject to an average power constraint P0. We can formulate this problem as the optimization program P = maxE [ log ( 1+ hp(h) N0 )] s.t. E [p(h)] ≤ q0. (2) In most cases the fading channel h takes on a continuum of values. Therefore, solving (2) requires the determination of a power allocation function p : R+ → R+ that maps nonnegative fading coefficients to nonnegative power allocations. This means that (2) is an infinite dimensional optimization problem which in principle could be difficult to solve. Nevertheless, the solution to this program is easy to derive and given by waterfilling as we already mentioned. The widespread knowledge of the waterfilling solution masks the fact that is is rather remarkable that (2) is easy to solve and begs the question of what are the properties that make it so. Let us then review the derivation of the waterfilling solution in order to pinpoint these properties. To solve (2) we work in the dual domain. To work in the dual domain we need to introduce the Lagrangian, the dual function, and the dual problem. Introduce then the nonnegative dual variable λ ∈ R+ and define the Lagrangian associated with the optimization problem in (2) as L(p, λ) = E [ log ( 1+ hp(h) N0 )] + λ [q0 − E [p(h)]] .