The performance of the proposed APD algorithm is compared to the dual decomposition-based algorithm proposed in [13], denoted as WSRmax, as well as to the optimal algorithm based on exhaustive search. The WSRMax algorithm uses a bisection search method to update the dual variable [13, Section IV]. For initializing the bisection search interval, , we exploit the fact that the subgradient of the dual function can be analytically computed. Since the dual function is convex, the sign of its subgradients changes as we pass through the minimum point of the dual function [13, equation ()]. Therefore, we use a grid search (with step size ) to identify the interval in which the subgradient of dual function changes its sign, and it is used as initial bisection search interval. Thus, the interval is guaranteed to contain the optimal value of the the dual function and the width of the initial interval is one, that is, . The proposed APD algorithm is initialized by allocating equal power to all subcarriers.

In what follows, we compare the convergence behavior of the APD and the WSRMax algorithms. For a fair comparison, we define the following metric:

the average normalized weighted sum-rate deviation, where is the optimal weighted sum-rate value obtained using optimal exhaustive search, is the estimated objective value from either the APD algorithm or the WSRMax algorithm, and expectation is taken w.r.t. channel realization. An OFDMA system with subcarriers and a uniform power delay profile with channel taps is considered. We assume,, and define SNR per subcarrier as.

Figure 3 shows the convergence behavior of the considered algorithms with SNR dB for and users. The weights of the users are for and for . The floor of the curves is due to the suboptimality of the algorithms. The results show that the APD algorithm converges faster than the WSRMax algorithm and provides smaller average normalized weighted sum-rate deviations. Specifically, for both cases, and , the APD algorithm requires only iterations on average to achieve an average normalized weighted sum-rate deviation of whilst the WSRMax algorithm requires around iterations to reach the same accuracy level. It is intuitively obvious that the number of iterations required by the APD algorithm is sensitive to the nature of the surface of the objective function of problem (7), for example, see Figure 2. In general, it is a hard to quantify the number of iterations before convergence (or any bounds on the number of iterations) due to the nonconvexity of problem (5). However, the numerical results suggest that the APD algorithm often converges very fast in practice. It should be emphasized that the number of iterations required in the initialization of the WSRmax algorithm (i.e., the number of iterations required to find the initial bisection search interval) is not considered when drawing the curves. In particular, for the initialization process, the WSRMax algorithm requires a several number of steps (each step has complexity of ) and the proposed APD algorithm requires none. Moreover, it is hard to find good initialization methods for the WSRMax algorithm (i.e., initialization for bisection search method) compromising between the number of steps required in the initialization and the width of the initial searching interval . Consequently, additional precautions are required and therefore, in practical implementations the APD algorithm is more favorable compared to the WSRMax algorithm.

In the sequel, we compare the behavior of the APD and the WSRMax algorithms using the following metric:

the probability of missing the global optimal, where is a small number which quantifies the maximum admissible deviation between and . It is considered that the global optimum is missed if is more than away from .

Figure 4 uses the same simulation setup as that in Figure 3 and depicts the variation of probability of missing the global optimal, with the number of iterations. The floor of probability is again due to the suboptimality of both algorithms. The influence of on is totally indistinguishable in case of the APD algorithm. This behavior shows that the proposed algorithm APD can arrive very close-to-optimal solutions within a very small number of iterations and then it remains there. The results further show that the evaluated using the WSRMax algorithm is highly dependent on . That is, the smaller the deviations in the from the optimal , the larger the number of iterations required by the WSRMax algorithm to reach the expected target value . Therefore, independent from the , the APD algorithm allows to find a suboptimal solution within a small number of iterations at the expense of a slight increase in . These observations are very useful in practice since they carry significant information in the system design point of view. For example, consider a design requirement . Here, the WSRmax algorithm requires 18 iterations. If we tight more the design requirement as , then the number of iterations required by the WSRmax increases to 24. In contrast, the APD algorithm always requires just one iteration.

Figure 5 shows the rate region (The standard way to characterize the boundary points in the 2-user rate region is by solving problem (4) for and , where [30].) computed by using all considered algorithms. The same simulation setup as in [13] was used where , , and the channel SNR vectors for users and are and , respectively. Although the computational complexity of the proposed algorithm is much smaller compared to that of optimal exhaustive search-based method, Figure 5 indicates that the rate region obtained by the APD algorithm almost coincides with the optimal rate region. This behavior is expected since the average normalized weighted sum-rate deviation, (20) is in the order of as shown in Figure 3.

In the following we compare the behavior of the APD and the WSRmax algorithm for large number of subcarriers and users. Since, for large number of users and subcarriers the complexity of evaluating is prohibitively high, the metrics defined in (20) and (21) are not used. The behavior of the APD algorithm is compared with that of the WSRmax algorithm.

In Figure 6, the evolution of the expected weighted sum-rate provided by the APD algorithm is compared to the resulting expected weighted sum-rate from the WSRmax algorithm, where the expectation is taken w.r.t. channel realization. An OFDMA system with subcarriers, a uniform power delay profile with channel taps, and users is considered. The weights of the users are taken from the sequence , (e.g., when , weights are []). The SNR is assumed to be dB. The results show that even for a large number of carriers, the APD algorithm converges very fast as compared to the WSRMax algorithm independent of the number of users.