Optimal Power Allocation in Downlink NOMA

07/14/2021 ∙ by Sepehr Rezvani, et al. ∙ Carleton University uOttawa European Union 0

Power-domain non-orthogonal multiple access (NOMA) has arisen as a promising multiple access technique for the next-generation wireless networks. In this work, we address the problem of finding globally optimal power allocation strategies for the downlink of a generic single-cell NOMA system including multiple NOMA clusters each operating in an isolated resource block. Each cluster includes a set of users in which the well-known superposition coding (SC) combined with successive interference cancellation (SIC) technique (called SC-SIC) is applied among them. Interestingly, we prove that in both the sum-rate and energy efficiency maximization problems, network-NOMA can be equivalently transformed to a virtual network-OMA system, where the effective channel gain of these virtual OMA users are obtained in closed-form. Then, the latter problems are solved by using very fast water-filling and Dinkelbach algorithms, respectively. The equivalent transformation of NOMA to the virtual OMA system brings new insights, which are discussed throughout the paper. Extensive numerical results are provided to show the performance gap between fully SC-SIC, NOMA, and OMA in terms of system outage probability, BS's power consumption, users sum-rate, and system energy efficiency.

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I Introduction

I-a Evolution of NOMA: From Fully SC-SIC to Hybrid-NOMA

The rapid growing demands for high data rate services along with energy constrained networks necessitates the characterization and analysis of wireless communication systems. It is proved that the capacity region of degraded single-input single-output (SISO) Gaussian broadcast channels (BCs) can be achieved by performing linear superposition coding (SC) at the transmitter side combined with coherent multiuser detection algorithms, like successive interference cancellation (SIC) at the receivers side [1, 2, 3]. We call the latter technique as fully SC-SIC (FSIC), where the signal of all the users are multiplexed. The achievable rate region of FSIC, frequency division multiple access (FDMA), and time division multiple access (TDMA) are analyzed in [4, 5].

The SIC complexity is cubic in the number of multiplexed users [6]. Another issue is error propagation, which increases with the number of users [6]. Hence, FSIC is still impractical for a large number of users, and can be considered as a benchmark for performance comparison. To tackle the above practical limitations, a low-complexity, yet suboptimal, technique called non-orthogonal multiple access (NOMA) is proposed in [7] which is based on SC-SIC and orthogonal multiple access (OMA) such as FDMA/TDMA. In NOMA, users are grouped into multiple clusters, and SC-SIC is performed among users within each cluster. The clusters operate in isolated resource blocks, based on FDMA/TDMA111The spatial domain multiple access (SDMA) can also be introduced on NOMA, where clusters are isolated by zero-forcing beamforming [6].. NOMA is considered as a promising candidate solution for the beyond-5G (B5G)/6th generation (6G) wireless networks [6, 7, 8, 9, 10, 11]. The overall achievable rate of users can be increased by introducing the multicarrier technology to the fading channels. The concept of NOMA is also introduced on multicarrier systems, where SC-SIC is combined with OFDMA, called OFDMA-NOMA, multicarrier NOMA or Hybrid-NOMA. In Hybrid-NOMA, each user can occupy more than one subcarrier, and SC-SIC is applied to each isolated subcarrier. As a result, all the users can benefit from the multiplexing gain [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22].

I-B Related Works and Open Problems

The resource allocation in downlink single-cell NOMA under individual minimum rate demands of users can be classified into the following three categorizes:

I-B1 Fsic

The FSIC technique is also referred to as single-carrier/single-cluster NOMA [23, 24, 25]. In our previous work [25], we derived the closed-form of optimal powers for -user FSIC under the individual minimum rate demands of users to maximize users sum-rate as well as minimize total power consumption. The work in [26] addresses the problem of simultaneously maximizing users sum-rate and minimizing total power consumption defined as a utility function in FSIC. However, the analysis in [26] is affected by a detection constraint for successful SIC which is not necessary, since SISO Gaussian BCs are degraded. The closed-form of optimal power coefficients for the well-known fractional energy efficiency (EE) maximization problem is still an open problem.

I-B2 Noma

The joint power allocation and user clustering in NOMA is proved to be strongly NP-hard [13, 19, 20]. In this way, the latter two problems are decoupled in most of the prior works. For any given set of clusters, the optimal power allocation for sum-rate/EE maximization in NOMA is more challenging compared to FSIC. In NOMA, there exists a competition among multiple clusters to get the cellular power. Actually, the optimal power allocation in NOMA includes two components: 1) Inter-cluster power allocation: optimal power allocation among clusters to get the cellular power budget; 2) Intra-cluster power allocation: optimal power allocation among multiplexed users to get cluster’s power budget. From the optimization perspective, the analysis in [25] is also valid for NOMA with predefined (suboptimal) power budget for each cluster, e.g., [27, 28]. In this case, the intra-cluster power allocation can be equivalently decoupled into multiple single-carrier NOMA subproblems. To the best of our knowledge, the only work that address the globally optimal power allocation in NOMA is [29]. In [29], each cluster has only two users, and all the analysis is based on allocating more power to each weaker user to guarantee successful SIC, which is not necessary, due to the degradation of SISO Gaussian BCs [3, 10]. Therefore, the globally optimal power allocation for sum-rate maximization in NOMA under the individual users minimum rate demands is still an open problem.

The fractional EE maximization problem in NOMA is addressed by [30, 31, 32, 33, 34, 29]. The EE maximization problem is solved by using the suboptimal difference-of-convex (DC) approximation method [30], Dinkelbach algorithm with Fmincon optimization software [31], and Dinkelbach algorithm with subgradient method [32, 33, 34]. Despite the potentials, the closed-form of optimal powers for the EE maximization problem is not yet obtained by the prior works. The EE maximization problem was also addressed by [29] for only two-user case per cluster, however the formulated closed-form of optimal power coefficients among users within each cluster is derived based on the condition allocating more power to the weaker user which is not necessary [10]. Besides, the Karush–Kuhn–Tucker (KKT) optimality conditions analysis for EE maximization problem is not yet addressed by the literature. It is still unknown what are the optimal power coefficients among multiplexed users, and which user deserves additional power rather than its minimum rate demand?

I-B3 Hybrid-NOMA

The works on Hybrid-NOMA mainly focus on achieving the maximum multiplexing gain, where each user receives different symbols on the assigned subcarriers. The works in [12, 13, 14, 15, 16, 17, 18, 19, 20] address the problem of weighted sum-rate/sum-rate maximization without guaranteeing individual per-user minimum rate demands over all the assigned subcarriers. In Hybrid-NOMA with minimum rate constraints, [21] proposes a suboptimal power allocation strategy for the EE maximization problem based on the combination of the DC approximation method and Dinkelbach algorithm. Also, a suboptimal penalty function method is used in [22]. Finding the optimal power allocation for sum-rate/EE maximization problem in Hybrid-NOMA is challenging, and we consider it as our future work222The main challenge is the nonconvexity of the minimum rate constraints in Hybrid-NOMA. A suboptimal scheme is to define a specific minimum rate demand for each user on each subchannel, e.g., [32, 33, 34]. In this case, Hybrid-NOMA can be equivalently transformed to a NOMA system, however the solution is still suboptimal..

I-C Our Contributions and Paper Organization

In this work, we address the problem of finding optimal power allocation for the sum-rate and EE maximization problems of the general downlink single-cell NOMA systems including multiple clusters each having an arbitrary number of users. Our main contributions are listed as follows:

  • We show that for the three main objective functions as total power minimization, sum-rate maximization and EE maximization, only the cluster-head333The user with the highest decoding order which cancels the signal of all the other multiplexed users. users deserve additional power while all the other users get power to only maintain their minimal rate demands444For the total power minimization problem, the cluster-head users also get power to only maintain their minimal rate demands..

  • The feasible power allocation region in NOMA is defined as the intersection of closed boxes along with affine cellular power constraint. Then, the optimal value for the power minimization problem is obtained in closed form.

  • For the sum-rate/EE maximization problem, we show that the NOMA system can be transformed to an equivalent virtual OMA system. Each cluster acts as a virtual OMA user whose effective channel gain is obtained in closed form. Moreover, each virtual OMA user requires a minimum power to satisfy its multiplexed users minimum rate demands, which is obtained in closed form.

  • A very fast water-filling algorithm is proposed to solve the sum-rate maximization problem in NOMA. The EE maximization problem is solved by using the Dinkelbach algorithm with inner Lagrange dual with subgradient method or barrier algorithm with inner Newton’s method. Different from [30, 31, 32, 33, 34], the closed-form of optimal powers among multiplexed users is applied to further reduce the dimension of the problems, thus reducing the complexity of the iterative algorithms, as well as increase the accuracy of the solution, which is a win-win strategy.

  • We propose a necessary and sufficient condition for the equal inter-cluster power allocation strategy to be optimal. In contrast to FDMA, the equal inter-cluster power allocation strategy in NOMA can be suboptimal even if all the users in different clusters have the same channel conditions while different minimum rate demands.

  • We propose a sufficient condition such that the EE maximization problem in NOMA turns into the sum-rate maximization problem where the BS operates at its maximum power budget. The latter condition is characterized according to the effective channel gain of the virtual OMA users as well as constant which is derived from the closed-form of optimal powers among multiplexed users in each cluster .

Extensive numerical results are provided to evaluate the performance of FSIC, NOMA with different maximum number of multiplexed users, and FDMA in terms of outage probability, minimum total power consumption, maximum sum-rate and EE. The performance comparison between NOMA and FSIC brings new insights on the suboptimality-level of NOMA due to user grouping based on FDMA/TDMA. In this work, we answer the question "How much performance gain can be achieved if we increase the order of NOMA clusters, and subsequently, decrease the number of user groups?" for a wide range of number of users and their minimum rate demands. The latter knowledge is highly necessary since multiplexing a large number of users would cause high complexity cost at the users’ hardware. The complete source code of the simulations including a user guide is available in [35].

The rest of this paper is organized as follows: The system model is presented in Section II. The globally optimal power allocation strategies are presented in Section III. The numerical results are presented in Section IV. Our concluding remarks are presented in Section V.

Ii NOMA: OMA-Based SC-SIC

Ii-a Network Model and Achievable Rates

Consider the downlink channel of a multiuser system, where a BS serves users with limited processing capabilities in a unit time slot of a quasi-static channel. The set of users is denoted by . The maximum number of multiplexed users is . FSIC is infeasible when . In this case, NOMA is applied, where users are grouped into clusters555In NOMA, the condition implies that .. We assume that each cluster operates in an isolated subchannel based on FDMA. To this end, the total bandwidth (Hz) is equally divided into isolated subchannels with the set , where the bandwidth of each subchannel is . The set of multiplexed users on subchannel is denoted by , in which is the binary channel allocation indicator, where if user occupies subchannel , we set , and otherwise, . In NOMA, each user belongs to only one cluster [27, 28, 29, 30, 31, 32, 33, 34]. As a result, we have , or equivalently, . According to the above, we have , where indicates the cardinality of a finite set. The exemplary models of FSIC, NOMA, and FDMA are illustrated in Figs. 1(a)-1(c).

(a) FSIC: Capacity-achieving, thus optimal, when , and infeasible when .
(b) NOMA: Suboptimal but feasible. SC-SIC is applied among users within each cluster.
(c) FDMA: Suboptimal but feasible with no SC-SIC. Each cluster has a single user.
(d) Equivalent model of FSIC: A single virtual OMA user.
(e) Equivalent model of NOMA: FDMA with virtual OMA users.
(f) FDMA: Real OMA users.
Fig. 1: 1(a)-1(c): Exemplary models of FSIC, NOMA, and FDMA for a user distribution. (a). User is cluster-head; (b): Users , and are cluster-head; (c): All the users are cluster-head. 1(d)-1(f): The equivalent models of FSIC, NOMA, and FDMA including virtual OMA users (see Remark 1).

Each subchannel can be modeled as a SISO Gaussian BC. The transmitted signal by the BS on subchannel is formulated by where and are the modulated symbol from Gaussian codebooks, and transmit power of user on subchannel , respectively. Obviously, . The received signal at user on subchannel is

(1)

where is the (generally complex) channel gain from the BS to user on subchannel , and is the additive white Gaussian noise (AWGN). We assume that the perfect channel state information (CSI) is available at the BS as well as users666The extension of the analysis in this paper to the imperfect CSI scenario with outage constraints [32] is straightforward..

In NOMA, SC-SIC is applied to each multiuser subchannel according to the optimal CNR-based decoding order [1, 2, 3, 4, 5]. Let . Then, the CNR-based decoding order is indicated by , where represents that user fully decodes (and then cancels) the signal of user before decoding its desired signal on subchannel . Moreover, the signal of user is fully treated as noise at user . In summary, the SIC protocol in each isolated subchannel is the same as the SIC protocol of FSIC. We call the stronger user as the user with higher decoding order in the user pair . In each subchannel , the index of the cluster-head user is denoted by . When , the single user can be defined as the cluster-head user on subchannel since it does not experience any interference. Therefore, NOMA is a combination of SC-SIC and OMA. In each subchannel, the cluster-head user does not experience any interference. The signal-to-interference-plus-noise ratio (SINR) of each user for decoding the desired signal of user on subchannel is [3]. User is able to fully decode the signal of user if and only if , where is the SINR of user for decoding its own signal . According to the Shannon’s capacity formula, the achievable rate (in bps) of user after successful SIC is given by [2, 3, 25]

where

is the vector of allocated powers to all the users on subchannel

. For the user pair with , the condition or equivalently holds independent of . Accordingly, at any , the achievable rate of each user on subchannel is equal to its channel capacity formulated by [3]

(2)

Ii-B Optimization Problem Formulations

Assume that the set of clusters, i.e., is predefined. The general power allocation problem for maximizing users-sum-rate under the individual minimum rate demand of users in NOMA is formulated by

(3a)
s.t. (3b)
(3c)
(3d)

where (3b) is the per-user minimum rate constraint, in which is the individual minimum rate demand of user . (3c) is the cellular power constraint, where denotes the maximum available power of the BS. (3d) is the maximum per-subchannel power constraint, where denotes the maximum allowable power on subchannel777We do not impose any specific condition on . We only take into account in our analysis to keep the generality, such that , as special case. . For convenience, we denote the general power allocation matrix as .

The overall system EE is formulated by , where constant is the circuit power consumption [36, 37]. The power allocation problem for maximizing system EE under the individual minimum rate demand of users in NOMA is formulated by

(4)

Iii Solution Algorithms

In this section, we propose globally optimal power allocation algorithms for the sum-rate and EE maximization problems. The closed-form of optimal powers for the total power minimization problem is also derived to characterize the feasible set of our target problems.

Iii-a Sum-Rate Maximization Problem

Here, we propose a water-filling algorithm to find the globally optimal solution of (3). The sum-rate of users in each cluster, i.e., is strictly concave in , since its Hessian is negative definite [25]. Due to the isolation among subchannels, the overall sum-rate in (3a) is strictly concave in . Besides, the power constraints in (3c) and (3d) are affine, so are convex. The minimum rate constraint in (3b) can be equivalently transformed to the following affine form as Accordingly, the feasible set of (3) is convex. Summing up, problem (3) is convex in . Let us define as the power consumption of cluster . Problem (3) can be equivalently transformed to the following joint intra- and inter-cluster power allocation problem as

(5a)
s.t. (5b)
(5c)
(5d)
(5e)

In the following, we first convert the feasible set of (5) to the intersection of closed-boxes along with the affine cellular power constraint.

Lemma 1.

The feasible set of (5) is the intersection of , and cellular power constraint , where the lower-bound constant is given by in which

Proof.

The feasibility of (5) can be determined by solving the power minimization problem as

(6)

Assume that the feasible region of (6) is non-empty. The problem (6) is also convex with affine objective function. The strong duality holds since (5c) and (5e) hold with strict inequalities. In the KKT optimality conditions analysis in Appendix C of [25], we proved that in FSIC, the maximum power budget does not have any effect on the optimal powers obtained in the power minimization problem when the feasible region is nonempty. And, at the optimal point, all the multiplexed users get power to only maintain their minimal rate demands. In this system, if the strong duality holds, the problem can be equivalently decoupled into single-carrier power minimization problems. In each subchannel , regardless of the clusters power budget vector , the optimal power of each user can be obtained in closed form given by where . Since the solution of (6) is unique, it can be easily shown that the optimal powers are indeed component-wise minimum. In other words, for any feasible , we have . Therefore, is indeed the lower-bound of . According to (5e), we guarantee that any satisfying (5c) is feasible, and the proof is completed. ∎

The feasibility of (5) can be directly determined by utilizing Lemma 1. In the following, we find the closed-form of optimal intra-cluster power allocation as a linear function of .

Proposition 1.

For any given feasible , the optimal intra-cluster powers can be obtained by

(7)

and

(8)

where , .

Proof.

For any given feasible , (5c) and (5e) can be removed from (5). Then, problem (5) can be equivalently divided into intra-cluster convex power allocation subproblems, where for each subchannel , we find the optimal powers according to Appendix B in [25]. It is proved that at the optimal point of sum-rate maximization problem in FSIC, all the multiplexed users with lower decoding order get power to only maintain their minimum rate demands. And the rest of the available power will be allocated to the cluster-head user. ∎

In contrast to FSIC, in NOMA, there is a competition among cluster-head users to get the rest of the cellular power. It is worth noting that at the optimal point, the cellular power constraint is active, meaning that all the remaining cellular power will be distributed among cluster-head users. According to (8), the optimal power of the cluster-head user can be obtained as a function of given by

(9)

where , and are nonnegative constants. According to Proposition 1 and (9), the optimal value of (5) for the given can be formulated in closed form as

(10)

Hence, (5) is equivalently transformed to the following inter-cluster power allocation problem

(11a)
s.t. (11b)
(11c)

In the objective function (11a), we substituted the closed-form of optimal intra-cluster powers as a function of . Constraint (11b) is the cellular power constraint, and (11c) is to ensure that the individual minimum rate demands of all the users within each cluster is satisfied. The objective function (11a) is strictly concave, and the feasible set of (11) is affine. Accordingly, problem (11) is convex. For convenience, let , where . Then, (11) is equivalent to the following convex problem as

(12a)
s.t. (12b)

where , , , and . The equivalent OMA problem (12) can be solved by using the water-filling algorithm [38, 39]. After some mathematical manipulations, the optimal can be obtained as

(13)

such that satisfies (12b). Also, is the dual optimal corresponding to constraint (12b). For more details, please see Appendix A. The pseudo-code of the bisection method for finding is presented in Alg. 1.

1:  Initialize tolerance , lower-bound , upper-bound , and maximum iteration .
2:  for  do
3:     Set .
4:     if    then
5:          Set .
6:     else      Set .
7:     end if
8:     if    then
9:          break.
10:     end if
11:  end for
Algorithm 1 The bisection method for finding in (13).

After finding , we obtain by . Then, we find the optimal powers according to Proposition 1.

Remark 1.

In the transformation of (3) to (12), the NOMA system is equivalently transformed to a virtual OMA system including a single virtual BS with maximum power , and virtual OMA users operating in subchannels with maximum allowable power . Each cluster is indeed a virtual OMA user whose channel gain is , where is a function of the minimum rate demand of users with lower decoding order in cluster , and the channel gain of the cluster-head user, whose index is . Each virtual user has also a minimum power demand in order to guarantee the minimum rate demand of its multiplexed users in . For any given virtual power budget , the achievable rate of each virtual OMA user is the sum-rate of its multiplexed users.

Based on the definition of virtual users for the sum-rate maximization problem in NOMA and the KKT optimality conditions analysis, the exemplary systems in Figs. 1(a)-1(c) are equivalently transformed to their corresponding virtual OMA systems shown in Figs. 1(d)-1(f). Note that FDMA is a special case of NOMA, where each subchannel is assigned to a single user. Hence, each OMA user acts as a cluster-head user, and subsequently, the virtual users are identical to the real OMA users, i.e., , , and . As a result, each user in FDMA deserves additional power. It can be shown that the analysis for finding the optimal power allocation in NOMA and FDMA are quite similar, and the only differences are and . Finally, both of them can be solved by using the water-filling algorithm for any number of users and clusters.

Iii-B Equal Inter-Cluster Power Allocation

The inter-cluster power allocation is necessary when , i.e., there is at least one cluster which is not allowed to operate at its maximum power . In this case, the distributed intra-cluster power allocation leads to violating the cellular power constraint (3c), since in the distributed power allocation among clusters, the constraint (3d) will be active. Alternatively, when , we guarantee that . There are a number of works, e.g., [27, 28], assuming , i.e., equal inter-cluster power allocation. In this case, the optimal intra-cluster power allocation can be obtained by Proposition 1.

It can be shown that in FDMA, the equal power allocation strategy is optimal if and only if 1) it is feasible; 2) all the users have exactly the same channel gains normalized by noise.

Remark 2.

In NOMA, the equal inter-cluster power allocation strategy is optimal if and only if 1) it is feasible; 2) all the virtual OMA users have the same channel gains, i.e., .

Proposition 2.

Consider two clusters with the same number of multiplexed users, where the corresponding users according to the decoding orders in different clusters have exactly the same channel gains. The equal inter-cluster power allocation strategy may not be optimal if the corresponding users with lower decoding orders have different minimum rate demands.

Proof.

The optimality condition in Remark 2 implies that

(14)

The equivalent channel gain , and thus (14) only depends on the minimum rate demand of users with lower decoding order and the channel gain of the cluster-head users (see Remark 1). Based on (14), for the case that the cluster-head users have exactly the same channel gains, i.e., , the equal inter-cluster power allocation strategy is optimal if and only if . According to the definition of in (9), one simple case that for some is considering different minimum rate demands for the users with lower decoding order. ∎

Corollary 1.

The equal inter-cluster power allocation strategy can be optimal for completely diverse channel gains of users in different clusters and heterogeneous minimum rate demands of users. The unique condition for the optimality of equal inter-cluster power allocation strategy is given by (14).

In contrast to OMA, the individual minimum rate demand of users with lower decoding order plays an important role on the performance of equal inter-cluster power allocation strategy, while their channel conditions do not significantly affect the performance of this strategy.

Iii-C Energy Efficiency Maximization Problem

In this subsection, we find a globally optimal solution for problem (4). According to Lemma 1, the feasible set of (4) is affine, so is convex. Moreover, the sum-rate function in the numerator of the EE function in (4) is strictly concave in . The denominator of the EE function is an affine function, so is convex. Therefore, problem (4) is called a concave-convex fractional program with a pseudoconcave objective function [36, 37]. The pseudoconcavity of the objective function in (4) implies that any stationary point is indeed globally optimal and the KKT optimality conditions are sufficient if a constraint qualification is fulfilled [36]. Hence, the problem (4) can be solved by using the iterative Dinkelbach algorithm [36, 37]. In this algorithm, we iteratively solve the following problem

(15)

where is the fractional parameter, and is strictly concave in . This algorithm is described as follows: We first initialize parameter such that . At each iteration , we set , where is the optimal solution obtained from the prior iteration . After that, we find by solving (III-C) in which . We repeat the iterations until , where is a tolerance tuning the accuracy of the optimal value. The pseudo-code of the Dinkelbach algorithm for solving (4) is presented in Alg. 2.

1:  Initialize parameter satisfying , tolerance (sufficiently small), and .
2:  while  do
3:     Set . Then, solve (III-C) and find .
4:     if   then
5:         break.
6:     end if
7:  end while
Algorithm 2 The Dinkelbach method for solving the energy efficiency maximization problem.

It is verified that Alg. 2 converges to the globally optimal solution with polynomial time complexity depending on the number of variable and constraints in (III-C). Similar to the transformation of (3) to (5), we define as the power consumption of cluster . The main problem (III-C) can be equivalently transformed to the following joint intra- and inter-cluster power allocation problem as

(16)

The feasible set of problems (5) and (16) is equal, thus the feasibility of (16) can be characterized by Lemma 1.

Proposition 3.

For any given feasible , the optimal intra-cluster power allocation in problem (16) can be obtained by using (7) and (8).

Proof.

When is fixed, the second term in (16) is constant. Hence, the objective function of (16) can be equivalently rewritten as maximizing users sum-rate given by , which is independent of . Hence, for any given feasible , problems (16) and (5) are identical. Accordingly, Proposition 1 also holds for any given and in (16). ∎

According to Proposition 3, the KKT optimality conditions analysis for the sum-rate maximization problem (3) holds for the EE maximization problem (4). Subsequently, similar to the equivalent transformation of (5) to (11), problem (16) can be equivalently transformed to the following parametric form of the inter-cluster power allocation problem as

(17a)
s.t. (17b)

where and are defined in (9). Note that since and are constants, the term can be removed from (16), so is removed in (17a) during the equivalent transformation. The differences between problems (11) and (17) are the additional term in , and also inequality constraint (17b). We prove that the activation of cellular power constraint (17b) depends on the value of , such that at the optimal point, the BS may not operate at its maximum power budget.

Proposition 4.

The convex problem (17) can be solved by using Alg. 1 if and only if .

Proof.

The optimal solution of (17) is unique if and only if the objective function (17a) is strictly concave. For the case that the concave function in (17a) is increasing in , we can guarantee that at the optimal point, the cellular power constraint (17b) is active. In other words, for the case that , for any , the optimal satisfies . In this case, the cellular power constraint (17b) can be replaced with , thus the optimization problem (17) can be equivalently transformed to the sum-rate maximization problem (11) whose globally optimal solution is obtained by Alg. 1. In the following, we find a sufficient condition, where it is guaranteed that , for any . The condition can be rewritten as After some mathematical manipulations, the latter inequality is rewritten as

(18)

The right-hand side of (18) is a constant providing an upper-bound for the region of such that . The inequality in (18) holds for any , if and only if , and the proof is completed. ∎

For the case that Proposition 4 does not hold, Alg. 1 is suboptimal for (17) since the objective function in (17a) is decreasing in a subset of the feasible domain of . In this case, similar to the transformation of (11) to (12), we define , where . Then, the problem (17) is rewritten as

(19a)
s.t. (19b)

where , , , and . The equivalent convex problem (19) can be solved by using the Lagrange dual method with subgradient algorithm or interior point methods (IPMs) [40]. The derivations of the subgradient algorithm for solving (19) is provided in Appendix B. Moreover, the derivations of the barrier algorithm with inner Newton’s method for solving (19) is provided in Appendix C. According to the above, depending on the value of at each Dinkelbach iteration, (III-C) can be solved by using Alg. 1 or subgradient/barrier method. The pseudo-codes of our proposed algorithms for solving (III-C) in Step 3 of Alg. 2 based on the subgradient and barrier methods are presented in Algs. 3 and 4, respectively.

1:  Calculate .
2:  if   then
3:      Find by using the water-filling Alg. 1.
4:  else  
5:      Initialize Lagrange multiplier , step size , and iteration index .
6:      repeat
7:          Set .
8:              Find by using .
9:          Update .
10:      until convergence of .
11:      Find by using .
12:  end if
Algorithm 3 The mixed water-filling/subgradient method for solving problem (III-C).
1:  Calculate .
2:  if   then
3:      Find by using the water-filling Alg. 1.
4:  else  
5:      Initialize , , , , , , and .
6:      repeat
7:          Set .
7:          Set .
8:          if   then
9:              break
10:          end if
11:          Initialize .
12:        while , or do
13:              
14:          end while
15:          while do
16:              
17:          end while
18:          Set .
19:          if   then
20:              break
21:          end if
22:          Set .
23:  end if
Algorithm 4 The mixed water-filling/barrier method for solving problem (III-C).

After finding via Algs. 3 or 4, we find the optimal intra-cluster power allocation by using (7) and (8). The duality gap in Alg. 4 after iterations is , where is the initial , and is the stepsize for updating in the barrier method. Therefore, after exactly barrier iterations, Alg. 4 achieves -suboptimal solution [40].

Corollary 2.

In both the sum-rate and EE maximization problems, in each cluster, only the cluster-head user deserves additional power, and all the other users get power to only maintain their minimal rate demands. The analysis proves that in the sum-rate maximization problem, the BS operates at its maximum power budget. However, for the EE maximization problem, the BS may operate at lower power depending on the condition in Proposition 4.

According to Corollary 2, Remark 1 and Fig.