Multiuser Joint Energy-Bandwidth Allocation with Energy Harvesting - Part I: Optimum Algorithm & Multiple Point-to-Point Channels
In this paper, we develop optimal energy-bandwidth allocation algorithms in fading channels for multiple energy harvesting transmitters, each may communicate with multiple receivers via orthogonal channels. We first assume that the side information of both the channel states and the energy harvesting states is known for time slots a priori, and the battery capacity and the maximum transmission power in each time slot are bounded. The objective is to maximize the weighted sum-rate of all transmitters over the time slots by assigning the transmission power and bandwidth for each transmitter in each slot. The problem is formulated as a convex optimization problem with constraints, where is the number of the receivers, making it hard to solve with a generic convex solver. An iterative algorithm is proposed that alternatively solves two subproblems in each iteration. The convergence and the optimality of this algorithm are also shown. We then consider the special case that each transmitter only communicates with one receiver and the objective is to maximize the total throughput. We develop efficient algorithms for solving the two subproblems and the optimal energy-bandwidth allocation can be obtained with an overall complexity of . Moreover, a heuristic algorithm is also proposed for energy-bandwidth allocation based on causal information of channel and energy harvesting states.
With the rapid development of energy harvesting technologies, a new paradigm of wireless communications that employs energy harvesting transmitters has become a reality . The renewable energy source enables the flexible deployment of the transmitters and prolongs their lifetimes . State-of-the-art techniques can provide fairly accurate short-term prediction of the energy harvesting process, which can be used to assist energy scheduling . To make the best use of the harvested energy for wireless communications, many challenging research issues arise . In particular, optimal resource (energy, bandwidth, etc.) scheduling is key to the design of an efficient wireless system powered by renewable energy sources.
For a single transmitter with energy harvesting, a number of works addressed the energy scheduling problem with non-causal channel state information. For static channels,  proposed a shortest-path-based algorithm for the energy scheduling.  analyzed the optimality properties based on the energy causality and the optimal energy scheduling algorithm was also provided. For fading channels, a staircase water-filling algorithm was proposed in  for the case of infinite battery capacity; with finite battery capacity,  studied the energy flow behavior with an energy harvesting device and proposed a directional water-filling method. Taking the maximum transmission power into account,  proposed a dynamic water-filling algorithm to efficiently obtain the energy schedule to maximize the achievable rate. Energy scheduling for multiuser systems with energy harvesting transmitters has also been considered. In , the general capacity region for a static multiple-access channel (MAC) was characterized without considering the constraints on the battery capacity and the maximum transmission power.  discussed the optimal power policy for energy harvesting transmitters in a two-user Gaussian interference channel. In , the optimal energy scheduling algorithm was proposed for a static broadcast channel with finite battery capacity constraint. Considering both the finite battery capacity and the finite maximum transmission power, the iterative dynamic water-filling algorithm was extended to the fading MAC channel . Moreover, the scheduling problem in the Gaussian relay channel with energy harvesting was discussed in .
In this paper, we consider a multiuser system with multiple transmitters, each powered by a renewable energy source. Each transmitter communicates with its designated receivers and is constrained by the availability of the energy, the capacity of the battery, and the maximum (average) transmission power. Moreover, a frequency band is shared by all transmitters and we assume orthogonal channel access to avoid interference. We aim to obtain the optimal joint energy-bandwidth allocation over a fixed scheduling period based on the available information on the channel states and energy harvesting states at all transmitters, to maximize the weighted sum of the achievable rate.
Consider the special case of equal weights and each transmitter communicates with only one receiver. Then, without energy harvesting, TDMA is optimal for the maximum unweighted sum-rate, i.e., at any time the link with the highest rate takes all bandwidth. However, for energy harvesting transmitters, TDMA is no longer optimal. This is because the finite battery capacity leads to energy discharge and waste by some transmitters that are not scheduled to transmit in a time slot. Therefore, to make the best use of the harvested energy, multiple transmitters should split the frequency band and transmit in a same slot. In this paper, we assume that the channel is flat fading and therefore each transmitter only needs to be allocated a portion of the total bandwidth.
We first consider the non-causal case, i.e., the energy harvesting and the channel fading can be predicted for the scheduling period, and formulate a convex optimization problem with variables and constraints, where is the number of receivers and is the number of scheduling time slots. Since the computational complexity of a generic convex solver becomes impractically high when the number of constraints is large , we will develop an iterative algorithm that alternates between energy allocation and bandwidth allocation. We will show that this algorithm converges to the optimal solution of the joint energy-bandwidth scheduling problem. For the special case that each transmitter only communicates with one receiver and all weights are equal, optimal algorithms to solve the energy and bandwidth allocation subproblems are also the optimal energy-bandwidth allocation algorithm that obtained a computational complexity of . We then consider the causal case, where the harvested energy and the channel gain can only be observed at the beginning of the corresponding time slot. We propose a suboptimal energy-bandwidth allocation algorithm that follows a similar structure of the noncausal optimal solution. Simulation results demonstrate that both the proposed non-causal and causal algorithms achieve substantial performance improvement over some heuristic scheduling policies.
The remainder of the paper is organized as follows. In Section II, we describe the system model and formulate the joint energy-bandwidth scheduling problem. In Section III, we propose an iterative algorithm to obtain the optimal energy-bandwidth allocation and prove its convergence and optimality. Optimal algorithms for energy allocation and bandwidth allocation, respectively, are further developed in Section IV for the special case that each transmitter only communicates with one receiver and all weights are equal. In Section V, a suboptimal causal algorithm is proposed. Simulation results are provided in Section VI. Finally, Section VII concludes the paper.
2System Model and Problem Formulation
Consider a network consisting of transmitters and receivers sharing a total bandwidth of Hz, where and each transmitter may communicate with multiple receivers. We assume a scheduling period of time slots and no two transmitters can transmit in the same time slot and the same frequency band. Denote as the normalized bandwidth allocation for link in time slot . We consider a flat and slow fading channel, where the channel gain is constant within the entire frequency band of Hz and over the coherence time of seconds, which is also the duration of a time slot. Assuming that each time slot consists of time instants, we denote as the symbol sent to the receiver of link at instant in slot . The corresponding received signal at receiver is given by
where represents the complex channel gain for link in slot , and is the i.i.d. complex Gaussian noise. We denote and denote as the transmission energy consumption for link in slot . Without loss of generality, we normalize both and to ; then, and become the transmission power and the allocated bandwidth of link in slot , respectively. For link , the upper bound of the achievable channel rate in slot can be written as . Moreover, we denote as the scheduling period, as the set of transmitters, and as the set of receivers. Further, we denote as the set of receivers of transmitter , where for all .
Assume that each transmitter is equipped with an energy harvester and a buffer battery, as shown in Figure 1. The energy harvester harvests energy from the surrounding environment. We denote as the total energy harvested up to the end of slot by transmitter . Since in practice energy harvesting can be accurately predicted for a short period , we assume that the amount of the harvested energy in each slot is known. Moreover, the short-term prediction of the channel gain in slow fading channels is also possible . Therefore, we assume that and are known non-causally before scheduling. Note that such non-causal assumption also leads to the performance upper bound of the system. We will relax this assumption and consider causal knowledge of the channels and energy harvesting in Section V.
For transmitter , assuming that the battery has a limited capacity and is empty initially, then the battery level at the end of slot can be written as
where represents the energy discharge (waste) for transmitter in slot . Moreover, must satisfy for all .
We assume that each transmitter has a maximum per-slot transmission energy consumption, , such that for . With the maximum transmission energy and the limited battery capacity, some of the harvested energy may not be able to be utilized, and is therefore wasted, i.e., may necessarily be strictly positive in some slots. Then, the constraints on the battery level can be written as
Moreover, we denote as the discharge allocation. Note that, we assume controllable energy discharge, i.e., the energy can be discharged and wasted anytime, even when the battery is not full.
Define . We use upper bounds of the achievable channel rate over a weighted sum of the links and slots as the performance metric, given by
where is the energy allocation, is the bandwidth allocation, and is the weight set. In particular, when for all , becomes the throughput of the network.
Note that, both and can be zero in . However, if , the channel rate of link in slot is zero, even if the energy allocation , thus is actually wasted. However, we still treat the pair as feasible as long as .
We formulate the following energy-bandwidth allocation problem:
for all and .
2.3Optimal Energy Discharge Allocation
To efficiently solve the problem in -, we consider a two-stage procedure. In the first stage, we obtain the optimal energy discharge allocation such that
with the constraints in . And in the second stage, we use and define the energy expenditure for transmission as
Then we solve the following problem:
for all and .
We consider the following greedy strategy to obtain the energy discharge allocation by assuming that each transmitter transmits at the maximum power in each slot, i.e.,
for all .
Note that, following , the total discharged energy is minimized and thus the amount of the energy used for transmission is maximized. Intuitively, this way the feasible domain becomes the largest, providing the best performance for transmission energy scheduling. Specifically, given a feasible bandwidth allocation , the achievable rate of each link is non-decreasing with respect to the transmission energy, and the battery of each transmitters operates independently. Therefore, following the same lines of the proof in , the optimality of can be established. In particular, using any feasible energy discharge corresponding to the minimal energy wastage, the optimal value of is same, which is no less than the optimal value under any feasible energy discharge allocation with non-minimal energy wastage.
Note that, is continuous and jointly concave with respect to and for . Then, the problem in - is a convex optimization problem and can be solved by a generic convex solver, whose complexity becomes impractically high when the number of constraints is large , which in this case is . To reduce the computational complexity, we will develop an efficient algorithm in this paper, which exploits the structure of the optimal solution.
2.4K.K.T. Conditions for Non-Zero Bandwidth Allocation
The problem in - is a convex optimization problem with linear constraints. When the objective function is differentiable in an open domain, the K.K.T. conditions are sufficient and necessary for the optimal solution . Note that, is non-differentiable at . To use the K.K.T. conditions to characterize the optimality of the problem in -, we consider the following approximation:
for all , where is a small positive number. In particular, is the original problem in -.
By introducing the auxiliary variables , , and and converting the constraints in into the Lagrangian multiplier, we can define the Lagrangian function for as
Then, the following K.K.T. conditions, which are sufficient and necessary for the optimal solution to the convex optimization problem in -, are obtained from the Lagrangian function:
together with the constraints in , and for all , , and , where in
3Iterative Algorithm and its Optimality
In this section, we will first decompose the energy-bandwidth allocation problem in - into two subproblems, and then propose an iterative algorithm to solve . We will prove that the iterative algorithm converges to the optimal solution to the problem in -.
To efficiently solve problem in -, we first decompose it into two groups of subproblems, corresponding to energy allocation and bandwidth allocation, respectively.
Given the bandwidth allocation , for each , obtain the energy allocation by solving the following subproblem:
Given the energy allocation , for each , obtain the bandwidth allocation by solving the following subproblem:
To obtain the optimal solution to the original problem in -, we propose an iterative algorithm that alternatively solves EP for all and BP for all , with a diminishing over the iterations. To perform the algorithm, we initially set , and solve to obtain the initial . In each iteration , we first solve BP to update for all , where is a pre-specified positive value; with the updated , we then solve EP to update for all .
The proposed iterative algorithm is summarized in Algorithm 1 and its block diagram is shown in Figure 2.
|, , , Choose any , Solve EP for all to generate the initial|
|Specify the maximum number of iterations , the convergence tolerance|
|Solve BP to update for all|
|Solve EP to update for all|
In the next subsection, we will show that Algorithm 1 converges and the pairwise optimal and can be obtained, which is also the optimal solution to the problem in -.
We note that, is a convex optimization problem with variables and constraints. The computational complexity of using the generic convex solver is non-linear with respect to the number of the variables and constraints, which may be impractically high when and become large. Using Algorithm 1, the optimal solution to can be obtained by solving convex optimization subproblems which contains or variables and constraints. Therefore, the overall computational complexity can be significantly reduced with Algorithm 1 for large and .
3.2Proof of Optimality
We first give the following proposition.
Given a pair , if is the optimal solution to EP for all given , and is the optimal solution to BP for all given , we say that and are pairwise optimal for . We also note that, for each subproblem, its K.K.T. conditions form a subset of those of given the other primal variables, where any two subsets contain no common dual variable. Then, if the primal variables are pairwise optimal, the K.K.T. conditions in each corresponding subset are satisfied and hence all K.K.T. conditions of are satisfied, i.e., the pairwise optimal solution is also the optimal energy-bandwidth allocation for .
We note that, for , when , the objective value is zero for all feasible bandwidth allocations. Therefore, we can fix as the optimal bandwidth allocation for this case in Algorithm 1. Then, by Proposition ?, we have that the optimal solution to each subproblem in Algorithm 1 is unique. The next theorem establishes the optimality of Algorithm 1. The proof is given in Appendix A.
Note that, the convergence is due to the expansion of the feasible domain by reducing resulting in the increasing objective value over iterations. The optimality can be proved by first verifying the pairwise optimality of the solution upon convergence and then showing it cannot be suboptimal.
4Special Case: Throughput Maximization for Multiple Point-to-Point Channels
In this section, we consider the special case that each transmitter can only communicate with one receiver and all links have the same weight, i.e., and for all . The energy and bandwidth allocation subproblem can be rewritten as
4.1Solving Ep: Discounted Dynamic Water-Filling
Given , since EP is a subproblem of the problem in -, its K.K.T. conditions form a subset of those of the original problem, given by , and .
To develop an efficient algorithm, we first rewrite the K.K.T. condition in as
Since the energy allocation must satisfy , can be further written as
Theorem ? gives the necessary and sufficient conditions for the optimal energy allocation given any bandwidth allocation. Given the set of BDP/BFPs corresponding to the optimal energy allocation, the optimal energy allocation and the corresponding the water level for the segment between two adjacent BDP/BFPs, and , can be written as
where is the water level of a segment such that
with being an indicator function given by
Specifically, - represent the water-filling operation in a segment between two optimal BDP/BFPs, as mentioned in Theorem ?. Also, ensures that with the energy allocation the boundary points and are the desired BDP/BFPs.
In , to obtain the optimal energy schedule for a single energy harvesting transmitter, a dynamic water-filling algorithm is proposed that recursively performs the “forward search” and “backward search” operations on a water-filling attempting basis, i.e., in the above two operations we perform conventional water-filling on each “target” segment (some consecutive slots). Using the dynamic water-filling algorithm, the set of the optimal BDP/BFPs can be obtained and the corresponding water level between two adjacent BDP/BFPs satisfies the optimality conditions for the energy scheduling problem. Then, it is easy to verify that, by replacing the conventional water-filling in the dynamic water-filling algorithm by the water-filling operations defined in -, we can still obtain the optimal BDP/BFPs set for the energy allocation subproblem and thus the optimal energy allocation can be finally obtained. We call the new algorithm the discounted dynamic water-filling algorithm, whose computational complexity is .
4.2Solving Bp: Bandwidth Fitting Algorithm
We first note that, when and , the channel rate achieved by transmitter in slot is zero. Therefore, in a slot such that , any feasible bandwidth allocation is optimal, achieving the maximum channel rate . However, in a slot where , in order to maximize the channel rate, the transmitter with zero energy allocation must be allocated with the minimal bandwidth, i.e., . We denote the energy allocation as the non-zero energy allocation and, in the remainder of this subsection, we will obtain the optimal bandwidth allocation given a non-zero energy allocation.
Since BP is a subproblem of the problem given in -, its K.K.T. conditions form a subset of those of the original problem, given by , and .
Given a non-zero energy allocation, since we know that the transmitter with zero energy allocation should be allocated with the minimal bandwidth, we rewrite the K.K.T. condition related to the transmitters with the non-zero energy allocation in as
for all , and all such that .
Denoting as the solution to the following equation,
from -, we then have
By , we have that must be zero when . Then we divide into two sets: and . Then, can be rewritten as
Also, we have
Substituting and into the constraints , we further have
Next, we characterize the optimality condition for the bandwidth allocation problem BP given the non-zero energy allocation, as follows.
Intuitively, Theorem ? states that the optimal bandwidth allocation should be proportional to the transmission “condition”, i.e., . In particular, if the desired bandwidth allocation for transmitter is less than the minimal requirement , should be set as the minimal requirement . Based on Theorem ?, we propose the following iterative bandwidth fitting algorithm. Initially, we set and . In each iteration, we calculate the bandwidth allocation by with the current and . We denote as a “violation set”, containing the elements in that violate the definition of . Then, we move all from to . This iterative process ends when is empty. Finally, with the obtained and , the optimal allocation can be calculated by -.
The procedure of the algorithm is summarized as follows:
|FOR such that|
|Calculate by for all|
|Set the violation set|
|Move all from to|
|Obtain with and by - for all such that|
Note that, Algorithm 2 will terminate in at most iterations since the elements transfer between and is one-directional. Moreover, for all such that , at the end of the last iteration, since is empty, we have that the condition in is satisfied by all and we have for all . Also, for all other , obviously we have . If is also satisfied, we can further claim that with the obtained and , Algorithm 2 gives an optimal bandwidth allocation.
The next result shows that, at the end of each iteration (including the last iteration), with the obtained , is satisfied. The proof is given in Appendix B.
With Proposition ?, we conclude that the bandwidth allocation obtained by Algorithm 2 is optimal. Moreover, since the number of iterations is bounded by , the computational complexity of Algorithm 2 is .
5Suboptimal Algorithm with Causal Information
In Section III, we proposed an iterative algorithm to obtain the optimal energy-bandwidth allocation with non-causal information of the channel gains and the harvested energy, whose performance can also serve as an upper bound on the achievable rate. In this section, we consider the case that the channel fading and energy harvesting are not predicable, i.e., their realizations can only be observed causally at the beginning of the corresponding slot. We will propose a heuristic algorithm to obtain the suboptimal energy-bandwidth allocation that follows the structure of the optimal solution. For simplicity, we still focus on the throughput maximization problem for point-to-point channels considered in Section IV.
We first give the structure of the optimal solution for the problem in -.
Lemma ? provides the structure of the optimal energy-bandwidth allocation, in which the water level is the only parameter affected by the future channel fading and energy harvesting. Specifically, if the energy harvesting and channel gains are predictable, then the optimal can be obtained, as in the proposed non-causal algorithm, where if , and if . In other words, in , does not affect the value of when the optimal water level is given. However, when the energy harvesting and channel fading processes are unpredictable, the optimal is hard to obtain. Note that essentially acts as an adjusting factor to mitigate the energy waste caused by the non-optimality of , i.e., if the suboptimal water level is lower than the optimal one and therefore causes the energy waste, we can try to utilize the wasted energy for transmission. Then, we use the potentially wasted energy as the adjuster, given by
where is the energy harvested energy in slot . Specifically, becomes the actual energy wastage if the water-filling fashion in is followed using the water level .
Based on Lemma ?, we design an adaptive water-filling algorithm, aiming to obtain a suboptimal energy-bandwidth allocation, which follows the structure of the optimal solution given in Lemma ?. With the proposed algorithm, except for the calculation of the water levels, all other optimality conditions are approached by the obtained energy-bandwidth allocation. Specifically, to avoid the use of the future information, the water levels are calculated by a heuristic method.
The proposed algorithm is an online algorithm. Initially, we set a water level for each transmitter . At the beginning of slot , we check the battery level of each transmitter. If the battery is empty or full, we decrease or increase the water level by a factor, e.g., . Otherwise, we keep the water level unchanged. Then, based on the water level , we calculate the energy allocation and bandwidth allocation by solving the equations , and . In particular, substituting into , there are two equations and two variables, which can be solved numerically.
Moreover, we propose the following choices of the initial water level and the factor ,
The algorithm is summarized as follows:
|Current water level and battery level|
|Updated water level and battery level|
|3:||At the beginning of each slot|
|Solve the equation group of , and to obtain|
Suppose that there are transmitters in the network and each communicates with one receiver, and we assume for . We set the scheduling period as slots. For each transmitter , we set the initial battery level and the maximum battery capacity units. Assume that the harvested energy follows a truncated Gaussian distribution with mean and variance of , and the fading channel parameter follows the standard complex Gaussian distribution, i.e., , so that .
For comparison, we consider three scheduling strategies, namely, the greedy policy, the TDMA greedy policy, and the equal bandwidth policy. For the greedy policy, each transmitter tries to consume the harvested energy as much as possible in each slot, as calculated by . Then, the central controller allocates the bandwidth to each transmitter by using the iterative bandwidth fitting algorithm (i.e., Algorithm 2). For the TDMA greedy policy, each transmitter uses the maximum possible energy to transmit in each slot, and the central controller allocates the entire bandwidth to the transmitter with the maximum . For the equal bandwidth policy, the central controller allocates each transmitter equal bandwidth and then each transmitter uses the optimal energy allocation.
To evaluate the performance of different algorithms, we consider two scenarios, namely, the energy-limited scenario, where the maximum transmission power is units per slot, and the power-limited scenario, where the maximum transmission power is units per slot. Moreover, the convergence threshold in Algorithm 1 is set as , and the initial water level and the parameter in Algorithm 3 are set as and , respectively. We compare the achievable rates of different algorithms under different mean values of the harvested energy. In the energy-limited scenario, the transmitter has more freedom to schedule the harvested energy to be consumed in each slot because of the large maximum transmission power. One the other hand, in the power-limited scenario, the harvested energy would be consumed in the future slots since the maximum transmission power is reached more frequently. Furthermore, for both scenarios, when the energy harvesting parameter is small, it corresponds to the “energy-constrained” condition, where the scheduling is mainly constrained by the energy availability. And when is large, it corresponds to the “power-constrained” condition, where the scheduling is more constrained by the maximum transmission power.
Before we compare the performance of different algorithms, we first illustrate the convergence behavior of Algorithm 1 in Figure 3 with . It is seen that Algorithm 1 converges (the relative error is less than ) within and iterations for and , respectively. Next, we set and and give the -slot snapshots (slot 20 - slot 40) of the obtained energy-bandwidth allocation in Figure 4, Figure 5 and Figure 6. Specifically, Figure 4 and Figure 5 illustrate the relationship among the water level , transmission energy and the battery level obtained by Algorithm 1 and Algorithm 3, respectively, showing that although both algorithms follow the water-filling structure with the dynamic water levels, their water levels vary according to different rules, based on the dynamic of the battery. Moreover, the optimal bandwidth allocation obtained by Algorithm 1 is illustrated in Figure 6, and we can see that most of the time the channel is shared by multiple transmitters to maximize the sum-rate.
We then run the simulation times to obtain the rates given by various scheduling strategies, as well as by the optimal schedule solved by a general convex solver, shown in Figure 7 and Figure 8, for the energy-limited scenario and the power-limited scenario, respectively. It is seen from that for both scenarios the proposed non-causal iterative algorithm (Algorithm 1) achieves the same performance as that corresponding to the optimal energy-bandwidth allocation solved by the generic convex solver, corroborating the optimality of Algorithm 1 as stated by Theorem 2. Also, the proposed causal algorithm (Algorithm 3) performs worse than the optimal policy but still better than the other heuristic policies. Moreover, for all policies, the performance is improved as the mean of the harvested energy increases.
From Figure 7, for the energy-limited scenario, the performance gap between the TDMA greedy policy and the optimal solution increases as the mean of the harvested energy increases. It is because when the mean of the harvested energy is high, due to the maximum transmission power and battery capacity constraints, the single-user transmission of TDMA results in significant energy waste by the non-transmitting transmitters in each slot.
On the other hand, from Figure 8, for the power-limited scenario, the performance gap between the optimal solution and some of the suboptimal algorithms (Algorithm 3 and the greedy policy) decreases as the mean of the harvested energy increases. It is because when the harvested energy is ample, the optimal energy allocation achieves the maximum transmission power more frequently and approaches the greedy policy. Also, in the power-limited scenario, the TDMA greedy policy performs significantly worse than other algorithms since the low maximum transmission power results in a lot of energy waste in the absence of any channel sharing.
Moreover, as expected, the performance in the energy-limited scenario is better than that in the power-limited energy for all policies. This is because the lower maximum transmission power restricts the flexibility of the energy scheduling and causes waste of energy due to the limited battery capacity.
In this paper, we have considered the joint energy-bandwidth allocation problem for multiple energy harvesting transmitters over time slots. This problem is formulated as a convex optimization problem with variables and constrains, where is the number of the receivers and is number of the slots in a scheduling period, which is hard to solve with a generic convex tool. We have proposed an energy-bandwidth allocation algorithm that iterates between solving the energy allocation subproblem and the bandwidth allocation subproblem, and the convergence and the optimality of the iterative algorithm have been shown. When each transmitter communicates with one receiver and the sum-rate is unweighted, the discounted dynamic water-filling algorithm and the bandwidth fitting algorithm are proposed to optimally solve the energy and bandwidth allocation subproblems, respectively. Moreover, a heuristic algorithm is also proposed to obtain the suboptimal energy-bandwidth allocation causally and efficiently, by following the structure of the optimal energy-bandwidth solution. In a companion paper, we will consider multiple broadcast channels under the joint energy-bandwidth allocation framework and develop efficient algorithms for solving the two subproblems under both orthogonal and non-orthogonal access.
8Proof of Theorem
We note that, the feasible domain of expands with iterations while the feasible domain of remains unchanged. Since we successively solve the maximization problems EP and BP in iteration , we have that the objective value is non-decreasing over the iterations. On the other hand, the objective function is upper bounded by therefore the algorithm converges. Since the feasible domain of is a closed set for , at the converged point , we can find the corresponding and which are pairwise optimal for otherwise can be increased by performing another iteration. Specifically, if is reached within finite iterations , and are pairwise optimal for ; otherwise, and are pairwise optimal for .
We first consider the case that is reached within finite iterations . Since is reached within finite iterations , we have that and are pairwise optimal for both and . Then, by Theorem ?, is the optimal solution to and . Note that, since the feasible domain of is expanded from that of by decreasing , is not on the boundary of , i.e., the equality of does not hold. Therefore, continually expanding the feasible domain of by decreasing from to , remains at a local optimal point and thus also a global optimal point according to the domain’s convexity.
We then consider the case that can only be approached with infinite iterations. For this case, we have that and are pairwise optimal for . However, we note that, even so, is not necessarily optimal solution to when for some and . To show the optimality of , we use the proof by contradiction. Suppose that and are pairwise optimal for but is not an optimal solution to . Denote as the set of the links with zero bandwidth allocation. Since and are pairwise optimal for , we have that all K.K.T. conditions hold except for the links , i.e., excluding the links in , is optimal for by Theorem ? (excluding the links in , the problem