Sensor Physical Interpretation, Signal and Artificial Intelligence Processing
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Ying Luo, Min Zeng, Hong Jiang, Bin Han, "EnergyEfficient TimeDomain Equilibrium Scheduling and Optimization Scheme for Energy HarvestingPowered D2D Communication", Journal of Sensors, vol. 2020, Article ID 8839681, 16 pages, 2020. https://doi.org/10.1155/2020/8839681
EnergyEfficient TimeDomain Equilibrium Scheduling and Optimization Scheme for Energy HarvestingPowered D2D Communication
Abstract
Energy Harvesting (EH) powered DevicetoDevice (D2D) Communication underlaying Cellular Network (EHDCCN) has been deemed as one of the basic building blocks of Internet of Things due to its green energy efficiency and adjacent communication. But available energy will be one of the biggest obstacles when implementing EHDCCN due to the immaturity of EH technology and the volatility of environmental energy resources. To improve energy utilization, this study investigates an efficient scheduling and power allocation scheme about transmission load equilibrium in the time domain. Accordingly, a shortterm Sum Energy Efficiency (stSEE) maximization problem for EHpowered D2D communication is modelled, while ensuring a fundamental transmission rate requirement of cellular users. Consequently, the optimization problem is a nonconvex mixed integer nonlinear programming problem. Thus, we propose a twolayer convex approximation iteration algorithm which can obtain a feasible quasioptimal solution for the stSEE problem. Simultaneously, a twostep heuristic algorithm in a slotbyslot fashion is also developed to acquire a suboptimal solution without requiring statistical knowledge of channel and energy arrival processes. Simulated analysis indicates that the shortterm scheduling strategy can obtain better performances in terms of energy efficiency and transmission rate than conventional realtime scheduling scheme. Besides, the maximum scheduled number of EHD2D pairs underlaying one cellular user under different EH efficiency is analysed, which can give us a theoretical reference about the deployment of future EHDCCN.
1. Introduction
The dawn of Internet of Things (IoT) installed for industry production, smart housing, and environmental monitoring has given birth to billions of mobile wireless devices [1]. Ericsson predicts that there will be about 15 billion mobile devices in 2021, and most of them will be lowpower devices with shortrange communication [2]. The everincreasing proliferation of wireless devices, together with an exponential rise in users’ data demand, is already creating an urgent need for wireless cellular networks to design some new technologies that can attain desired transmission rates and, meanwhile, can achieve green communications [3, 4]. Recent emphasis on green communications has generated great interest in the investigations of energy harvesting (EH) powered wireless networks [5–7]. EH technique can harvest energy from environmental energy resources to prolong the lifetime of wireless communication devices with low power consumption and limited battery life. On the other hand, DevicetoDevice (D2D) communication has been viewed as a promising paradigm that can offload traffic from cellular networks by communicating directly with each other in close proximity via multiplexing the spectrum resources being assigned to cellular users (CUs) [8, 9]. Accordingly, EHpowered D2D Communication underlaying Cellular Network (EHDCCN), which combines the characteristics of shortrange transmission and green communication, will be an attractive way to keep a better transmission service quality with green energy resources. Nevertheless, this sustainable energy support technique will result in intermittent energy supply issue, which do not exist in the conventional D2D communication systems with fixed energy sources. Consequently, how to efficiently utilize and manage unreliable energy so as to satisfy different transmission demand is the most urgent challenge to achieve the green D2D communication paradigm.
1.1. Related Works
A lot of research has paid their attentions to the challenges caused by the uncertain energy supply technique from different aspects, such as access control and resource management, for the EHDCCN.
In view of access control, Darak et al. design an online learning handover scheme between D2D mode and Radio Frequency EH (RFEH) mode based on subband statistics in D2DRFEH communication [10]. For cognitive and EHbased D2D transmission, Sakr and Hossain propose two spectrum access policies for the cellular network, namely, random and prioritized access policies, to evaluate transmission and outage probability for D2D and cellular users [11]. The authors of [12] develop a D2D communication provided by EH Heterogeneous cellular Network (D2DEHHN), where User Equipment Relays (UERs) harvest energy from an access point to support D2D communication. This paper derives the proper distribution of RFEHpowered UER and proposes an efficient UER selection method.
To adapt the uncertainty in energy supply, researchers have also devoted their efforts to the design of efficient resource management schemes in terms of power allocation, spectrum matching, and time sharing for EHDCCN. Tutuncuoglu and Yener study the power allocation policies for EHsupported transmitters by optimizing the sum rate [13], which is similar to a onetoone spectrum sharing model in D2D communication underlaying cellular network [14]. Yet, the Quality of Service (QoS) requirement, which mainly refers to transmission rate demand, is not involved in the proposed policy. Similarly, the sum rate maximization problem for D2D communication under a downlink resource multiplex system in the presence of multiple CUs and EHpowered D2D links is studied in [15]. To ensure energyefficient spectrum resource assignment, Ding et al. investigate the energy cost minimization problem [16]. Considering the power allocation and timesharing spectrum occupation management, HadziVelkov et al. maximize the overall cellular network transmission rate based on the statistical average of the harvested energy [17].
The above research works have addressed many challenges caused by the unstable and unreliable power supply of EH from different aspects. Nevertheless, those works are mainly based on a onetoone spectrum sharing mode where one CU’s radio resource is multiplexed by one D2D pair. In this sharing mode, the CU’s spectrum will be vacant when the available energy of the EHpowered D2D pair (EHDP) cannot meet the energy consumption requirement. Hence, by taking the high spectrum efficiency demand into account, onetomultiple sharing mode, namely, one CU and multiple EHDPs sharing one radio spectrum, is needed. In this way, the cellular spectrum gap caused by the energy deficiency of the single EHDP can be filled up. In practice, multiple D2D pairs can be allowed to share the same resource with the cellular users as long as the interference of D2D communications is not harmful to the cellular links [18–20]. However, under the onetomultiple sharing mode, due to the variation of EH efficiency, environmental energy resources, and channel status, the transmission requests, which are relied on available energy, among EHDPs at different time slots may be unbalanced. Since the traffic is delaytolerant, the mutual interference among users can be efficiently decreased by balancing the transmission requests among time slots and finally improve energy efficiency. The following section will take two EHpowered D2D pairs underlaying cellular network as a simple example to illustrate our main motivations.
1.2. Motivations
Before describing our motivations, we must make some important statements. We only consider that the D2D user (DU) has EH capability [21]. Meanwhile, for simplicity, suppose that the traffic pattern of each user is full buffer, and all users operate in a timeslotted fashion and are synchronous.
When multiple EHDPs multiplex the same spectrum resource, the transmission request of EHDPs at the different time slots may be unbalanced. As illustrated by Figure 1, two EHDPs and are permitted to share a radio spectrum with one CU. From Figure 1(a), the available energy of users is different due to various channel interference conditions and energy conversion efficiency. When the available energy of EHDP reaches its transmission power threshold , EHDPs will launch the transmission request. Thus, when the two EHDPs and initiate the transmission request at the same time slots, such as time slot 1, the conventional realtime transmission strategy will let them transmit data at the same time by the corresponding power allocation scheme under the interference constraint. However, each of them may not multiplex the spectrum resource in the next time slot (e.g., time slot 2) because of energy supply or energy consumption. However, in the shortterm timedomain equilibrium strategy, as depicted in Figure 1(b), either of the two EHpowered D2D pairs can be assigned to multiplex the spectrum resource of CU in the time slot 2. As a result, the interference between the two D2D links in the time slot 1 will be eliminated. Hence, to avoid unnecessary consumption of the harvested energy, the interference among users is required to be appropriately managed in onetomultiple sharing scenarios [5].
As mentioned above, by fully considering the available energy and channel status (channel status mainly refers to mutual interference conditions among users (including CU and EHDPs) in this study), how to balance the transmission loads among different time slots under EHDCCN with onetomultiple sharing mode so as to improve the performance of EHDCCN is the key concern of this study. As far as we know, the considered shortterm timedomain energyefficient equilibrium program is the first attempt to do so in EHDCCN.
1.3. Contributions and Organizations
As previously described, this study focuses on designing an energyefficient transmission scheduling and power allocation scheme so as to increase the performance of the EHDCCN with the onetomultiple radio resource sharing mode. Thus, our main contributions can be divided into three main areas:(i)Firstly, this study optimizes a shortterm Sum Energy Efficiency (stSEE) problem about EHpowered D2D communication to realize the energyefficient scheduling scheme. Simultaneously, the available energy and transmission rate constraints of both CUs and EHDPs are also considered in the optimization problem(ii)Subsequently, a twolayer convex approximation iteration algorithm (CAIA), which consists of an outerlayer iteration algorithm (OLIA) and an innerlayer convex approximation (ILCA) algorithm, is proposed to obtain a feasible quasioptimal solution for the modelled stSEE maximization problem which is a nonconvex mixed integer nonlinear programming (MINLP) problem(iii)Thirdly, a twostep heuristic algorithm, the timedivision scheduling scheme (TDSS), is also developed to acquire a suboptimal solution without requiring statistical knowledge of channel and EH processes. Remarkably, TDSS not only can acquire a suboptimal solution for the stSEE problem but also has a lower computational complexity
The rest of this study is organized as follows. In Section 2, we describe the system model in detail and formulate the stSEE maximization problem. The two feasible algorithms, CAIA and TDSS, are elaborated by Sections 3 and 4, respectively. The numerical simulation performance results and the computational complexity of the proposed algorithms are presented and analysed in Section 5. In Section 6, we conclude this study.
2. System Description and Problem Formulation
This section introduces the system model and formulates the considered resource scheduling problem. To facilitate the understanding, some important notations in this study are listed in Table 1.

2.1. System Model
In what follows, we assume that spectrum matching has already finished. This is to say that multiple EHDPs have already been allocated to one dedicated CU in some particular optimization conditions, e.g., EE maximization [22]. Thus, as shown in Figure 2, a typical single cellular network consisting of a Base Station (BS) and EHDP/CU groups is considered. Suppose that the system utilizes a certain number of orthogonal spectra, then we can divide the spectra into the same number of EHDP/CU groups. Namely, the communication links in the same group transmit on the same spectrum, and the communication links in different groups use the orthogonal one. Let represent BS, denote CU in the th EHDP/CU group, and be a pair of D2D users in the EHDP set of the th EHDP/CU group. In the th EHDP/CU group, EHDPs share the uplink transmission link of the th CU () to transmit. According to the energy assumption in Section 1.2, the transmitter of each EHDP is supplied by EH technique and has a battery to store harvested energy. Meanwhile, the available power of the receiver of each EHDP is deemed as unlimited due to the lowpower property of the decoding process.
Generally speaking, as illustrated by the EHDP/CU group 2 of Figure 2, each EHDP transmission will simultaneously cause interference to receivers of cellular and EHDP links. Likewise, cellular transmission will generate interference to the EHDP receivers. Assume that the entire system executes on a slotbyslot basis. Accordingly, in any time slot, the instantaneous transmission rate of the cellular and EHDP links can be given by and , respectively,where is the indicator parameter, 1 indicates the th D2D pair chosen to transmit in time slot , and 0 indicates the th D2D pair not chosen. and are the corresponding transmission power of CUs and EHDPs in time slot , respectively. denotes the channel gain between nodes and . means the noise power and equals to , where is the density of noise power and is the uplink channel bandwidth of each group.
2.2. Energy Model
As demonstrated in Figure 3, at time slot , the transmitter of each EHDP needs to harvest energy from the environmental energy resources, to store the energy in a battery, and to use the available energy to finish transmission. We study the condition that the energy arrival process in each EHDP is i.i.d. For the th EHDP, units of energy can be harvested in time slot , where . is the time sequence of harvested energy in time slots and obeys an i.i.d. Bernoulli process:
and are called EH efficiency of the th EHDP. Notably, the concepts of terms of energy and power in this study are equivalent in the unit time slot.
In Figure 3, units of energy are harvested by EH technique and added to the battery at each time slot. Accordingly, units of energy will be consumed for data transmission of the th device. The existing energy of the th EHDP in a battery is defined as . Thus, a cumulative power constraint can be expressed as follows:
Suppose that the harvested energy can be stored without any loss and used for only communication purposes from the battery. Meanwhile, the battery capacity is large enough to hold every quanta of harvested energy. This assumption is especially valid for the current state of technology in which batteries have very large capacities compared to the energy harvesting efficiency [23]. Furthermore, assume that all state information including Channel State Information (CSI) and Energy State Information (ESI) can be obtained by BS so that BS has the control capability in terms of transmission scheduling and power allocation [24, 25].
2.3. Mathematical Model
The shortterm Sum Energy Efficiency (stSEE) for EHpowered D2D communication optimization problem is formulated as :where is the indicator parameter set about whether the th EHDP is allowed to transmit or not in an instantaneous time slot . and are the sets of transmission power of CU and EHDPs, respectively. To avoid serious mutual interference, as represented by (4a) and (4d), the maximal transmission power in each time slot should be limited at the CU and EHDP side, respectively. In this study, multiple EHDPs can share the CU’s uplink channel resource to transmit. Thus, the mathematical model should guarantee the minimum QoS of CU. So, (4b) defines a threshold about transmission rate demand for CU. Similarly, as (4e) shows, the chosen EHDPs allowed to transmit in the th time slot must have a minimum transmission rate requirement. At last, (4c) denotes the available energy constraint of EHDP.
2.4. Problem Decoupling
The maximization problem can be decoupled into subproblems according to the spectrum orthogonality. Hence, for any of EHDP/CU groups , we have following optimization problem :where , , and .
3. TwoLayer Convex Approximation Algorithm (TLCA)
As described in , some of the variables (the components of and ) can be realvalued, whereas the other variables (the components of ) are binaryvalued. Furthermore, the optimization utility function (5) and restraints (4b) and (4e) depend on and , which have nonconvex feature (the simple proof of the nonconvex ofandcan be seen in Appendix A). So, (5) is a nonconvex MINLP problem, by which computational complexity is NPhard. An intuitive proof of NPhardness is that MINLP includes ILP problem (Formula (5) can be reduced to an ILP problem when the power allocation variablesandare fixed), which has been proved to be NPhard [26, 27]. Based on the above discussion, we design a twolayer convex approximation iteration algorithm (CAIA), which contains an outerlayer iteration algorithm (OLIA) and an innerlayer convex approximation (ILCA) algorithm, to obtain a feasible quasioptimal solution. The OLIA first equivalently transform the fractional programming problem. Secondly, the ILCA is implemented to approximately convert the nonconvex MINLP optimization into a convex one.
3.1. OuterLayer Iteration Algorithm (OLIA)
First of all, the target of , (5), is a nonlinear fractional programming paradigm [28], which can be transformed into an equivalent multiobjective program by the Dinkelbach method. For easier description, we use to represent the feasible solution set of problem (5). Let denote the maximum stSEE of EHDP communication in the th EHDP/CU group. Then, we have the following definition:
Accordingly, the following theorem can be ready to present.
Theorem 1. The maximum stSEE can be achieved if and only if
Proof. The proof is similar to the proof in [28].
Hence, formula (7) can be addressed by an iterative process, which is demonstrated by Algorithm 1. Define as the number of iterations, as the instantaneous EE for the th EHDP/CU group in the th iteration, and as the convergence threshold.

Although problem (Algorithm 1) is equivalent to problem (5), which is mainly transferred by Dinkelbach’s theorem, problem (Algorithm 1) is also a nonconvex MINLP formulation. Hence, to handle this situation, we propose an innerlayer convex approximation algorithm to convert (Algorithm 1) into a convex one.
3.2. InnerLayer Convex Approximation (ILCA)
For convenience, let represent in each iteration of OLIA. ILCA should perform the following three steps to convert the nonconvex MINLP problem (Algorithm 1) into a convex one. For the first step, the value of is relaxed into a continuous interval , and is substituted by :where and shown by (9) and (10) are the equivalent transformation functions of and according to , respectively, and are related to and :
According to the above equivalent substitution, the optimization problem (Algorithm 1) can be equivalently solved by finding solutions about variables of , , and .
For the second step, as inequality (11) expresses, ILCA introduces the same convex approximation formula as [29] used to acquire an approximate transmission rate about the original one:
The bound of approximation rate is proven to be tight and has low complexity in a highSINR regime (i.e., ). At this moment, and , [29]. To obtain the tightened lower bound, we need an iteration algorithm (such as step 2∼step 6 in p.3751 of [29]) to make the approximation reach a highSINR one.
For the third step, we perform some equivalent substitution of variables by equations of . Consequently, according to the above three steps, problem (Algorithm 1) can be approximately transformed into the following convex optimization formulation:where and are the tight approximation about and after the second and third steps and denoted by (13) and (14), respectively,where the updating of , and , is the same. Obviously, according to the convexity of logsumexp [29, 30], problem (12) can easily proof to be a convex one. As a result, we can utilize one of the typical convex optimization algorithms to solve it easily and efficiently. When the solutions of problem (12) are obtained, we can convert the solving variables of original problem (5) back by using equations when is no less than zero, and otherwise, .
Even though CAIA can obtain a quasioptimal solution for the original problem, there are two key obstacles to practically implement the proposed algorithm. Firstly, the iteration complexity of CAIA is one of the key obstacles to implement in the LTE (LongTerm Evolution) system which requires the scheduling period in milliseconds [31]. Secondly, the overall CSI and ESI during a period of time are hard to obtain practically. Thus, we propose a heuristic algorithm, which is named timedivision scheduling scheme (TDSS), to obtain a suboptimal solution with low computational complexity.
4. The TimeDivision Scheduling Scheme (TDSS)
The complexity of CAIA and the difficulty obtained in the overall ESI and CSI promote us to design a heuristic algorithm to solve the stSEE problem. Although the ESI of all D2D pairs and the CSI of all involved communication links during a period of time are hard to obtain, the latest CSI and ESI (such as the next time slot) can be acquired through some prediction algorithms. For example, some environmental sources’ (e.g., solar and wind) behavior can be predicted through the expected availability at a given time within some error margin [32]. Similarly, channel prediction is feasible and accurate if the predicted frequency is much higher than the channel changing time [33]. It is important to note that the prediction algorithms are not in our consideration.
With the CSI and ESI of the latest time slot, we can simply balance the transmission requirements between two adjacent time slots and hence increase the network performance. Accordingly, a heuristic algorithm, which is called TDSS, is proposed. TDSS can decouple the stSEE problem into two steps: D2D Pairs Choosing Strategy (DPCS) and power allocation strategy (PAS). As the pseudocode of Algorithm 2 shows, firstly, DPCS determines the corresponding EHDPs in the set of to multiplex the channel resource in each time slot for the th EHDP/CU group, where . After that, PAS allocates the corresponding power for the CU and the chosen EHDPs. In other words, in each time slot, the twostep scheme firstly determines the indicative factor for EHDPs in the th group and then allocates and for the CU and the chosen EHDPs. In the following two subsections, the detailed algorithm steps of DPCS and PAS will be described.

4.1. D2D Pairs Choosing Strategy (DPCS)
The DPCS is the first procedure in TDSS is and aimed at choosing the proper EHDPs to multiplex the channel resource of CU at each time slot with the intention of load balance. The purpose of load balance is to schedule the transmission requirements between two adjacent time slots so as to decrease the interference between EHDPs.
The design of DPCS is inspired by a basic characteristic of the optimization problem, which comes from the transmission rate restraints of CU and EHDP. It means that those chosen EHDPs in each time slot should satisfy a basic available energy threshold . If not, the EHDP cannot be a candidate to multiplex the channel resource. Based on this, is expressed by Corollary 4 and is derived in Appendix A.
Corollary 4. In order to ensure the PAS has a feasible solution set, the available power of the chosen EHDP battery must satisfy a minimum value:wherewhere .
Proof. The proof of this corollary is provided in Appendix B.
Therefore, based on the minimum energy threshold and the latest ESI prediction value , the proposed DPCS has three key steps to determine the candidate EHDPs for each EHDP/CU group as illustrated in Figure 4 and Algorithm 3.
Step 1. With the current available energy and minimum energy consumption threshold , we can pick out candidate EHDPs from the set in an instantaneous time slot for the th EHDP/CU group. As Figure 4 illustrates, the EHDPs with the red color are candidates which meet the above basic energy demand.
Step 2. If candidate EHDPs can multiplex the channel resource to transmit at the th time slot, unit power will be consumed at least. According to the power update rule , candidate EHDPs can also be picked out from the set at time slot according to and .
Step 3. The number of candidate D2D pairs between the two time slots can be balanced for the purpose of decreasing the interference among D2D pairs. For example, in Figure 4, the current and next EHDP candidates are 3 and 1, respectively. Thus, the number of transmitted D2D pairs of the two adjacent time slots can be evenly assigned to 2. In other words, if the number difference of candidate D2D pairs between the current time slot and the adjacent next time slot is larger than or equal to 2, we can execute the load balance procedure. Notably, we can only schedule the condition that the service requirements in the current time is larger than the next due to the storeanduse characteristic of energy harvesting [34]. After that, in time slot , DPCS will choose the average assignment number of D2D candidates to multiplex CU channel according to the principles of lower interference and larger transmission rate.

After the DPCS, EHDPs can be selected from to multiplex the channel resource of CU in th EHDP/CU group. Therefore, , where is the set of the selected EHDPs in , is set to 1. In the same way, is set to 0.
4.2. Power Allocation Strategy (PAS)
After determining the binary indicator variables , the optimal power should be allocated for each transmission node (CUs and chosen EHDPs) by maximizing the EE of the D2D communication, while guaranteeing the CUs’ transmission service quality. Thus, the optimization problem will become an EE maximization problem as stated in :where and are the relative transmission rate equations about and , respectively, and can be obtained by (9) and (10). Notably, the EHDPs belong to the set of at this time. The constraint (17c) is the available energy of each chosen D2D pair in battery. (17d) and (17e) are the constraints of maximum transmission power and minimum transmission rate of the relative choosing EHDPs, respectively. The similar restraints for CU are (17a) and (17b).
Remarkably, the objective function is a fractional nonconvex optimization problem because the numerator of objective function of and the constraints (17b) and (17e) are nonconvex. Thus, it is difficult to find a solution for the objective optimization problem. However, we can utilize the same convex approximation approach as CAIA used to obtain a tight lowerbound convex approximation of the numerator in the nonconcave formula (17). Hence, inequality (11) and equality substitution can convert to a lower bound which is illustrated by :where and are shown by (19) and (20), respectively. Similarly, and can be seen in (13) and (14), respectively. As shows, it is a fractional optimization problem. As we know, logsumexp is convex. So, the function is jointly concave about parameters and . Besides, the function can be easily proved to be jointly convex about parameters and . Likewise, the constraints (18b) and (18e) are also jointly convex functions. Thus, is a typical fractional optimization problem and can be solved by the Dinkelbach algorithm [35]. So, the main power allocation algorithm flow of PAS can be summarized by Algorithm 4.
5. Simulation Results
In this section, our goals are to verify the effectiveness of our proposed algorithms and study the impact of EH efficiency factors on system performance. Consequently, we will present numerical results to evaluate the proposed algorithm (CAIA) and the suboptimal heuristic algorithm (TDSS) in aspects of average (avg.) achievable EE and transmission rate of EHDP. Furthermore, to assess the proposed algorithms, we will compare the proposed algorithms with the realtime transmission strategy (RTS), the Exhaustive Searching Scheme (ESS), and the Learning Approach (QLA).
In each time slot, once the EHDP has enough energy to satisfy its transmission power demand, the RTS scheme will let EHDP transmit directly by executing the power allocation algorithm for the CUs and the chosen EHDPs. ESS can enumerate all possible solutions during the shortterm time horizon and thus attain an optimal solution. QLA, a wellknown reinforcement learning program, is widely used to solve some longterm or shortterm utilities [36, 37]. To better assess the effectiveness of the proposed algorithms, we implement QLA as a centralized one.
5.1. Simulation Setup
The performance of the compared methods and the proposed algorithms in this study is evaluated via simulations. Above all, the considered cellular network with radius of 800 meters is demonstrated in Figure 1. The central controller, BS, has the capability of acquiring all users’ position, and it is always located at the centre of this cellular area. Suppose that there exists EHDP/CU groups. For the th group (), EHDPs can multiplex the uplink channel radio resource of CU , where is randomly selected from 2 to 8. All users are randomly deployed in the cellular zone. And meanwhile, the distance between transmitter and receiver of each EHDP pair is randomly selected between [20, 50] meters [21]. Remarkably, to avoid serious mutual interference between each other, a minimum distance threshold, which equals to 200 meters, should be obeyed between CU and EHDPs [38, 39]. Similarly, the distance among EHDPs in each group must be larger than 100 meters. The energy arrival process for every EHDP is assumed to be i.i.d. Bernoulli sample, which conforms to formula (2). The other network parameters used in this study are listed in Table 2.

We repeat each simulation scenario with different energy arrival probabilities; for example, is 0.3 among EHDPs, 100 times and average the results.
5.2. Complexity Comparison
The computational complexity is an important aspect to better assess the effectiveness of the above algorithms. First of all, there are two important things that should be mentioned.(i)The procedure of algorithms (CAIA, TDSS, and RTS) includes convex optimization of nonlinear programming. So, we use the approximate solution to measure the computational complexity. It means that the computational complexity of algorithms (CAIA, TDSS, and RTS) is the needed iteration times when the solution reaches condition, such as Algorithms 1 and 4(ii)We calculate and express the worstcase computational complexity of all algorithms for a fair comparison
It is hard to have a thorough and correct analysis of complexity of convex nonlinear programming problems. However, generally speaking, the complexity is related to the space required to store input data and to the running time of the algorithm until a solution is found [40]. Besides, Vidal et al. [41] produced that the complexity of an approximate solution for the continuous convex problem is , where is the number of variables, is the number of constraints, and is the constraint bound. Moreover, as we all know, the complexity of QLA is related to the size of stateaction space [42]. Thus, the complexity of the abovementioned algorithms can be depicted in Table 3.
