# Optimal Power Allocation for Cooperative Pattern Division Multiple Access Systems

Authors: Xiurong Zhang; Shaoli Kang; Xueqian Fu
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

This study focuses on the power allocation (PA) problem in code division multiple access (PDMA) networks with amplify-and-forward (AF) relays. We provide two wireless communication technical performance indexes, which are system throughput and outage probability. A multiobjective version of particle swarm optimization is proposed to solve Pareto-optimal solutions for optimal PA problems. A novel programming model is built based on improvement of constraint functions, and the accuracy and efficiency of solution can be improved via compact constraints. The technique for order preference by similarity to an ideal solution is proposed to balance the performance indicators, which include outage probability and system throughput. It has been calculated that the proposed approach for optimal PA is verified and performs better than a genetic algorithm approach.


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## Body

## 1. Introduction

Renewable energy has become a trend in smart grid development. Wind power and photovoltaic are representatives of renewable energy. To address the randomness of wind power, robust sliding mode control [1], maximum power point tracking control [2, 3], and passivity-based sliding-mode control [4] are presented to maximize the availability of wind power. Maximum power point tracking is also used in the photovoltaic system, which can get the maximum power output under partial shading condition [5, 6]. The reliability of the renewable energy communication system has significant engineering significance for optimization and control in distribution networks integrated with renewable energy.With the development of 5G networks, wireless communication technology will support the intellectualization of the renewable energy system. 5G networks can deliver superfast speeds and support extremely low-latency connections. Power communication adopts both optical fiber communication and wireless communication, which are applied to new energy monitoring systems. 5G networks can meet the real-time data acquisition and transmission and realize data measurement, state awareness, and scheduling control. Nonorthogonal multiple access (NOMA) has emerged as a key component of 5G networks and has attracted extensive attention and research for its superior spectral efficiency performance [7]. The core concepts of NOMA are to service multiple users on the same resource simultaneously by utilizing power allocation (PA) and to separate multiplexed users at the receiver side by using the successive interference canceller (SIC) technique [8]. Different from the traditional orthogonal transmission, NOMA uses nonorthogonal transmission at the transmitter and actively introduces interference information. At the receiving end, the correct demodulation is realized by a SIC technology. Huawei, ZTE, and Datang Telecom have presented their own multiple access technologies, which are called sparse code multiple access (SCMA) [9], multiuser shared access (MUSA) [10], and pattern division multiple access (PDMA) [11], respectively. Although these technical details are different, they basically belong to NOMA schemes. SCMA, MUSA, and PDMA [11] have been studied in both academia and industry. PDMA is a multicarrier NOMA scheme that has attracted significant attention due to its unequal diversity design at the transmitting side, and its ability to obtain almost equal or approximately equal diversity benefits at the receiving side. PDMA combined with relay technology can enhance system performance [12]. PDMA is a significant NOMA technology, which defines sparse mapping from data to a great sum of resources. Compared with other multiple access technologies, such as SCMA and MUSA, the advantage of PDMA is that PDMA patterns provide unequal classification to meet the communication requirements of different users [13]. Different diversity orders are provided for multiple users to alleviate the error propagation problem of the SIC receiver in PDMA-based systems.The programming problem is a common problem in wireless communication systems, including optimal parameter setting and optimal power allocation (OPA). Bae and Han joined power and time allocation to minimize system outage probability for two-way cooperative NOMA [14]. Cao et al. [15] presented two PA schemes to improve the secrecy of wireless transmissions in a NOMA system. One scheme jammed eavesdropping attempts, and the other optimized the PA power. Zhu et al. presented a PA method to achieve max-min fairness, weighted sum rate maximization, sum rate maximization with quality of service (QoS) constraints, and energy efficiency maximization with weights or QoS constraints under a given channel assignment in NOMA systems [16]. Cui et al. investigated PA in multicell multicarrier NOMA (MC-NOMA) networks by considering maximizing the sum mean opinion scores (MOSs) of users and proposed a low-complexity suboptimal approach based on successive convex approximation techniques, which attained a good computational complexity-optimality trade-off [17]. Ni et al. developed a centralized minimum power control algorithm to minimize the total transmit power by considering the user data rate requirements for multicell MC-NOMA networks where the user assignment is fixed [18]. Khan et al. presented an iterative local optimal solution to calculate the optimal PA and enhance the sum capacity, which considered the transmission power in sending node, the user PA, and the minimum rate requirements per user [19]. Zhao et al. presented a joint problem of spectrum allocation and power control to maximize the sum rate of small-cell users in a NOMA network by considering user fairness. The PA problem used sequential convex programming to update the result iteratively. The spectrum allocation problem used a many-to-one matching game with peer effects [20]. Song et al. presented a PA algorithm based on a bisection search algorithm and a gradient value to maximize the energy efficiency in downlink NOMA networks with imperfect channel state information. The nonconvex problem was transformed to a convex problem by sequential convex programming, and the PA factors were attained by the Lagrangian multiplier method [21]. He et al. presented a deep reinforcement learning framework where PA and channel assignment were combined to allocate resources to users in a near optimal way. Specifically, an attention-based neural network was exploited to perform the channel assignment [22]. Xiao et al. presented an improved PSO algorithm to improve the energy efficiency of the systems while guaranteeing the spectral efficiency [23]. Except GA, particle swarm optimization (PSO) is another artificial intelligence algorithm that can be used for wireless communication system optimization [24]. From the point of view of the programming solution method, the following comparison can be summarized [24], and the proposed methods are artificial intelligence algorithms. The main difference between this study and [24] is that, in this study, OPA is a multiobjective optimization problem rather than a single-objective optimization problem as in [24]. High system throughput is a major target in [24]. Both high system throughput and low outage probability are targets in this study. We not only improve the constraint functions in [24] but also propose an effective method for multiobjective programming.The existing OPA methods only consider a single performance objective, that is, the system throughput, but single-objective optimization cannot meet the requirements of multiple performance indicators of wireless communication. Beside system throughput, outage probability is another important index to measure the performance of the communication system. Multiobjective optimization can make power allocation meet the requirements of different performance indexes of PDMA at the same time, that is, the significance of this work. Multiobjective optimization is of vital importance in a wireless communication system, but it is more complex and time-consuming than single-objective optimization. This motivates us to find an OPA algorithm in amplify-and-forward (AF) relaying with PDMA (AF-PDMA) networks to improve multiple system performances. The first challenge is that OPA is a nonconvex problem, which cannot be solved effectively by the gradient descent method. The second challenge is that how to balance the optimization needs of different system performance indicators. It was proved that different performance objectives make the optimization problem become a constrained multiobjective optimization problem in PDMA systems [25].The contributions of this study can be outlined as follows. (1) We present a multiobjective PSO algorithm considering a Pareto-optimal solution to solve the complicated PA solution problems in an AF-PDMA network. The benefit is that the feasible region of the PSO algorithm is greatly reduced by modifying the original constraint functions, and the convergence efficiency is greatly improved. The iteration in the infeasible region is also avoided. (2) We analyse the relationship of user outage probabilities and system throughput from which a trade-off between the outage probability of a single user and system throughput is attained. The benefit is that different requirements of users and systems on OPA are taken into consideration, and the proposed multiobjective evaluation theory can be extended to other wireless communication systems for equilibrium optimization.Notation: the notation⊙ represents the dot multiple. ⋅−1 denotes the matrix inversion. ⋅H denotes the matrix conjugate transpose. IN represents an identity matrix with an N×N vector.


## 2. System Model

In this section, we consider an AF-PDMA downlink network as outlined in [26]. A PDMA pattern matrix GPDMAN,K=g1,g2,…,gK denotes a K pattern with gi mapping on N resources, where gi is the ith user’s PDMA pattern.The transmission powers for the BS and relaying node are equal (Ps=Pr=P). The channel coefficients among the BS, the kth user, and the AF relaying node are denoted as hmnk,mnk∈buk,br,ruk, in which hmnk∼CN0,μmnk. nbr∼CN0,N0 and nbuk∼CN0,N0 represent additive white Gaussian noise at the relaying node and the kth user, respectively. γ≜P/N0 is a transmit SNR.The received signals atkth user from the BS or relaying node are denoted as follows:(1)ybuk=hbuk⊙∑i=1KαiPgixi+nbuk=∑i=1KαiPh˜bukxi+nbuk=Hx+nbuk,yruk=Gh˜ruk⊙hbr∑i=1KαiPgixi+nbr+nruk,where ybuk and yruk are the signals from the BS or relaying node, respectively. nbuk and nruk are the noises from the BS and relaying node, respectively. h˜buk and h˜ruk are the channel responses from the BS and relaying node, respectively. h˜buk=hbuk⊙gk, h˜ruk=hruk⊙gk, and H˜ is the PDMA equivalent channel response matrix. xi is the ith user’s signal. αi is the PA coefficient, where α1+α2+⋯+αK=1. x=α1Px1α2Px2⋯αKPxKT is the modulated symbol. G is the amplifying gain factor in which G=P/Ph˜br2+N0.The corresponding SINR for thekth user who receives a signal from the BS is represented as follows:(2)γbuk=Pkh˜bukHN0IN+∑i=k+1KPih˜ih˜iH−1h˜buk.Pk=αkP denotes the kth user’s transmit power.The corresponding SINR for thekth user who receives the signal from the relaying node is represented as follows:(3)γruk=αkγ2h˜ruk⊙h˜brkHKZk−1h˜ruk⊙h˜brk,where Zk is the noise plus the interference from the relaying node. KZk is the covariance of Zk, which is calculated as follows:(4)KZk=γ2∑i=k+1Kαih˜rui⊙h˜brih˜rui⊙h˜briH+γh˜rukh˜rukH+γh˜brh˜brH+IN.The metrics are defined as follows. Generally, outage probability is defined as the probability by which SINR is smaller than a specified threshold value in both BS-to-user and relay-to-user channels. An outage occurs if neither the direct nor the relay transmission succeeds [24]. System throughput is defined as the product of the transmission rate and the probability of successful transmission [27].To simplify the outage expressions, we define functionGkx as (5).(5)Gkx=1−e−x/γ1/μbr+1/μruk4xx+1μbrμrukγK14ϕx+1μbrμrukγ,x≠0,0,x=0,where K1⋅ is the first-order modified Bessel function of the second type, and μmnk is the channel coefficient where mnk∈buk,br,ruk.The matrix with two rows (N = 2) and three columns (k = 3) is the most representative matrix with the minimum degree of unequal diversity. Other larger dimension matrices can be extended similarly. The PSO algorithm has a good applicability. When the number of users and channels change, we only need to modify the PSO parameters, constraints, and objective functions, rather than the algorithm mechanism. Taking the PDMA pattern matrix GPDMA2,3 as an example, the outage expressions for users are as follows:(6)O1=1−e−τ1/μbu1+e−τ2/μbu1×1+G1τ1−G1τ2,ifa<0,e−τ1/μbu1−e−τ2/μbu1×G1τ2−G1τ1,ifa>0,O2=1−e−ς1/μbu2+e−ς2/μbu2×1+G2ς1−G2ς2,ifa<0,1−e−ς/μbu2+e−ς1/μbu2−e−ς2/μbu2×G2ς−G2ς1+G2ς2,ifa>0,O3=1−e−ξ1/μbu3+e−ξ2/μbu3×1+G3ξ1−G3ξ2,ifa<0,1−e−ξ/μbu3+e−ξ1/μbu3−e−ξ2/μbu3×G3ξ−G3ξ1+G3ξ2,ifa>0,where τ1=max0,x1, τ2=max0,x2, ς=ϕ2/α2γ,ς1=maxϕ2/α2γ,x1, ς2=maxϕ2/α2γ,x2, ψ1=maxx1,ϕ2/α2γ,ϕ3/α3γ,ψ2=maxx2,ϕ2/α2γ,ϕ3/α3γ, ψ=maxϕ2/α2γ,ϕ3/α3γ,anda=α1α2+α3−α2α3ϕ1γ2.ϕk is the targeted SNR of the kth user; x1 and x2 stand for different values of hbu12, and they can be obtained by solving the following equation.(7)α1α2+α3−α2α3ϕ1γ2hbu14+2α1−α2+α3ϕ1γhbu12−ϕ1=0.The AF-PDMA system throughput expression is denoted as(8)ℛsum=ℛ11−O1+ℛ21−O2+ℛ31−O3.


## 3. Optimal Power Allocation

The optimal PA is mathematically equivalent to the optimal PA factor setting. Multiobjective programming is an effective way to solve complicated PA problems.


### 3.1. Programming Model

Two typical objectives, based on system throughput and outage probability, have been proposed for optimal PA in a downlink network. Hence, the PA allocation problem is an optimization problem with two objective functions, and it can be formulated as(9)minfα,θ=O3α,θ,−Rsumα,θ,where α is the three PA coefficients (i.e., α1, α2, and α3), θ is a system parameter vector in the downlink network, O3 is the outage probability of the third user, the significance ensures the reliability of single-user communication, and Rsum is the system throughput.The inequality constraints are given as follows:(10)1>α1>α2>α3>0.The equality constraint is given as follows:(11)α1+α2+α3=1.The challenge of PSO is that once the initial value and update particles are not in the feasible region, all the particles may be updated in the infeasible region. We propose a new approach to address this issue by revising the existing constraints through logic. After the constraint conditions are modified, the decision variables are changed from three coefficients to two coefficients. The third coefficient is calculated as a parameter by (11). The inequality constraints are improved as follows:(12a)13<α1<1,(12b)0<α2<12,(12c)0<α2<α1,(12d)1−α1−α2<α2.Equations (13) and (14) are broad constraints, but (12a)–(12d) make the constraints narrow. The point is that the constraint conditions are very important for the efficiency and accuracy of the evolutionary algorithm. With respect to PSO, the programming model based on (13) and (14) is easy to iterate in the infeasible region, but the model based on (12a)–(12d) can avoid this problem.


### 3.2. PSO Theory

PSO exhibits strong randomness and covers most solution spaces to avoid reaching a local optimum [28]. The advantage of an artificial intelligence algorithm is that it is amenable to almost all stochastic programming models. Its disadvantage is that it cannot guarantee that a global optimal solution can be found. Even if the global optimal solution is found, it is impossible to provide a strict mathematical proof. In addition, adjusting the parameters, including the step size and penalty coefficients, is difficult. Reasonable parameters are necessary conditions for obtaining the optimal solution, and the PSO parameters in this study are listed in Table 1.Table 1
Simulation parameters for PSO.
ItemValueThe PSO particle size30The PSO iteration number15The penalty function factor106


### 3.3. Pareto Frontier

The Pareto frontier is a key to solving the multiobjective programming problem [29]. A Pareto-optimal solution can include a noninferior solution set, a nondominated solution set, and a nondominant solution set. Based on the Pareto dominance, we present a multiobjective PSO algorithm considering a Pareto-optimal solution. The procedure of the multiobjective PSO algorithm is given as follows: Algorithm 1.Algorithm 1: Process for multiobjective optimization.
Step 1. Single-objective optimization(i)
Find the optimal solution of the maximum system throughput using the PSO algorithm(ii)
Find the optimal solution of the minimum outage probability of user3 using the PSO algorithmStep 2. Probability distribution calculation(i)
Calculate the system throughput using the solutions, which ensures the minimum outage probability of user3(ii)
Generate a row vector ofm linearly equally spaced points between the optimal system throughput and the system throughput, which is solved in the last stepStep 3. Obtain the Pareto frontier(i)
Add inequality constraintRsum (α, θ)>a, which is an element of the above equally spaced points. For each solution, a constraint is added in order.(ii)
Find the optimal solution of the minimum outage probability of user3 using the PSO algorithm based on the above added inequality constraint(iii)
Repeat the above processm times to obtain m groups of optimization target values


### 3.4. TOPSIS

The technique for order preference by similarity to an ideal solution (TOPSIS) is an effective double benchmark evaluation method [30] that can be used to balance the outage probability of a single user and system throughput. TOPSIS can calculate the positive and negative ideal solutions of each index (i.e., outage probability of a single user or system throughput). The relative closeness of each evaluation object to positive and negative ideal solutions can be used to evaluate the comprehensive performance of the system. The optimal TOPSIS should be close to the positive ideal solution, and the distance from the negative ideal solution should be large. The decision problem of two objectives is shown in Figure 1.Figure 1
Positive ideal solution and negative ideal solution.A+ and A− denote the positive ideal solution and negative ideal solution, respectively. The feasible solution A1 is closest to the ideal solution A+ but not the farthest solution from the negative ideal solution A−. The feasible solution A2 is further away from the negative ideal A−. The disadvantage of TOPSIS is that the weight is not reflected in the distance calculation. The calculated Euclidean distance can be weighted using an entropy-weighing method to overcome this shortcoming.The calculation steps of an entropy TOPSIS method are as follows. A standard treatment of the evaluation matrix data using 0-1 transformation is implemented.When the index is positive, the following formula is used for standardized transformation.(13)bij∗=bij−minjbijmaxjbij−minjbij,1≤i≤m,1≤j≤n,where bij is an optimal system throughput in step 3 in section C, and n is equal to two. When the index is an inverse index, the following formula is used for standardized transformation.(14)bij∗=maxjbij−bijmaxjbij−minjbij,1≤i≤m,1≤i≤n,where bij is an optimal outage probability in step 3 in section C.According to the standardized decision matrix, the characteristic proportion of thejth index and the ith evaluation sample can be calculated using(15)cpij=bij∗∑i=1mbij∗.The entropy of the index is calculated using(16)ej=−1lnm∑i=1mcpijlncpij,1≤j≤n.Note that when cpij = 0, we set ln (cpij) equal to zero. The entropy weight formula is(17)wj=1−ej∑i=1n1−ej,1≤j≤n.From here, it follows that(18)cij=wj×bij,i=1,2,…,m;j=1,2,…,n,C+=maxcij,1≤i≤m,C−=mincij,1≤i≤m,where C = (cij)m×n is the weighted normal matrix, C+ is the set of positive ideal solutions, and C− is the set of negative ideal solutions.Then, the weighted Euclidean distances from each evaluation object to the positive and negative ideal solutions are calculated separately.(19)di+=∑j=1nwj2cij−cj+2,i=1,2,…,m,di−=∑j=1nwj2cij−cj−2,i=1,2,…,m,fi=di−di++di−,i=1,2,…,m,where fi is the evaluation result of the comprehensive performances. One can find the best solution at the Pareto frontier by finding the greatest fi.The implementation process of the proposed algorithm is shown in Figure2.Figure 2
Implementation process of the proposed algorithm.


## 3.1. Programming Model

Two typical objectives, based on system throughput and outage probability, have been proposed for optimal PA in a downlink network. Hence, the PA allocation problem is an optimization problem with two objective functions, and it can be formulated as(9)minfα,θ=O3α,θ,−Rsumα,θ,where α is the three PA coefficients (i.e., α1, α2, and α3), θ is a system parameter vector in the downlink network, O3 is the outage probability of the third user, the significance ensures the reliability of single-user communication, and Rsum is the system throughput.The inequality constraints are given as follows:(10)1>α1>α2>α3>0.The equality constraint is given as follows:(11)α1+α2+α3=1.The challenge of PSO is that once the initial value and update particles are not in the feasible region, all the particles may be updated in the infeasible region. We propose a new approach to address this issue by revising the existing constraints through logic. After the constraint conditions are modified, the decision variables are changed from three coefficients to two coefficients. The third coefficient is calculated as a parameter by (11). The inequality constraints are improved as follows:(12a)13<α1<1,(12b)0<α2<12,(12c)0<α2<α1,(12d)1−α1−α2<α2.Equations (13) and (14) are broad constraints, but (12a)–(12d) make the constraints narrow. The point is that the constraint conditions are very important for the efficiency and accuracy of the evolutionary algorithm. With respect to PSO, the programming model based on (13) and (14) is easy to iterate in the infeasible region, but the model based on (12a)–(12d) can avoid this problem.


## 3.2. PSO Theory

PSO exhibits strong randomness and covers most solution spaces to avoid reaching a local optimum [28]. The advantage of an artificial intelligence algorithm is that it is amenable to almost all stochastic programming models. Its disadvantage is that it cannot guarantee that a global optimal solution can be found. Even if the global optimal solution is found, it is impossible to provide a strict mathematical proof. In addition, adjusting the parameters, including the step size and penalty coefficients, is difficult. Reasonable parameters are necessary conditions for obtaining the optimal solution, and the PSO parameters in this study are listed in Table 1.Table 1
Simulation parameters for PSO.
ItemValueThe PSO particle size30The PSO iteration number15The penalty function factor106


## 3.3. Pareto Frontier

The Pareto frontier is a key to solving the multiobjective programming problem [29]. A Pareto-optimal solution can include a noninferior solution set, a nondominated solution set, and a nondominant solution set. Based on the Pareto dominance, we present a multiobjective PSO algorithm considering a Pareto-optimal solution. The procedure of the multiobjective PSO algorithm is given as follows: Algorithm 1.Algorithm 1: Process for multiobjective optimization.
Step 1. Single-objective optimization(i)
Find the optimal solution of the maximum system throughput using the PSO algorithm(ii)
Find the optimal solution of the minimum outage probability of user3 using the PSO algorithmStep 2. Probability distribution calculation(i)
Calculate the system throughput using the solutions, which ensures the minimum outage probability of user3(ii)
Generate a row vector ofm linearly equally spaced points between the optimal system throughput and the system throughput, which is solved in the last stepStep 3. Obtain the Pareto frontier(i)
Add inequality constraintRsum (α, θ)>a, which is an element of the above equally spaced points. For each solution, a constraint is added in order.(ii)
Find the optimal solution of the minimum outage probability of user3 using the PSO algorithm based on the above added inequality constraint(iii)
Repeat the above processm times to obtain m groups of optimization target values


## 3.4. TOPSIS

The technique for order preference by similarity to an ideal solution (TOPSIS) is an effective double benchmark evaluation method [30] that can be used to balance the outage probability of a single user and system throughput. TOPSIS can calculate the positive and negative ideal solutions of each index (i.e., outage probability of a single user or system throughput). The relative closeness of each evaluation object to positive and negative ideal solutions can be used to evaluate the comprehensive performance of the system. The optimal TOPSIS should be close to the positive ideal solution, and the distance from the negative ideal solution should be large. The decision problem of two objectives is shown in Figure 1.Figure 1
Positive ideal solution and negative ideal solution.A+ and A− denote the positive ideal solution and negative ideal solution, respectively. The feasible solution A1 is closest to the ideal solution A+ but not the farthest solution from the negative ideal solution A−. The feasible solution A2 is further away from the negative ideal A−. The disadvantage of TOPSIS is that the weight is not reflected in the distance calculation. The calculated Euclidean distance can be weighted using an entropy-weighing method to overcome this shortcoming.The calculation steps of an entropy TOPSIS method are as follows. A standard treatment of the evaluation matrix data using 0-1 transformation is implemented.When the index is positive, the following formula is used for standardized transformation.(13)bij∗=bij−minjbijmaxjbij−minjbij,1≤i≤m,1≤j≤n,where bij is an optimal system throughput in step 3 in section C, and n is equal to two. When the index is an inverse index, the following formula is used for standardized transformation.(14)bij∗=maxjbij−bijmaxjbij−minjbij,1≤i≤m,1≤i≤n,where bij is an optimal outage probability in step 3 in section C.According to the standardized decision matrix, the characteristic proportion of thejth index and the ith evaluation sample can be calculated using(15)cpij=bij∗∑i=1mbij∗.The entropy of the index is calculated using(16)ej=−1lnm∑i=1mcpijlncpij,1≤j≤n.Note that when cpij = 0, we set ln (cpij) equal to zero. The entropy weight formula is(17)wj=1−ej∑i=1n1−ej,1≤j≤n.From here, it follows that(18)cij=wj×bij,i=1,2,…,m;j=1,2,…,n,C+=maxcij,1≤i≤m,C−=mincij,1≤i≤m,where C = (cij)m×n is the weighted normal matrix, C+ is the set of positive ideal solutions, and C− is the set of negative ideal solutions.Then, the weighted Euclidean distances from each evaluation object to the positive and negative ideal solutions are calculated separately.(19)di+=∑j=1nwj2cij−cj+2,i=1,2,…,m,di−=∑j=1nwj2cij−cj−2,i=1,2,…,m,fi=di−di++di−,i=1,2,…,m,where fi is the evaluation result of the comprehensive performances. One can find the best solution at the Pareto frontier by finding the greatest fi.The implementation process of the proposed algorithm is shown in Figure2.Figure 2
Implementation process of the proposed algorithm.


## 4. Results and Discussion

### 4.1. Subheadings

We use a step-by-step validation method to demonstrate the power optimization method for AF-PDMA in a downlink network. First, we compare GA with the proposed PSO algorithm. Then, system throughput performance and the minimum outage probability of the third user are taken as planning objectives, and different solutions for OPA are derived. To balance the need for system throughput with the desire to minimize outage probabilities, a Pareto frontier consisting of a set of possible solutions is found. The Pareto solutions are further evaluated with the TOPSIS approach to help the decision-maker find the desired OPA solution. The simulation in this study is performed on a Lenovo laptop with an Intel(R) Core (TM) i7-1065G7 CPU operating at 1.30 GHz with 15.7 GB of available memory.An internal function of MATLAB, i.e., a GA solver for mixed-integer or continuous-variable optimization, constrained or unconstrained, is used to solve the OPA problem. Because GA internal function is installed in Matlab, the solution performance is reliable and can be used to verify the accuracy and effectiveness of the proposed PSO algorithm.PSO and GA are used to solve the OPA problem to obtain the maximum system throughput. As shown in Figures3–5, the performances of this method are better than those of the GA. It should be noted that it is important to adjust parameters in an artificial intelligence algorithm according to the programming model. The simulation results do not demonstrate that PSO is more efficient and accurate than GA but demonstrate that the solution performance of the artificial intelligence algorithm depends on the parameter setting. Noted that the PSO algorithm uses the modified constraints, the GA uses the constraints before modification. The feasible region of GA solution is original and broad, and the solution is limited by (13)-(14). The feasible region of the PSO algorithm is modified and compact, and the solution is limited by (12a)–(12d). It will greatly improve the efficiency and accuracy of the solution via transforming (10)-(11) to (12a)–(12d). With respect to the proposed method, the feasible regions of α1 and α2 have been reduced to improve the calculation efficiency. α3 is no longer used as a solution, and it is directly calculated using α1 and α2. The benefit of such a treatment is that the equality constraint (11) is avoided when generating fitness function. This result shows that rigorous mathematical derivations and reductions of the feasible region are important for an intelligent optimization algorithm.Figure 3
System throughput (BPCU) versus SNR (dB) using different solutions.Figure 4
Sum of outage probabilities versus SNR (dB) using different solutions.Figure 5
Outage probabilities versus SNR (dB) using different solutions.Figures3–5 show the different evaluation metrics, but their optimization objectives are the same. System throughput is taken as the single-objective function. According to the formulas for outage probabilities and system throughput, there is no contradiction between them. When the outage probability is very low, the system throughput performance can be guaranteed. However, this does not mean that the outage probability of each user is completely consistent with the system throughput performance of the downlink network. It has been calculated that the individual user may be sacrificed to maximize the system throughput. With the increasing of outage probability (power reduction) of the individual user, the total outage probability and system throughput may be improved. Although some GA results are better than the PSO results for user3, system throughput of GA optimization is definitely worse than that of the proposed PSO optimization. There exist balance and game problems between system throughput performance and a user outage probability. To illustrate this problem, the outage probability of user3 is taken as the objective function, and the proposed PSO method is used to solve the OPA.As shown in Figures6–8, the system performances vary depending on the target functions. It can be seen that the optimal system throughput performance does not guarantee the best performance for every user. Even if multiple users have the lowest total outage probability, there is no guarantee that each user will have the lowest outage probability. With regards to the solution results, it should be noted that it is difficult for an artificial intelligence algorithm to find the global optimal solution. In terms of few individual points, the system throughput of the curve targeting the best system throughput can be less than that of the curve targeting the best user3. Also, these locally optimal solutions are few and reasonable when PSO is used. To solve the biobjective programming problem, we use the weighted ideal method to evaluate Pareto frontier using 500 solutions, which is shown in Figure 9.Figure 6
System throughput (BPCU) versus SNR (dB) considering different optimization objectives.Figure 7
Sum of outage probabilities versus SNR (dB) considering different optimization objectives.Figure 8
Sum of outage probabilities versus SNR (dB) considering different optimization objectives.Figure 9
Pareto frontier for user3 outage probability versus system throughput with SNR = 10.At the Pareto frontier, user3 outage probability and system throughput performances need to be balanced. OPA should balance the two to achieve a balance of the two indicators.The index of outage probability is an inverse index and has a cost type attribute. The smaller the outage probability value is, the better the system performance. The index of system throughput is positive and has a benefit type attribute. The larger the index value, the better the system performance. According to formulas (9) and (10), the system throughput and outage probability values in Figure 9 are standardized. The weight matrix is determined by the entropy weight method, and the matrix is [0.1408, 0.8592]. The TOPSIS approach is a double benchmark that can reduce the probability of the same evaluation results. We can obtain the best power distribution scheme using the method combining entropy weight and TOPSIS, and the solution of PA coefficients is [0.7, 0.2, 0.1]. The optimal system throughput and outage probability of TOPSIS is listed in Table 2. It can be seen that each index of TOPSIS is not the best, but there is a balance between the two indexes.Table 2
System performances of TOPSIS.
ItemSystem throughputOutage probabilityLower boundary3.880.0010Upper boundary3.970.0078TOPSIS3.960.0011Although PSO can be applied to solve nonconvex multiobjective optimization problems, it cannot avoid the complexity of parameter setting and program debugging. The setting of PA initial values and particle moving speed will affect the PSO performance. In practical programming, the biggest challenge is that the particle swarm can be updated in the infeasible region. With respect to the proposed optimization method, the updating of particle swarm is strictly limited in the feasible region via putting strict constraints on the feasible region instead of setting the penalty term in the fitness function. Of course, multiobjective will bring about the complexity problem, which can be dealt with by Pareto frontier and an effective comprehensive evaluation method.


## 4.1. Subheadings

We use a step-by-step validation method to demonstrate the power optimization method for AF-PDMA in a downlink network. First, we compare GA with the proposed PSO algorithm. Then, system throughput performance and the minimum outage probability of the third user are taken as planning objectives, and different solutions for OPA are derived. To balance the need for system throughput with the desire to minimize outage probabilities, a Pareto frontier consisting of a set of possible solutions is found. The Pareto solutions are further evaluated with the TOPSIS approach to help the decision-maker find the desired OPA solution. The simulation in this study is performed on a Lenovo laptop with an Intel(R) Core (TM) i7-1065G7 CPU operating at 1.30 GHz with 15.7 GB of available memory.An internal function of MATLAB, i.e., a GA solver for mixed-integer or continuous-variable optimization, constrained or unconstrained, is used to solve the OPA problem. Because GA internal function is installed in Matlab, the solution performance is reliable and can be used to verify the accuracy and effectiveness of the proposed PSO algorithm.PSO and GA are used to solve the OPA problem to obtain the maximum system throughput. As shown in Figures3–5, the performances of this method are better than those of the GA. It should be noted that it is important to adjust parameters in an artificial intelligence algorithm according to the programming model. The simulation results do not demonstrate that PSO is more efficient and accurate than GA but demonstrate that the solution performance of the artificial intelligence algorithm depends on the parameter setting. Noted that the PSO algorithm uses the modified constraints, the GA uses the constraints before modification. The feasible region of GA solution is original and broad, and the solution is limited by (13)-(14). The feasible region of the PSO algorithm is modified and compact, and the solution is limited by (12a)–(12d). It will greatly improve the efficiency and accuracy of the solution via transforming (10)-(11) to (12a)–(12d). With respect to the proposed method, the feasible regions of α1 and α2 have been reduced to improve the calculation efficiency. α3 is no longer used as a solution, and it is directly calculated using α1 and α2. The benefit of such a treatment is that the equality constraint (11) is avoided when generating fitness function. This result shows that rigorous mathematical derivations and reductions of the feasible region are important for an intelligent optimization algorithm.Figure 3
System throughput (BPCU) versus SNR (dB) using different solutions.Figure 4
Sum of outage probabilities versus SNR (dB) using different solutions.Figure 5
Outage probabilities versus SNR (dB) using different solutions.Figures3–5 show the different evaluation metrics, but their optimization objectives are the same. System throughput is taken as the single-objective function. According to the formulas for outage probabilities and system throughput, there is no contradiction between them. When the outage probability is very low, the system throughput performance can be guaranteed. However, this does not mean that the outage probability of each user is completely consistent with the system throughput performance of the downlink network. It has been calculated that the individual user may be sacrificed to maximize the system throughput. With the increasing of outage probability (power reduction) of the individual user, the total outage probability and system throughput may be improved. Although some GA results are better than the PSO results for user3, system throughput of GA optimization is definitely worse than that of the proposed PSO optimization. There exist balance and game problems between system throughput performance and a user outage probability. To illustrate this problem, the outage probability of user3 is taken as the objective function, and the proposed PSO method is used to solve the OPA.As shown in Figures6–8, the system performances vary depending on the target functions. It can be seen that the optimal system throughput performance does not guarantee the best performance for every user. Even if multiple users have the lowest total outage probability, there is no guarantee that each user will have the lowest outage probability. With regards to the solution results, it should be noted that it is difficult for an artificial intelligence algorithm to find the global optimal solution. In terms of few individual points, the system throughput of the curve targeting the best system throughput can be less than that of the curve targeting the best user3. Also, these locally optimal solutions are few and reasonable when PSO is used. To solve the biobjective programming problem, we use the weighted ideal method to evaluate Pareto frontier using 500 solutions, which is shown in Figure 9.Figure 6
System throughput (BPCU) versus SNR (dB) considering different optimization objectives.Figure 7
Sum of outage probabilities versus SNR (dB) considering different optimization objectives.Figure 8
Sum of outage probabilities versus SNR (dB) considering different optimization objectives.Figure 9
Pareto frontier for user3 outage probability versus system throughput with SNR = 10.At the Pareto frontier, user3 outage probability and system throughput performances need to be balanced. OPA should balance the two to achieve a balance of the two indicators.The index of outage probability is an inverse index and has a cost type attribute. The smaller the outage probability value is, the better the system performance. The index of system throughput is positive and has a benefit type attribute. The larger the index value, the better the system performance. According to formulas (9) and (10), the system throughput and outage probability values in Figure 9 are standardized. The weight matrix is determined by the entropy weight method, and the matrix is [0.1408, 0.8592]. The TOPSIS approach is a double benchmark that can reduce the probability of the same evaluation results. We can obtain the best power distribution scheme using the method combining entropy weight and TOPSIS, and the solution of PA coefficients is [0.7, 0.2, 0.1]. The optimal system throughput and outage probability of TOPSIS is listed in Table 2. It can be seen that each index of TOPSIS is not the best, but there is a balance between the two indexes.Table 2
System performances of TOPSIS.
ItemSystem throughputOutage probabilityLower boundary3.880.0010Upper boundary3.970.0078TOPSIS3.960.0011Although PSO can be applied to solve nonconvex multiobjective optimization problems, it cannot avoid the complexity of parameter setting and program debugging. The setting of PA initial values and particle moving speed will affect the PSO performance. In practical programming, the biggest challenge is that the particle swarm can be updated in the infeasible region. With respect to the proposed optimization method, the updating of particle swarm is strictly limited in the feasible region via putting strict constraints on the feasible region instead of setting the penalty term in the fitness function. Of course, multiobjective will bring about the complexity problem, which can be dealt with by Pareto frontier and an effective comprehensive evaluation method.


## 5. Conclusion

The entropy method is used to determine the weight of a system index (system throughput and outage probability), which avoids the subjectivity of multifactor weight determination. Through the analysis of the comprehensive evaluation results of the system performance of a downlink network, the effectiveness and rationality of the TOPSIS approach for OPA are verified. TOPSIS can calculate the weighted distance and overcome the shortage of comprehensive evaluation results caused by the use of a single standard. The simulation results ensure that the outage probability of a single user is low, and the system throughput performance is good.



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*Source: 5581811-2021-02-26.xml*