A novel coevolving differential evolution and its application in intelligent device-to-device communication systems

2.1 Resource allocation of D2D communication

In D2D communication system, the two UEs nearby can directly communicate with each other under the control of eNB. In normal conditions, the D2D-pair users (DP) may reuse the uplink resources of cellular users (CU). The transmission of DP transmitter (DT) could cause interference to eNB, and the transmission of CU could also interfere the DP receiver (DR). As a result, optimally allocate the uplink resources of CU to DP is of great importance to the system spectrum efficiency and QoS. In recent years, with the fast development of D2D communication technology, the resource allocation models and optimization algorithms are widely developed. Zhao et al. developed a resource management method by considering both power and channel allocation, in order to overcome the problems of spectrum efficiency promotion and interference controlling caused by spectrum reuse in hybrid communication scenario [9]. Mohamad et al. proposed a dynamic sectorization method in which eNB can vary the number of sectors dynamically and allocated the resource block to D2D users. According to their study, the signal-to-interference-noise-ratio (SINR) and the network overall performance are improved [10]. Sultana et al. introduced the sparse code multiple access (SCMA) technology to a D2D enabled cellular network, in order to utilize the overloading feature of SCMA to support a massive device connectivity while enhancing the overall network performance [11]. Wang et al. proposed a matching-based social-aware resource allocation algorithm for multicast D2D communication underlaying certain cellular network. According to their simulation results, the proposed allocation algorithm achieved promising performance gain compared with some other state-of-the-art algorithms [12]. Amin et al. proposed a resource allocation algorithm based on the so-called Q-learning, in which the multi-agent learners from multiple D2D users were created, and the system throughput was determined by the state-learning of Q value list. According to their study, the system throughput was effectively maximized by controlling the D2D users’ power, and a fine QoS of cellular users was also ensured [13].

In D2D communication, the resource allocation is always considered together with power control strategy for obtaining promising performance on spectrum efficiency. For example, based on the multi-agent reinforcement learning methodology, Wei et al. studied the channel resource allocation and power control strategy for improving the system resource utilization and network throughput [14]. Sobhi-Givi et al. proposed a resource block allocation scheme for reducing the co-channel interference by providing and maintaining adequate distance between D2D UE and CU. According to the experimental results reported in their study, the system spectral efficiency can be significantly improved by optimizing the D2D UE power. Thus, the system throughput was much higher than that of the random resource allocation strategy, and also outperformed some other compared methodologies [15].

In recent years, with the fast development of artificial intelligence (AI), the AI-based techniques were widely introduced into the resource allocation of D2D communication and achieved promising performance. For example, Hamdi et al, studied the resource allocation strategy using the swarm intelligence algorithms, that is, the hybrid genetic algorithm (GA) and particle swarm optimization (PSO) [16]. Takshi et al. proposed a GA-based method to minimize the interference, and maximize the spectral efficiency at the same time [17]. By allowing two DPs to share the same frequency resource of one CU, Sun et al. studied the algorithms for maximizing system throughput based on PSO and hybrid PSO-GA. According to their simulation results, the system performance could be effectively guaranteed with the proposed methods [18].

2.2 Swarm intelligence-based optimization

In this study, the swarm intelligence-based optimization algorithms (SIOA) are developed and employed for solving the proposed ORA model. With the development of AI theory and technology, the SIOA attracts attentions of worldwide scholars in recent years, and are also successfully applied in solving real-world engineering problems [19–22]. Some famous SIOA are listed as follows: the PSO [23], GA [24], differential evolution (DE) [25], artificial bee colony algorithm (ABC) [26], ant colony optimization (ACO) [27], woodpecker mating algorithm (WMA) [28–31] and so on. However, due to the increasing complexity of real-world problems, the basic SIOA always loses efficiency and fails to find the global optimum. As a result, a lot of SIOA-variants are widely developed to solve the complex problems with certain characteristics. For example, Tang et al. introduced the human simulated properties in to PSO and proposed a so-called human-brain simulated particle swarm optimization algorithm [32]. For optimizing the binary variables, Nguyen et al. proposed a dynamic sticky binary PSO by developing a dynamic parameter control strategy based on an investigation of exploration and exploitation in the binary search spaces [33]. For the multimodal optimization problem, Lin et al. developed an algorithm named the FBK-DE, in which the formulation, balance and keypoint of species for multimodal optimization problems are considered [34]. Li et al. divided the original population of DE into four sub-populations, and most of the computing resources were allocated to the best one. According to their experimental results, the proposed MPMDE algorithm obtained competitive performance in both optimization accuracy and convergence speed [35].

As discussed above, in order to obtain promising performance on spectrum efficiency improvement in D2D communication, the SIOA-based algorithm is a good choice to solve the resource allocation problem. However, the performance of SIOA-based resource allocation methodology is deeply relied on the allocation model and efficacy of SIOA adopted. In the following sections, the ORA model with certain constraints is established to accurately describe resource allocation problem, and an effective DE-variant is developed to optimize the ORA model to obtain robust allocation performance.

3.1 System model

In this study, we consider a cellular network of LTE-Advance systems, in which the eNB locates at the center of a hexagonal region (denote the radius as R). The cellular users (i.e., CU) and D2D-pair users (i.e., DP) are randomly distributed within the hexagonal region. Denote C={CU _n | n = 1, 2,  ...  , N} and D={DP _m | m = 1, 2,  ...  , M} as the sets of CU and DP, respectively. Assume that the uplink resource of each CU can be reused by at most one DP, and each DP can reuse more than one CU uplink resource. The system model of the evaluated D2D communication system is illustrated as Fig. 1.

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Fig. 1

Schematic of D2D communication system.

In the evaluated D2D communication system, the optimization objective is to maximize the entire energy efficiency by effectively allocating the uplink resource of all the CUs. At the same time, the communication QoS of each DP should be also ensured. According to reference [36], the entire energy efficiency (denote as EE) to be maximized is formulized as Equation (1). EE = ∑ m = 1 M ∑ n = 1 N x m , n · R m , n ∑ m = 1 M ∑ n = 1 N x m , n · P m , n + P c(1) where x_m,n denotes whether the uplink resource of CU _n is reused by DP _m . The definition of x_m,n can be formulized as Equation (2); R_m,n denotes the transmission speed of DP _m when reuse the uplink resource of CU _n ; P_m,n denotes the transmission power of DP _m when reuse the uplink resource of CU _n ; P _c denotes the total circuit power consumption of DP _m .

x m , n = { 1 if the uplink resource of CU n is reused by DP m 0 , otherwise(2)

The transmission speed R_m,n in Equation (1) is related to the transmission power of DP _m when reuse CU _n , the channel gain from DP transmitter DT _m to DP receiver DR _m , the transmission power of CU _n , the channel gain from DT _m to CU _n and the channel noise power. According to reference [36], the transmission speed R_m,n can be formulized as R m , n = log 2 ( 1 + P m , n · H m P n · H m , n + n 0 )(3) where H _m denotes the channel gain from DP transmitter DT _m to DP receiver DR _m ; P _n denotes the transmission power of CU _n ; H_m,n denotes the channel gain from DT _m to CU _n ; n₀ denotes the channel noise power under the effect of white Gaussian noise.

According to Equation (2), x_m,n is defined as binary variable. In addition, as the uplink resource of each CU can be reused by at most one DP and each DP can reuse more than one CU uplink resource, the constraints for Equation (1) can be formulized as s . t . { x m , n ∈ { 0 , 1 } ∑ m = 1 M x m , n ⩽ 1 ( n = 1 , 2 , . . . , N ) ∑ n = 1 N x m , n ⩾ 1 ( m = 1 , 2 , . . . , M )(4)

As shown in Equation (4), the introduced constraints are used to ensure the allocation solutions satisfy the following conditions: (1) all the variables within a certain solution (i.e., x_m,n) are binary variables, in another word, the value of each optimization variable x_m,n should be equal to 0 or 1; (2) the uplink resource of each CU can be reused by at most one DP. That is to say, for the variables within a valid solution, at most one variable within x_1,n, x_2,n,  ...  , x_M,n (n = 1, 2,  ...  , N) is equal to 1; (3) each DP can reuse more than one CU uplink resource (but at least one). That is to say, in a valid allocation solution, at least one variable within x_m,1, x_m,2,  ...  , x_m,N (m = 1, 2,  ...  , M) is equal to 1.

In the resource allocation problem as described above, the variables to be optimized are the relationship between CU _n (n = 1, 2,  ...  , N) and DP _m (m = 1, 2,  ...  , M). That is to say, to decide each x_m,n (n = 1, 2,  ...  , N; m = 1, 2,  ...  , M) to be equal to whether 1 or 0, in order to obtain the maximum spectrum efficiency and satisfy the constraints listed in Equation (4). In this study, the optimization vector x → is defined as

x → = ( x 1 , x 2 , . . . , x N ) , x n ∈ { 1 , 2 , . . . , M } ( n = 1 , 2 , . . . , N )(5)

Note that in Equation (5), each x _n is a discrete variable, and x _n  = m (m = 1, 2,  ...  , M) indicates the uplink resource of CU _n is reused by DP _m . For example, assume that the number of cellular users N = 8 and the number of D2D-pair users M = 4. For a certain value of optimization vector, e.g., x → = { 1 , 1 , 3 , 4 , 2 , 3 , 1 , 2 } , it denotes that the 1st, 2nd and 7th CU uplink resource are allocated to the 1st DP, the 5th and 8th CU are allocated to the 2nd DP, the 3rd and 6th CU are allocated to the 3rd DP, and the 4th CU is allocated to the 4th DP. The coding strategy employed in this study is illustrated in Fig. 2. The corresponding resource allocation strategy for this example is listed in Table 1.

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Fig. 2

Coding strategy of optimization vector.

As shown in Equation (5), the upper and lower boundaries of optimization variable x _n are M and 1, respectively. In addition, considering that each x _n is a discrete variable, the coding strategy to connect the continuous algorithm and discrete variable x _n is defined as

x n = { 1 if x n * ∈ [ 0 , 1 ) 2 , if x n * ∈ [ 1 , 2 ) … N , if x n * ∈ [ N - 1 , N ) ( n = 1 , 2 , . . . , N )(6)

where x n * denotes the continuous variable of x _n after coding operation.

In order to optimize the model described in Section 3.1 using swarm-intelligence-based approach, the fitness function should be defined accurately. As discussed in Section 3.1, target of the resource allocation problem is to maximize the entire energy efficiency EE in Equation (1). According to reference [36], the maximization of EE is equal to the minimization of the following equation f min = ∑ m = 1 M ∑ n = 1 N x m , n · H n , m · H m , n H m · H n(7) where H _n denotes the channel gain from CU _n to eNB; H_n,m denotes the channel gain from CU _n to DR _m . The definitions of H _m and H_m,n are the same as Equation (3). The relationship between x_m,n and x _n is illustrated as Section 3.2. By taking shadowing effect into consideration, as suggested in reference [5], the channel gains here (H _n , H _m , H_n,m and H_m,n) are calculated as H = 10^{(- PL - shad ∘wing)/10}, where PL represents the pathloss, and shadowing represents the lognormal fading between transmitter and receiver.

Considering that each DP should be allocated with at least one CU uplink resource, the allocation solution that does not satisfy this constraint should be punished. As a result, the punish function is defined as f p = λ · ∑ m = 1 M ∏ n = 1 N ( m - x n ) 2(8) where λ denotes the penalty factor which is used for controlling the punishment intensity. Note that in Equation (8), the punish function f _p is equal to 0 only if each DP _m is allocated with at least one CU uplink resource. Otherwise, for a certain DP (say DP _m ), if DP _m is not allocated with any resource, then all the items (m-x _n )² (n = 1, 2,  ...  , N) are not equal to 0. Thus, the allocation solution will be punished by a large f _p value, and the solution is naturally given up by the swarm-intelligent algorithm.

4.1 Differential evolution

Differential evolution (DE) is one of the most effective swarm-intelligence-based optimization algorithms [37–39]. In DE, the mutation, crossover and selection operations are adopted to evolve the population and search the global optimum within search space. Individuals in DE are represented by n-dimensional optimization vectors x _i , (i = 1, 2,..., N _P ), where N _P denotes the population size.

1) Mutation operator v i = x i 1 + F · ( x i 2 - x i 3 )(9) where i, i₁, i₂ and i₃ ∈{1,  2,   . . . ,  N _P }, are mutually different; F is a positive coefficient used to control the differential variation; v _i denotes the ith newly created mutant individual; x _{i 1}, x _{i 2} and x _{i 3} denote the individuals in the current population related to the corresponding subscript.

2) Crossover operator u i ( j ) = { v i ( j ) , if rand j < CR or j = j rand x i ( j ) , otherwise(10) where rand _j is a uniformly distributed random number between 0 and 1; j _rand is a randomly selected index; CR ∈ (0, 1) represents the crossover rate; u _i denotes the ith newly created individual after the crossover operation.

3) Selection operator x i ′ = { u i , if f ( u i ) < f ( x i ) x i , otherwise(11) where x i ′ is the offspring of x _i in the next generation.

1) Cooperative coevolution in differential evolution

In DTC-DE, the effective cooperative coevolution (CC) framework is adopted to improve the convergence ability for high-dimensional problem [40]. That is to say, the D-dimensional original problem is decomposed into several sub-problems and each with a certain number of variables. Then, in each cycle of DTC-DE, all the sub-problems are evolved in turn. For computing the fitness function value of the individuals within each sub-problem, a D-dimensional vector is defined as the combination of the best individual in each sub-component, and is used as the so-called “context vector”. The CC framework in DTC-DE can be illustrated as the following steps:

➀ Initialize a D-dimensional swarm P with N _p individuals, and then decompose P into K groups P _i (i = 1, 2,  ...  , K), each group represents a sub-problem with D/K dimensionalities. For a D-dimensional individual x=(x¹, x²,  ...  , x ^K ), where x ⁱ represents the ith group of components;

➁ Denote y ˆ ( t ) as the context vector, and the ith sub-problem with y ˆ ( t ) can be formulized as min f i ( y , y ˆ ( t ) ) , y ∈ R S(12) where f i ( y , y ˆ ( t ) ) = f ( ( y ˆ ( t ) 1 , . . . , y ˆ ( t ) i - 1 , y , y ˆ ( t ) i + 1 , . . . , y ˆ ( t ) K ) ) . Start an evolution circle, in which all the sub-problems are optimized by DTC-DE in turn, and update y ˆ ( t ) after solving each sub-problem.

➂ Proceed another cycle if stopping criteria are not satisfied, otherwise stop the coevolution and output the final result.

2) Dynamic topology

In DTC-DE, we divide the original population P (include N _p individuals) into N _sp sub-populations and each with N _p /N _sp individuals (denote M _sp  = N _p /N _sp ). Denote the best individual of the jth sub-population (SP _j ) as Ibest _j (j = 1, 2,  ...  , N _sp ). Then, when evolve a certain individual x _i which belongs to the jth sub-population, the adopted context vector (see CC framework) is randomly selected from Ibest _j and all the Ibest_{j - topology}. Note that Ibest_{j - topology} denotes the neighbour sub-populations of SP _j under a certain topology. The concept of “dynamic topology” implies that all the N _sp sub-populations are connected in ring topology at beginning. Then, randomly connect each pair of non-neighbour sub-populations (denote as NSP-pair) at each iteration while the original ring topology keep constant. In the dynamic topology mechanism, we define a parameter P _dt  ∈ [0,  1] to control the connections between each NSP-pair, which is formulized as C ij = { 1 , if r ij ⩽ P dt and i ≠ j 0 , otherwise(13) where C _ij denotes the connection between SP _i and SP _j . C _ij  = 1 means connect SP _i and SP _j , and C _ij  = 0 means the opposite; r _ij is randomly generated with (0, 1). Note that P _dt  = 0 represents the original ring topology, and P _dt  = 1 represents the full-connection topology. In the proposed dynamic topology mechanism, Equation (13) is implemented independently for each NSP-pair in every iteration. That is to say, the topology structure dynamically changes during the entire evolving process.

The dynamic topology mechanism is illustrated as Fig. 3, in which the neighbour relationships are indicated by solid lines. For example, the neighbour sub-populations of SP₄ are SP₃, SP₅ and SP _Nsp .

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Fig. 3

Dynamic topology mechanism in DTC-DE.

3) “DE/current-to-dtbest/1” mutation operator

Mutation operator significantly affects the overall performance of DE. Over the last few years, some mutation operators with different properties are proposed, e.g., the “DE/rand/1”, “DE/best/1”, “DE/current-to-best/1”, “DE/current-to-rand/1”, “DE/current-to-pbest/1”, “DE/current-to-SP-best-ring/1” and so on [38]. In DTC-DE, a novel “DE/current-to-dtbest/1” mutation operator is proposed as v i = x i + F 1 · ( dt best i - x i ) + F 2 · ( x r 1 - x oea )(14) where “dtbest _i ” implies the best individual under dynamic topology connection, i.e., randomly selected from the best individual of the ith sub-population (Ibest _i ), and the best individuals of all its neighbour sub-populations (Ibest_{i - topology}); x _{r 1} denotes a random individual in the current population; x _oes denotes a randomly selected individual from the union of current population and the so-called “optional external archive”, which is composed of the inferior individuals from selection operation [41]; F₁ and F₂ are positive coefficients.

As shown in Equation (14), the proposed “DE/current-to-dtbest/1” is neither too greedy nor too stochastic. The introduction of “dtbest _i ” and “x _oea ” aims to provide a slower but more extensive communication mode for all the individuals to balance the global exploration and local exploitation.

As discussed above, the pseudo code of DTC-DE is illustrated as Table 2.

In this section, the performance of DTC-DE is empirically evaluated on a comprehensive set of benchmark functions, which are listed in Table 3. In the following experiments, the maximum number of fitness evaluations (FE _max ) is set to 1 × 10⁶. Dimensionality of each function is set as D = 100, and the population size is set to 50. The dynamic group size of DTC-DE is set as S ={2, 5, 10, 20}. The number of sub-population is set as N _sp  = 10. Settings of the mutation coefficients F₁ and F₂ in Equation (14), and the crossover rate CR are following reference [41]. In addition, some state-of-the-art swarm-intelligent algorithms are also employed for comparison, including the CPSO-S _{K  - rg - aw} [42], CCPSO2 [43], CCDE [44], CPSO-S _K , HSPSO-FI [32] and AMCCDE [39]. The parameter settings of these compared algorithms are following their original studies.∥In the following experiments, the results of 20 independent runs are recorded. Experimental results are compared in Fig. 4 and Table 4.

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Fig. 4

The average convergence graphs on f₁ to f₆ for 20 independent runs.

As shown in Fig. 4 and Table 4, the proposed DTC-DE outperforms the other compared algorithms on most of the benchmarks. For f₁ and f₂, DTC-DE is the best performer on both accuracy and convergence speed. The average errors on f₁ and f₂ obtained by DTC-DE are 8.38 × 10^-55 and 3.55 × 10^-15 respectively, which significantly outperform its competitors. For f₃, DTC-DE obtains the average optimization error of 2.71 × 10^-03, which slightly outperforms 7.63 × 10^-03 obtained by CCDE, 3.76 × 10^-03 obtained by CPSO-S _{K  - rg - aw}, 1.32 × 10^-02 obtained by CCPSO2 and 1.45 × 10^-02 obtained by AMCCDE, respectively. For f₄, the errors of DTC-DE, CCDE and AMCCDE are almost the same, but DTC-DE obtains the fastest convergence speed. For f₅, the outperformance of DTC-DE on optimization accuracy and convergence speed is significant: the best performance (1.27 × 10^-26) is obtained by DTC-DE, which is followed by 2.17 × 10^-16 obtained by CCDE. AMCCDE (1.76 × 10^-11) and CCPSO2 (4.73 × 10^-10) are the 3rd and 4^th best performers, respectively. However, HSPSO-FI performs the worst on f₅ and obtains the optimization error of 9.66 × 10⁺⁰⁰. Finally, for the benchmark f₆, all the algorithms lose the efficacy and obtain a relatively large error of more than 10 + ⁰⁵.∥In order to compare the optimization results more clearly, the interval graphs are further plotted in Fig. 5. Note that in Fig. 5 (c), as the results of some algorithms are 0, the ordinate of the figure is the absolute error without logarithmic transformation. As shown in the figures, the developed DTC-DE algorithm obtains promising performance on most of the benchmarks. For example, as shown in Fig. 5 (b), (c) and (d), DTC-DE obtains the best performance on accuracy and robustness for f₂, f₃ and f₄. For the benchmarks f₁ and f₅, as shown in Fig. 5 (a) and (e), although the robustness of DTC-DE is worse than its performance on f₂ to f₄, DTC-DE can still obtain the best optimization accuracy. Especially for f₅, the outperformance of DTC-DE is very significant.

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Fig. 5

The interval graphs on f₁ to f₆ for 20 independent runs.

5.1 Parameter setting and simulation analysis

In this section, the proposed DTC-DE is employed to optimize the aforementioned ORA model for obtaining the optimal resource allocation in D2D communication. For evaluating the effectiveness of DTC-DE, the CPSO-S_{K - rg - aw}, CCPSO2, CCDE, CPSO-S_K, HSPSO-FI and AMCCDE described in Section 4 are employed for comparison. In the following simulation, the numbers of CU and DP are set to 80 and 20, respectively. Assume all these users are randomly distributed within a hexagon region with a radius of 700 m. The distribution of these 120 users are shown as Fig. 6.

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Fig. 6

Distribution of CU and DP of the case study.

As suggested in reference [5], the communication model and parameters employed in the following simulation experiments are chosen in accordance with 3GPP LTE regulation for OFDMA system. The pathloss model is defined as Equation (15). { eNB - UE : PL = 33.65 + 23.47 log 10 ( d [ m ] ) UE - UE : PL = 36.67 + 19.54 log 10 ( d [ m ] )(15)

According to the coding strategy described in Section 3.2, the dimensionality of ORA model in this case is 80 as it is equal to the number of CU. As a result, the set of dynamic group size S in DTC-DE, CCPSO2 and CCDE is set as S ={2, 4, 8, 10, 20}. The penalty factor λ in Equation (8) is set to 100. The other parameters of these algorithms are set the same with Section 4. In the following simulation experiments, the results of 20 independent runs are recorded. Experimental results are compared in Fig. 7 and Table 5.

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Fig. 7

Convergence graphs for optimizing the ORA model.

As shown in Fig. 7 and Table 5, for the aforementioned ORA model in D2D communication system, the DTC-DE algorithm obtains the best optimization performance. To be specific, the average optimization result obtained by DTC-DE is 19.915, which significantly outperforms its competitors. AMCCDE and CCPSO2 are the second-best and third-best performers, which obtain the similar results of 24.197 and 25.814, respectively. Besides, the average results obtained by CCDE, CPSO-S_{K - rg - aw} and HSPSO-FI are 27.427, 32.472 and 46.540, respectively. Finally, CPSO-S_K performs the worst and its average optimization result is just 100.663. In addition, the standard deviation of the 20 independent runs obtained by DTC-DE is significantly smaller than the other compared algorithms. This implies that DTC-DE has good robustness performance on ORA model optimization. In addition, it can be seen from Fig. 7 that DTC-DE can also obtain the fastest convergence speed on optimizing the ORA model.

In order to further verify the effectiveness of the proposed methodologies, some comparisons for general cases are conducted in this section. For each case, the parameter settings of ORA model are listed in Table 6. Numerical results of the simulation experiments are listed in Table 7, in which the best performance is set in bold. The convergence graphs for each case are plotted in Fig. 8. Note that the ordinate in Fig. 8 is the fitness function value after logarithmic transformation.

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Fig. 8

Convergence graphs for Case 1 to 6.

As shown in Table 7 and Fig. 8, performance of the compared algorithms is almost the same when the model dimensionality is low. For example, in Case 1, the final fitness function values (listed in the first column of Table 7) and the convergence speed (reflected in Fig. 8 (a)) for the 7 algorithms are very close. However, the outperformance of DTC-DE increases with the growing of model dimensionality. For example, in Case 2, the final fitness function value obtained by DTC-DE is just 6.502, which is close to 8.847 obtained by CPSO-S_{K - rg - aw} and 9.595 obtained by AMCCDE, but significantly better than that of CCDE, CCPSO2, CPSO-S_K and HSPSOFI. Similarly, for Case 3 and Case 4, the final results obtained by DTC-DE are 21.019 and 33.426 respectively, which significantly outperform most of the competitors. Moreover, when the model dimensionality continuously increases to 150 (in Case 5) and 200 (in Case 6), the outperformance of DTC-DE becomes more significant. For example, in Case 6, the final result obtained by DTC-DE is just 85.652, which is significantly less than 160.368 of CCDE, 136.227 of CCPSO2, 265.518 of CPSO-S_{K - rg - aw}, 647.646 of CPSO-S_K, 356.111 of HSPSO-FI and 97.082 of AMCCDE. In addition, as shown in Fig. 8, DTC-DE can also obtain the fastest convergence speed in Case 3, 4, 5 and 6. In a word, results of the general case studies further show the effectiveness and robustness of the proposed DTC-DE algorithm, especially when employed in optimizing the high-dimensional ORA model of D2D communication system.