Efficient global optimisation of microwave antennas based on a parallel surrogate model‐assisted evolutionary algorithm
2018; Institution of Engineering and Technology; Volume: 13; Issue: 2 Linguagem: Inglês
10.1049/iet-map.2018.5009
ISSN1751-8733
AutoresBo Liu, Mobayode O. Akinsolu, Nazar Ali, Raed A. Abd‐Alhameed,
Tópico(s)Antenna Design and Optimization
ResumoIET Microwaves, Antennas & PropagationVolume 13, Issue 2 p. 149-155 Research ArticleFree Access Efficient global optimisation of microwave antennas based on a parallel surrogate model-assisted evolutionary algorithm Bo Liu, Corresponding Author Bo Liu liubo168@gmail.com orcid.org/0000-0002-3093-4571 School of Applied Science, Computing and Engineering, Wrexham Glyndwr University, Mold Road, Wrexham, UK Department of Electronic, Electrical and Systems Engineering, University of Birmingham, Edgbaston, Birmingham, UKSearch for more papers by this authorMobayode O. Akinsolu, Mobayode O. Akinsolu School of Applied Science, Computing and Engineering, Wrexham Glyndwr University, Mold Road, Wrexham, UK Department of Electronic, Electrical and Systems Engineering, University of Birmingham, Edgbaston, Birmingham, UKSearch for more papers by this authorNazar Ali, Nazar Ali Department of Electrical and Computer Engineering, Khalifa University, Al Zafranah, Abu Dhabi, UAESearch for more papers by this authorRaed Abd-Alhameed, Raed Abd-Alhameed School of Electrical Engineering and Computer Science, University of Bradford, Richmond Road, Bradford, UKSearch for more papers by this author Bo Liu, Corresponding Author Bo Liu liubo168@gmail.com orcid.org/0000-0002-3093-4571 School of Applied Science, Computing and Engineering, Wrexham Glyndwr University, Mold Road, Wrexham, UK Department of Electronic, Electrical and Systems Engineering, University of Birmingham, Edgbaston, Birmingham, UKSearch for more papers by this authorMobayode O. Akinsolu, Mobayode O. Akinsolu School of Applied Science, Computing and Engineering, Wrexham Glyndwr University, Mold Road, Wrexham, UK Department of Electronic, Electrical and Systems Engineering, University of Birmingham, Edgbaston, Birmingham, UKSearch for more papers by this authorNazar Ali, Nazar Ali Department of Electrical and Computer Engineering, Khalifa University, Al Zafranah, Abu Dhabi, UAESearch for more papers by this authorRaed Abd-Alhameed, Raed Abd-Alhameed School of Electrical Engineering and Computer Science, University of Bradford, Richmond Road, Bradford, UKSearch for more papers by this author First published: 15 November 2018 https://doi.org/10.1049/iet-map.2018.5009Citations: 30AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract Computational efficiency is a major challenge for evolutionary algorithm (EA)-based antenna optimisation methods due to the computationally expensive electromagnetic simulations. Surrogate model-assisted EAs considerably improve the optimisation efficiency, but most of them are sequential methods, which cannot benefit from parallel simulation of multiple candidate designs for further speed improvement. To address this problem, a new method, called parallel surrogate model-assisted hybrid differential evolution for antenna optimisation (PSADEA), is proposed. The performance of PSADEA is demonstrated by a dielectric resonator antenna, a Yagi-Uda antenna, and three mathematical benchmark problems. Experimental results show high operational performance in a few hours using a normal desktop 4-core workstation. Comparisons show that PSADEA possesses significant advantages in efficiency compared to a state-of-the-art surrogate model-assisted EA for antenna optimisation, the standard parallel differential evolution algorithm, and parallel particle swarm optimisation. In addition, PSADEA also shows stronger optimisation ability compared to the above reference methods for challenging design cases. 1 Introduction In recent years, evolutionary algorithms (EAs) are playing an important role in antenna design optimisation [[1]–[4]] because of their global optimisation capability, free of an initial design, generality, and robustness. However, the efficiency of the optimisation process is a major challenge due to the computationally expensive electromagnetic (EM) simulations. An effective efficiency improvement method is surrogate model-assisted evolutionary algorithms (SAEAs), which are employed and developed for antenna design optimisation [[5], [6]]. Using the antenna design parameters as the input and EM-simulated responses as the output, a computationally cheap surrogate model (which is often based on statistical learning techniques) is constructed and used to replace potentially numerous computationally expensive EM simulations in optimisation, so as to highly improve the efficiency. Another speed improvement method is parallel computation. Employing parallel solvers for a single antenna design simulation has already been widely applied. The computing capability (CPU, memory) of desktop workstations is continuously increasing, while the complexity of numerical techniques does not. In addition, with the development of cloud computing, more computing resources can be utilised, and carrying out multiple simultaneous simulations of several antenna designs (parallel solvers are applied to each of them) is a foreseeable tendency. Even at present, the above two-level parallel simulation is already available in some commercial software packages such as the CST Microwave Studio using multi-cores and shared memory [[7]]. To avoid confusion, the parallel computation in this paper refers to parallel simulations of two to four antenna candidate designs considering the capacity of a standard desktop workstation or the financial cost of using shared computing resources in a cloud. Availability of large-scale high-performance computing facilities is not considered. Parallel solvers are also not the concentration of this paper despite that they may be employed for simulating each antenna candidate design in the optimisation process. Clearly, the above parallelism has difficulty in providing substantial help for standard EA-based antenna optimisation methods, because the required population size (i.e. needed EM simulations in one iteration) is often not small (e.g. 30–50) and the computing overhead is still large even after parallelisation. Therefore, a natural idea is to introduce the above parallelisation into SAEA-based antenna optimisation methods. Introducing parallel simulations into modern SAEAs is not trivial. In terms of surrogate model management methods, SAEAs can generally be classified into offline SAEAs and online SAEAs. Offline SAEA-based efficient antenna design optimisation methods perform a one-shot initial sampling to build the surrogate model, which is then used as the substitute of the EM simulations. In the optimisation process, the surrogate model is not updated or only updated for local refinements [[8]]. In this kind of methods, the EM simulations involved in the one-shot initial sampling are independent of each other and are easy to be parallelly performed. However, although more than ten variables can be handled when employing both low- and high-fidelity EM models [[9], [10]], it is well known that most offline SAEAs have difficulty in handling problems with more than a few variables (e.g. 4–5) for the targeted problem in this paper [[11], [12]]. To address the scalability problem, online SAEA-based efficient antenna design optimisation methods are introduced. Siah et al. [[5]] employ the efficient global optimisation method [[13]] for antenna optimisation and a new online SAEA, called surrogate model-assisted hybrid differential evolution for antenna optimisation (SADEA), is proposed in [[6]] by the authors. However, the terminology 'online' indicates a sequential process: the surrogate model keeps updating, often a single EM simulation is performed in each iteration, and the selected candidate design for that simulation depends on the current surrogate model built by simulation data in previous iterations [[5], [6]]. Therefore, parallel simulations are difficult to be used. Although Liu et al. [[6]] show up to almost an order of speed improvement compared to standard EAs for various antennas, parallel candidate antenna design simulation, which is straightforward for standard EAs, is not taken into account. This paper aims to introduce small-scale parallel simulations into online SAEA-based antenna optimisation. A new method, called parallel surrogate model-assisted hybrid differential evolution for antenna optimisation (PSADEA), is proposed. The main innovations are the ensemble of three selected differential evolution (DE) mutation strategies and the self-adaptive employment of them, as well as a new surrogate modelling method. These innovations enable parallelisation with enhanced search capacity and surrogate model quality. The goal of PSADEA is to reduce the several weeks' optimisation time of a standard EA-based antenna optimisation method to a few hours in a small-scale parallel computing environment. An additional benefit is the stronger performance optimisation ability compared to popular standard EAs, which are often considered as the benchmark in terms of solution quality. The remainder of the paper is organised as follows: Section 2 provides the basic techniques. Section 3 elaborates the PSADEA method. Section 4 presents two typical antenna examples and three mathematical benchmark problems to show the performance of PSADEA. The concluding remarks are presented in Section 5. 2 Basic techniques 2.1 Gaussian process surrogate modelling The surrogate modelling method selected for PSADEA is Gaussian process (GP) [[14]]. Given a set of observations and , GP predicts a function value at some design point x by modelling as a Gaussian distributed stochastic variable with mean and variance . If the function is continuous, the function values of two points and should be close if they are highly correlated. In this work, we use the Gaussian correlation function to describe the correlation between two variables: (1) where d is the dimension of x and the correlation parameter which determines how fast the correlation decreases when moves in the l direction. The smoothness of the function is related to with respect to . To determine the parameters and , the likelihood function that at is maximised. The function value at a new point can be predicted as (2) where (3) (4) (5) The mean square error value of the prediction uncertainty is: (6) where (7) Several prescreening methods can be used to evaluate the quality of a candidate design with respect to the predicted value in (2) and the prediction uncertainty in (6) [[15]]. In PSADEA, the lower confidence bound (LCB) method [[16]] is used. Given the predictive distribution for , an LCB prescreening of can be defined as: (8) where is a constant, which is often set to 2 to balance the exploration and exploitation ability [[15]]. The ooDACE toolbox [[17]] is used for implementing the GP surrogate model. 2.2 DE algorithm The DE algorithm [[18]] is adopted as the search engine in PSADEA. DE outperforms many EAs for continuous optimisation problems [[18]] and is widely used in EM simulation-driven design optimisation. Suppose that P is a population. Let be an individual solution in P. To generate a child solution for x, a donor vector is first produced by mutation: (9) where , , and are three mutually exclusive solutions randomly selected from P (the current population); is the mutant vector in the population after mutation; is a control parameter, often called the scaling factor. The above mutation strategy in (9) is called DE/rand/1, which is one of the popular DE mutation strategies. Then the following crossover operator is applied to produce the child u: Randomly select a variable index . For each to d, generate a uniformly distributed random number from and set: (10) where is a constant called the crossover rate. In PSADEA, two other DE mutation strategies are used: Mutation strategy: DE/current-to-best/1 (11) where is the vector in the current population and the best candidate in the current population P. Mutation strategy: DE/rand/2 (12) where and are two different solutions randomly selected from P and are different from , , and . 3 PSADEA method 3.1 SADEA method PSADEA aims to introduce small-scale parallel simulation (simultaneous simulations of two to four candidate antenna designs) into the authors' SADEA method [[6]]. Hence, SADEA is firstly introduced as the foundation. SADEA shows up to an order of speed improvement compared to standard EAs for antenna optimisation (not considering parallel simulation of different candidate designs) and is scalable to around 30 design variables [[6], [11]]. SADEA works as follows: Step 1: Sample (often a small number) candidate designs from the design space (a and b are the lower and upper bounds of design variables, respectively; d is the number of design variables) using Latin hypercube sampling [[19]]. Evaluate the objective function values of all these solutions using EM simulations and let them form the initial database. Step 2: If a preset stopping criterion is met (e.g. a maximum number of allowed EM simulations is exceeded), output the best solution from the database; otherwise go to step 3. Step 3: Select the best solutions from the database to form a population P. Step 4: Apply the DE/best/1 mutation (13) and crossover operations [[18]] to P to generate new child solutions. Step 5: Select nearest candidate designs from the database (based on Euclidean distance in the design space) around the centroid of the child solutions. Construct a GP surrogate model [[14]] using the selected candidate designs (i.e. training data points in surrogate modelling). Step 6: Estimate the child solutions generated in step 4 using the GP model and the LCB method [[15]]. Step 7: Evaluate the EM simulation model at the estimated best child design candidate from step 6. Add this candidate design and its objective function value to the database. Go back to step 2. The DE/best/1 mutation strategy used in SADEA is: (13) 3.2 Exploration ability enhancement and local surrogate modelling To include parallel simulations in SADEA, a straightforward idea is to select the top n rather than the top 1 candidate designs (all based on prediction) in step 7 of each iteration and simulate them in parallel. Here, n depends on the computing resource and is often from 2 to 4 as mentioned in Section 1. If this framework is effective, the computational time should be reduced by approximately n times, like simulating n candidate designs simultaneously in standard EAs. However, experiments show far less than three times speed improvement when using various mathematical benchmark problems [[20]]. Two observations are: (i) in many iterations, the predicted best candidate is the true best candidate and there are often very few other candidates with similar optimality. (ii) The best candidate may not change for every 10–20 iterations. This explains why simply selecting n top-ranked candidates for parallel simulation is not effective: most top-ranked candidates do not have the required quality. Simulating them does not help much to obtain good quality solutions and improve the surrogate model quality of the optimal region. Hence, improving the exploration ability to obtain more good quality candidates in each iteration becomes the central issue. In SADEA, the exploration is realised by the DE/best/1 mutation strategy (13) [[18]] (step 4). The main consideration of using this strategy is its fast convergence property, while the cost is the reduced population diversity. Although experiments show that it can still jump out of local optima, various good quality candidates from different regions of the design space are difficult to be generated, and the parallel simulation of ' useful' candidates, therefore, cannot be realised. Our idea to address this challenge is to ensemble the DE/current-to-best/1 (11), DE/rand/1 (9), and DE/rand/2 (12) strategies. Each DE mutation strategy [(9)–(12)] trades off the exploration ability (population diversity) and convergence speed to a certain extent. DE/current-to-best/1 (11) has the least population diversity consideration (but still higher than DE/best/1), but the convergence speed is the fastest, DE/rand/2 (12) emphasises the promotion of population diversity but with the slowest convergence speed. DE/rand/1 (9) is in the middle in both respects. When employing the new DE mutation strategies, it is observed that various good quality candidate designs from different regions of the design space can be generated, but many of them are not correctly predicted by the GP model in step 5. SADEA constructs a single surrogate model in each iteration. When the diversity of the generated candidates becomes high thanks to the new DE mutation strategies, this single surrogate model does not have good performance. Therefore, in PSADEA, a local GP surrogate model is built for each child solution using nearest samples (based on Euclidean distance). This means that separate local GP models are built in each iteration. Pilot experiments show substantially improved prediction ability. 3.3 Self-adaptive DE mutation strategy selection and the PSADEA framework Given the three selected DE mutation strategies and the surrogate modelling method, the key problem becomes how to optimally use the selected mutation strategies. Experiments show that when evenly employing the three selected DE mutation strategies, although high-quality design solutions can be generated with improved speed, the speed improvement is not as high as n times when n candidate designs are simulated in parallel. Note that DE/rand/1 and DE/rand/2 strategies have low convergence speed. As said in Section 3.2, each mutation strategy has a different trade-off between the exploration ability (population diversity) and convergence speed. On the one hand, sufficient population diversity is needed. On the other hand, the convergence speed should be as fast as possible. However, the most appropriate trade-off depends on the particular antenna optimisation problem and is difficult to be analysed mathematically. To address this challenge, a self-adaptive method is proposed, which works as follows: For each child population : Step 1: If the algorithm is within the learning period (the current number of iterations is smaller than a threshold L), the rate of using DE/rand/1 (9), DE/current-to-best/1 (11), and DE/rand/2 (12) is 1/3. Otherwise, use the rates in step 5. Step 2: Perform a roulette wheel selection [[21]] based on the rates to determine a DE mutation method and generate a child population ( child solutions). Step 3: For each child solution, select nearest samples (based on Euclidean distance) as the training data points and construct a local GP surrogate model. Step 4: Compare the predicted value of each solution in and the current best solution (simulated value). Add the number of solutions that are better than the current best solution to [the number of successes of (9), (11), or (12)] and add to [the number of uses of (9), (11) or (12)]. Until all the n groups of child solutions are generated. Step 5: Update the rates of using DE/rand/1 (9), DE/current-to-best/1 (11), and DE/rand/2 (12) by . Update the number of iterations. It can be seen that the DE mutation strategies are employed self-adaptively and the most frequently fitted will be used with a greater chance. Experiments show higher than n times time reduction compared to SADEA. As shown by the flow diagram in Fig. 1, the PSADEA method can be constructed as follows: (i) steps 1–3 are the same as the SADEA method (Section 3.1). (ii) The above self-adaptive mutation method (new steps 1–5) is used to replace the original steps 4–6 of SADEA. (iii) Step 7 of SADEA is revised to select top n candidate designs from all of () the generated child solutions and simulate them in parallel. Fig. 1Open in figure viewerPowerPoint Flow diagram of the PSADEA method Compared to SADEA, there is only one additional algorithm parameter, L, in PSADEA. Empirically, L is suggested to be within [30, 50]. It is clear that L is not sensitive. It is shown that the SADEA method is not sensitive to parameter setting [[6]] through detailed experimental study. All the parameter setting rules are still useful in PSADEA. 4 Experimental results and comparisons PSADEA is tested by four real-world antennas, including a hybrid dielectric resonator antenna (DRA), a planar Yagi-Uda antenna (YUA), a broadband microstrip antenna, and an ultrawideband MIMO antenna array. The reference methods include SADEA, DE (DE/current-to-best/1 mutation), and particle swarm optimisation (PSO) (the CST Microwave Studio PSO optimiser is used). As SADEA outperforms several popular SAEAs [[6]], it is used as the reference method in this paper. For all of the test cases, PSADEA successfully obtains highly optimal designs. The comparison results among different methods in terms of solution quality and efficiency are similar except that DE and PSO fail to obtain a satisfactory design for the YUA. The tests are run on a workstation with Intel 4-core i7 CPU and 24 GB RAM. Ten independent runs are carried out for PSADEA and SADEA. Standard DE and PSO are carried out for three runs because multiple runs are not affordable (a single run costs ∼10 days for some test cases). SADEA does not support parallel simulations of different candidate designs [[6]]; for the parallel simulations, three candidate antenna designs are simulated in parallel for PSADEA, PSO, and DE (parallel solvers are used to each of them). This is decided by the optimal computing budget allocation algorithm of CST Microwave Studio according to the CPU and shared memory of the workstation. Note that the initial surrogate model in all test cases is constructed by 50 samples/simulations, so the starting time of PSADEA and SADEA is not 0. Based on the suggested values in [[6]], the following algorithm parameters are used for PSADEA and SADEA: , , , , (d is the number of design variables). L is set to 30. For DE, the same F, , and are used. For PSO, the same is used and default values set by the CST Microwave Studio optimiser are used for other parameters. In the following, the DRA and YUA are used to demonstrate PSADEA. The time consumption is wall clock time. 4.1 Example 1 The first example is a hybrid DRA and the layout is shown in Fig. 2a [[22]]. The DRA is modelled in CST Microwave Studio using a mesh density of 12 cells per wavelength and ∼22,000 mesh cells. It is implemented on a RO4003C substrate with a relative permittivity of 3.38, loss tangent of 0.0027, and thickness of 0.5 mm. The excitation mode is and each simulation costs from 20 s to 1 min. As shown in Table 1 and (14), the design exploration goal is to adjust the dimensions of the DR brick and slot, length of the microstrip slab, and position of the DR relative to the slot, so that the bandwidth of the DRA is to be centred at 5.5 GHz from 5.28 to 5.72 GHz. () is the geometric constraint used to ensure the slot remains under the DRA in all possible cases. To make all the methods converge, the computing budget is as follows: 360 parallel EM simulations for PSADEA, 1080 sequential EM simulations for SADEA (the same number of total EM simulations with PSADEA), and 2000 parallel EM simulations each for PSO and DE. (14) Fig. 2Open in figure viewerPowerPoint Layout of the DRA and a typical optimised performance by PSADEA (a) The DRA, (b) response Table 1. Ranges of the design variables (all sizes in mm) (example 1) Variables lower bound 6 12 6 6 0.5 4 2 upper bound 10 16 10 8 4 12 12 The results are shown in Table 2. A typical response of PSADEA is shown in Fig. 2b with the obtained optimal design of , , , , , , and . The following observations can be made: (i) in all the ten runs, PSADEA provides very satisfactory results even for the worst one, using only 2.8 h on average. (ii) PSADEA shows good robustness. (iii) PSADEA shows a clear advantage over SADEA, parallel DE, and parallel PSO in terms of solution quality. Table 2. Optimised in the operational bandwidth using different methods for example 1 Method Best Worst Mean Std PSADEA −27.32 −24.76 −26.14 0.89 SADEA −25.70 −22.05 −23.84 1.10 parallel DE −25.03 −23.15 −24.31 1.01 parallel PSO −22.58 −19.05 −21.34 1.99 The convergence trends of all methods are shown in Fig. 3. From Fig. 3a, it can be seen that to obtain the average best objective function value of the SADEA algorithm (−23.84 dB at 8.5 h), PSADEA needs 2 h. Hence, PSADEA shows 4.3 times speed improvement compared to SADEA when three candidate designs are simulated in parallel. To obtain the average best objective function values of parallel DE (−24.31 dB at 16.1 h) and parallel PSO (−21.34 dB at 13.2 h), PSADEA needs 2.1 h and 1.6 h, respectively. Hence, PSADEA offers 7.7 times speed improvement over parallel DE and 8.3 times speed improvement over parallel PSO, while obtaining better results. This speed improvement comes from both parallel simulations of three candidate designs and the self-adaptive SAEA framework. Fig. 3b shows the convergence trends for PSADEA and SADEA when not considering parallel simulations. (The parallel simulations of PSADEA are considered as sequential.) It can be seen that to obtain the average best objective function value of SADEA (−23.84 dB at 1059 sequential EM simulations), PSADEA needs 708 sequential EM simulations. This speed improvement is only contributed by the self-adaptive SAEA framework. Fig. 3Open in figure viewerPowerPoint Convergence trends for example 1 (a) Convergence trends for PSADEA and SADEA (average of ten runs), parallel DE, and parallel PSO (average of three runs), (b) Comparison of PSADEA (in sequential mode) and SADEA 4.2 Example 2 As shown in Fig. 4a, the second example is a planar YUA [[23]] modelled in CST Microwave Studio using a mesh density of 15 cells per wavelength and over 86,000 mesh cells. The YUA is implemented on an RT6010 substrate with the relative permittivity of 10.2, loss tangent of 0.0033, and thickness of 0.635 mm. The driven element of the YUA is fed by a microstrip-to-slot balun via a power divider and each simulation costs ∼2 min. As shown in Table 3 and (15), the design exploration goal is to adjust the dimensions of the YUA for an operational bandwidth of 10–11 GHz at an average gain not <6 (7.78 dB) over the bandwidth. To make all the methods converge, the computing budget is as follows: 220 parallel EM simulations for PSADEA, 660 sequential EM simulations for SADEA (the same number of total EM simulations with PSADEA), and 1500 parallel EM simulations each for PSO and DE. (15) Fig. 4Open in figure viewerPowerPoint Layout of the YUA and typical optimised performances by PSADEA (a) The YUA, (b) response, (c) Gain response Table 3. Ranges of the design variables (all sizes in mm) (example 2) Variables lower bound 3 1 5 2 2 2 1 1 upper bound 7 6 12 12 6 6 5 5 The results are shown in Table 4. Typical responses of PSADEA are shown in Figs. 4b and c with the obtained optimal design of , , , , , , , and . For this example, the following observations can be made: (i) PSADEA and SADEA obtain successful results, while parallel PSO and parallel DE fail to satisfy the constraints on gain and provide poor responses. (ii) In all the ten runs, PSADEA provides very satisfactory results even for the worst one, using only 6.3 h on average. (iii) PSADEA shows good robustness and reveals an added advantage over SADEA in terms of solution quality. Table 4. Optimised in the operational bandwidth using different methods for example 2 Method Best Worst Mean Std PSADEA −25.87 −22.82 −23.98 0.90 SADEA −24.53 −22.52 −23.63 0.70 parallel DE −6.27 −5.19 −5.57 0.61 parallel PSO −15.65 −7.53 −12.26 4.23 The convergence trends of all methods are shown in Fig. 5. From Fig. 5a, it can be seen that to obtain the average objective function value of the SADEA algorithm (−23.63 dB at 18.6 h), PSADEA needs 5.5 h. Hence, PSADEA shows over 3.4 times speed improvement than SADEA when three candidate designs are simulated in parallel. Again, when the parallel simulations of PSADEA are considered as sequential (Fig. 5b), to obtain the average best objective function value of SADEA (−23.63 dB at 645 sequential EM simulations), PSADEA needs 546 sequential EM simulations. This speed improvement is only contributed by the self-adaptive SAEA framework. Fig. 5Open in figure viewerPowerPoint Convergence trends for example 2 (a) Convergence trends for PSADEA and SADEA (average of ten runs), parallel DE, and parallel PSO (average of three runs), (b) Comparison of PSADEA (in sequential mode) and SADEA 4.3 Benchmark problem tests To complement the antenna examples, the performance of PSADEA is demonstrated by three mathematical benchmark problems [[20], [24]]. Mathematical benchmark problems are computationally cheap analytical functions. Using them, the algorithm performance can be directly evaluated becaus
Referência(s)