Engineering optimization by constrained differential evolution with nearest neighbor comparison

Vietnam Journal of Mechanics, VAST, Vol. 38, No. 2 (2016), pp. 89 – 101 DOI:10.15625/0866-7136/38/2/6568 ENGINEERING OPTIMIZATION BY CONSTRAINED DIFFERENTIAL EVOLUTIONWITH NEAREST NEIGHBOR COMPARISON Pham Hoang Anh National University of Civil Engineering, Hanoi, Vietnam E-mail: anhpham.nuce@gmail.com Received July 27, 2015 Abstract. It has been proposed to utilize nearest neighbor comparison to reduce the num- ber of function evaluations in unconstrained optimization. The nearest neighbor com- parison omits the function evaluation of a point when the comparison can be judged by its nearest point in the search population. In this paper, a constrained differential evolu- tion (DE) algorithm is proposed by combining the ε constrained method to handle con- straints with the nearest neighbor comparison method. The algorithm is tested using five benchmark engineering design problems and the results indicate that the proposed DE algorithm is able to find good results in a much smaller number of objective function eval- uations than conventional DE and it is competitive to other state-of-the-art DE variants. Keywords: Engineering optimization, differential evolution, ε constrained method, nearest neighbor comparison. 1. INTRODUCTION Engineering optimization problems arising from modern engineering design pro- cess often involve inequality and/or equality constraints. Most of these constrained opti- mization problems (COPs) are complex and difficult to solve by traditional optimization techniques [1]. Evolutionary algorithms (EAs) for the COPs have received considerable attention and have been successfully applied in many real applications [2–4]. Among different EAs, differential evolution (DE) [5] is considered as one of the most efficient algorithm and suitable for various engineering problems. The advantage of DE is that it has simple structure, requires few control parameters and highly supports parallel computation [6]. Together with the constraint-handling techniques, DE has been applied to the COPs [7–12]. However, one of the main issues in applying DE is its expensive computation re- quirement. It is from the fact that evolutionary algorithm (EA) often needs to evaluate c© 2016 Vietnam Academy of Science and Technology 90 Pham Hoang Anh objective function as well as constraints thousand times to get a well acceptable solu- tions. A simple method, the nearest neighbor comparison, has been proposed to reduce the number of function evaluations effectively [13]. This method uses a nearest neighbor in the search population to judge a new point whether it is worth evaluating, i.e. the function evaluation of a solution is omitted when the fitness of its nearest point in the search population is worse than that of the compared point. The nearest neighbor com- parison (NNC) method has been proposed for unconstrained optimization [13] and fuzzy structural analysis [14]. In this study, the NNC method is proposed to constrained optimization. In or- der to use the nature of NNC, the ε constrained method [15] is applied to handle con- straints. The ε constrained method can transform algorithms for unconstrained problems into algorithms for constrained problems using the ε level comparison that compares search points based on their pair of fitness value and their constraint violation. It has been shown that, the application of ε constrained method to DE (εDE) could solve con- strained problems successfully and stably [16–19], including engineering optimization problems [16]. The proposed constrained DE in this paper is defined by applying the NNC method to the ε level comparison. Thus, it is expected that both the number of fitness evaluations and the number of constraint evaluations can be reduced. The effec- tiveness of the proposed constrained DE is shown by solving five well-known benchmark engineering design problems and comparing the results with those of ε constrained DE and other state-of-the-art DE algorithms. In section 2, the ε constrained method for constrained optimization is briefly re- viewed. The new constrained DE with the NNC method, denoted as εDE-NNC, is de- scribed in section 3. In section 4, numerical results on the five engineering design prob- lems are shown. Conclusions are given in section 5. 2. THE ε CONSTRAINEDMETHOD 2.1. Constrained optimization problems In this work, we consider the following optimization problem with equality con- straints, inequality constraints and boundary constraints minimize f (x) subject to gj(x) ≤ 0, j = 1, . . . , q hj(x) = 0, j = 1+ q, . . . , m li ≤ xi ≤ ui, i = 1, . . . , n (1) where x is a n dimension vector, xi is the i-th decision variable of x, f (x) is an objective function, gj(x) ≤ 0 and hj(x) = 0 are q inequality constraints and m − q equality con- straints, respectively. The functions f , gj and hj are real-valued functions, can be linear or nonlinear. Values li and ui are the lower bound and upper bound of xi, respectively. To solve the above optimization problem using EAs, the constraints can be treated as follows: (1) Constraints are used to see if a search point is feasible (the death penalty method); (2) The sum of the violation of all constraint functions is combined with the ob- jective function to form an extended objective function (the penalty function method); (3) Engineering optimization by constrained differential evolution with nearest neighbor comparison 91 The constraints and the objective function are optimized by multi-objective optimization methods; (4) The constraint violation and the objective function are treated separately. It is seen that the methods in the last category show better performance than meth- ods in the other categories in many benchmark problems. Belonging to this category, the ε constrained method [15] is the recently developed approach, which can be applied to various unconstrained direct search algorithms to obtain constrained optimization algo- rithms. The ε constrained method is described briefly in the following. 2.2. The ε constrained method In the ε constrained method, the constraint violation is defined by the maximum of all constraints (Eq. (2)) or the sum of all constraints (Eq. (3)) φ(x) = max { max j {0, gj(x)}, max j ∣∣hj(x)∣∣} , (2) φ(x) =∑ j ∥∥max{0, gj(x)}∥∥p +∑ j ∥∥hj(x)∥∥p, (3) where p is a positive number. The ε constrained method uses the ε level comparison that is defined as an order relation on a pair of objective function value and constraint violation ( f (x), φ(x)). Let f1 ( f2) and φ1 (φ2) be the function values and the constraint violation at a point x1 (x2), respectively. Then, for any ε ≥ 0, ε level comparisons <ε and ≤ε between ( f1, φ1) and ( f2, φ2) are defined as follows ( f1, φ1) <ε ( f2, φ2)⇔ { f1 < f2, if φ1, φ2 < ε or φ1 = φ2 φ1 < φ2, otherwise (4) ( f1, φ1) ≤ε ( f2, φ2)⇔ { f1 ≤ f2, if φ1, φ2 ≤ ε or φ1 = φ2 φ1 < φ2, otherwise (5) When ε = ∞, the ε level comparisons <ε and ≤ε become the ordinary comparisons < and ≤ between function values. When ε = 0, <ε and ≤ε are equivalent to the lexico- graphic orders in which the constraint violation φ(x) precedes the function value f (x). Using the ε constrained method a constrained optimization problem is converted into an unconstrained one by replacing the ordinary comparison in direct search methods with the ε level comparison. 3. CONSTRAINED DEWITH THE NNCMETHOD 3.1. Differential evolution Differential Evolution (DE), which is originated by Storn and Price [20], is a population-based optimizer. DE creates a trial individual using differences within the search population. The population is then restructured by survival individuals evolu- tionally. Basic of DE (based on DE/rand/1/bin) is given in the following. We want to search for the global optima of an objective function f (x) over a con- tinuous space: x = {xi} , xi ∈ [xi,min, xi,max], i = 1, 2, . . . , n. For each generation G, 92 Pham Hoang Anh a population P of NP points xk, k = 1, 2, . . . , NP, is utilized. The initial population is generated as xk,i = xi,min + rand[0, 1].(xi,max − xi,min), i = 1, 2, . . . , n (6) where rand[0, 1] is a uniformly distributed random real value in the range [0, 1]. For each target point xk, k = 1, 2, . . . , NP, a perturbed point y is generated according to y = xr1 + F(xr2 − xr3), (7) with r1, r2, r3 are randomly chosen integers and 1 ≤ r1 6= r2 6= r3 6= k ≤ NP; F is a real and constant factor usually chosen in the interval [0, 1], which controls the amplification of the differential variation (xr2 − xr3). Crossover is introduced to increase the diversity, creating a trial point z with its elements determined by zi = { yi if (rand[0, 1] ≤ Cr) or (r = i) xk,i if (rand[0, 1] > Cr) and (r 6= i) (8) Here, r is randomly chosen integer in the interval [1, n]; Cr is user-defined crossover constant in the interval [0, 1]. The new point z is then compared with xk. If z is better than xk then z becomes a member in P of the next generation (G + 1); otherwise, the old value xk is retained. 3.2. Nearest neighbor comparison method It is desirable that only trial points which might better than the target point should be evaluated. A concept of possibly useless trial point is defined. A trial point with high possibility of being worse than the compared point is called possibly useless trial point (PUT point). To judge a trial point whether it is a PUT point, we use its nearest neighbor, xnn, in the population to compare with the target point. This method is named as nearest neighbor comparison (NNC). The point xnn nearest to the trial point z is searched in the current pop- ulation using distance measure. For this task, the following normalized distance measure is adopted. d(x, z) = √√√√√ n∑ i=1  xi − zi max k xk,i −min k xk,i 2, (9) where d(x, z) is distance between two points x and z. Thus, point xnn has smallest dis- tance to z. Comparison is then made between xnn and xk. If xnn is worse than xk, the trial point z is possibly not better than xk, and it is judged as PUT vector and evaluations of its fitness and constraint violation are not carried out. Engineering optimization by constrained differential evolution with nearest neighbor comparison 93 The NNC for constrained optimization using the ε constrained method can be writ- ten as follows If ( f (xnn), φ(xnn)) ≤ε ( f (xk), φ(xk)) Then Evaluate z; If ( f (z), φ(z)) ≤ε ( f (xk), φ(xk)) Then xk = z; End End where the true values at the nearest neighbor point ( f (xnn), φ(xnn)) and the parent point ( f (xk), φ(xk)) are known. Thus, the NNC can reject PUT points and omit several function evaluations. 4. SOLVING ENGINEERING OPTIMIZATION PROBLEMS 4.1. Test problems and experimental conditions In this section, five benchmark engineering design problems are solved to test the performance of εDE-NNC. The problems are: the welded beam design [21], the ten- sion/compression spring design [22], the pressure vessel design [23], speed reducer de- sign [24], and the 200-bar plane truss sizing [25]. Due to space limitation, the formulations of these problems are omitted here. The parameter setting for the ε level comparison is as follows: the constraint vi- olation φ is given by the sum of all constraints (p = 1) in Eq. (3) and the ε level is as- signed to 0. The binary crossover and random mutation with one pair of individuals (DE/rand/1/bin) is adopted as the base algorithm. The parameters of DE for welded beam design are: NP = 30, F = 0.8, Cr = 0.9; for spring design, pressure vessel design, and speed reducer design are: NP = 65, F = 0.8, Cr = 0.9; and for the 200-bar truss sizing are: NP = 50, F = 0.5, Cr = 0.9. In the first set of experiments, the welded beam, the tension/compression spring, the pressure vessel, and speed reducer problems are considered. Firstly, the εDE-NNC is compared with εDE. The stop condition for the optimization process is when the relative accuracy value, determined by the ratio between the standard derivative and the mean of objective function values in the population, is less than 1e-4. The average number of constraint evaluations and number of function evaluations over 50 random runs are given in Tab. 1. Secondly, εDE-NNC is compared with five other DE algorithms. The five DE algorithms are: (1) multiple trial vectors differential evolution (MDDE) [9], (2) differ- ential evolution with level comparison (DELC) [26], (3) constrained modified differential evolution (COMDE) [27], (4) multi-view differential evolution (MVDE) [28], and (5) im- proved constrained differential evolution (rank-iMDDE) [29]. In these experiments, the optimization termination is controlled by the maximum number of evaluations, MaxNEs. The optimal sizing of 200-bar plane truss presents a relative large-scale optimiza- tion with 29 design variables. The proposed εDE-NNC is compared with the adaptive differential evolution algorithm (ADEA) [25] and the adaptive differential evolution with optional external archive (JADE) [30]. Twenty runs are performed with termination cri- terion of maximum number of evaluations, MaxNEs = 20000. 94 Pham Hoang Anh Table 1. Average constraint evaluations and function evaluations over 50 random runs with the same stop condition (relative accuracy value < 1e-4) Problem Method fitness No. evaluation No. skip Omit (%)#const #func #skip Fail-skip rate (%) Welded εDE-NNC 1.72530398 3672 1315 3771 167 4.42 49.42 beam εDE 1.72520666 7260 2734 - - - - Spring εDE-NNC 0.01266614 6935 2104 7387 319 4.32 48.25 εDE 0.01266572 13402 4578 - - - - Pressure εDE-NNC 6060.10916 5822 2632 5514 376 6.82 45.78 vessel εDE 6059.91411 10738 5163 - - - - Speed εDE-NNC 2995.38015 7117 3193 5874 552 9.40 26.76 reducer εDE 2994.99483 9717 4242 - - - - Table 2. The best results obtained by εDE-NNC for each problem Problem Best solution Best fitness Welded beam 0.205729639786079, 3.470488665628002, 1.724852308597365 9.036623910357633, 0.205729639786080 Spring 0.051689031917057, 0.356717038149551, 0.012665232788377 11.289006887322081 Pressure vessel 0.8125, 0.4375, 42.0984455958548, 6059.714335048453 176.6365958424412 Speed reducer 3.5, 0.7, 17, 7.3, 7.715319912497795, 2994.471066247639 3.350214666225438, 5.286654465026051 200-bar truss 0.104094378466599, 0.974548756359698, 25267.90428363515 0.110520001966675, 0.120671688064705, 1.970068748667075, 0.231492528644432, 0.104293151372446, 3.147051903202999, 0.131659736442437, 4.142349528145140, 0.317129899931068, 0.116306449963694, 5.409951291487695, 0.116082739727849, 6.461642218197368, 0.481690461779939, 0.333020346450255, 8.000360212524122, 0.141619982705281, 8.992097253445209, 0.761567718928248, 0.200262604294582, 11.100410974573968, 0.167270637939214, 12.165852816206018, 0.925872725405440, 6.080582730332873, 10.892946939448695, 14.020052424656935 Engineering optimization by constrained differential evolution with nearest neighbor comparison 95 The best result obtained by εDE-NNC in each problem is listed in Tab. 2. 4.2. Experimental results and discussion 4.2.1. The welded beam design problem With the relative accuracy of 1e-4, Fig. 1a shows the plot of the best function values over the number of function evaluations. In the graphs, the solid line shows optimization process by εDE-NNC. The dashed line shows optimization process by εDE. It is clearly seen in the figure that εDE-NNC is faster than εDE. Fig. 1b shows the plot of success evaluation rate, defined by the rate of the success evaluations on actual evaluations. We can see that εDE-NNC has higher success rate than εDE. 0 1000 2000 3000 4000 1.5 2 2.5 3 3.5 Number of function evaluations O b je c ti v e f u n c ti o n v a lu e DE DE-NNC (a) 0 200 400 600 800 1000 1200 0 0.1 0.2 0.3 0.4 Number of success evaluations S u c c e s s r a te DE DE-NNC (b) Fig. 1. Optimization of welded beam problem with accuracy of 1e-4 Also, the average number of evaluations of the constraints and the objective func- tion when stop condition is met is listed in the columns labeled “#func” and “#const” of Tab. 1, respectively. It is noted that, the number of objective function evaluations is less than the number of constraint evaluations. The reason for this result is that in the ε constrained method, the objective function and the constraints are treated separately. So, when the order relation of the search points can be decided only by the constraint violation, the objective function is not evaluated. It can be seen that εDE-NNC can omit 49.42% evaluations, comparing with εDE. Moreover, there is only 4.42% of points skipped are good points, which implies that the judgment by the nearest neighbor comparison is quite accurate. It is important to point out that εDE has better performance than various methods on the same problem as shown in [8]. Tab. 3 shows the results obtained by εDE-NNC with 15000 constraint evaluations. The results of other algorithms are also listed. A result in boldface means a better (or best) solution obtained. From the results in Tab. 3, we can see that εDE-NNC obtains the optimal solution in all runs. Moreover, with the same MaxNEs, εDE-NNC gives smallest standard deviation value compared with other DE algorithms. The number of actual fitness evaluations is even much smaller as shown in the parentheses. 96 Pham Hoang Anh Table 3. Comparison on the results of welded beam design Algorithm Best Mean Worst Std MaxNEs MDDE [9] 1.725 1.725 1.725 1.00e-15 24000 DELC [26] 1.724852 1.724852 1.724852 4.10e-13 20000 COMDE [27] 1.724852309 1.724852309 1.724852309 1.6e-12 20000 MVDE [28] 1.7248527 1.7248621 1.7249215 7.88e-06 15000 rank-iMDDE [29] 1.724852309 1.724852309 1.724852309 7.71e-11 15000 εDE-NNC 1.724852308597 1.724852308597 1.724852308597 5.09e-15 15000 (5772) 4.2.2. The tension/compression spring design problem Fig. 2a shows the plot of the best function values corresponding to relative accuracy of 1e-4 over the number of function evaluations. The figure shows that εDE-NNC is faster than εDE. Fig. 2b shows the plot of success evaluation rate. The success rate of εDE-NNC is obviously higher than that of εDE. The average number of evaluations of the constraints and the objective function are given in Tab. 1. It is shown that εDE-NNC can omit 48.25% evaluations, comparing with εDE. Moreover, there is only few percents (4.32%) of points skipped are wrongly judged. 0 1000 2000 3000 4000 5000 6000 0 0.005 0.01 0.015 0.02 Number of function evaluations O b je c ti v e f u n c ti o n v a lu e DE DE-NNC (a) 0 500 1000 1500 2000 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Number of success evaluations S u c c e s s r a te DE DE-NNC (b) Fig. 2. Optimization of spring problem with accuracy of 1e-4 The optimization results with 20000 evaluations are compared with the results ob- tained by other DE algorithms in Tab. 4. We can observe that εDE-NNC can obtain the best solution with smallest standard deviation and gets the best mean value. With about one-half evaluations, εDE-NNC provides as good results as other algorithms, such as rank-iMDDE, COMDE, DELC. Engineering optimization by constrained differential evolution with nearest neighbor comparison 97 Table 4. Comparison on the results of tension/compression spring design Algorithm Best Mean Worst Std MaxNEs MDDE [9] 0.012665 0.012666 0.012674 2.00e-06 24000 DELC [26] 0.012665233 0.012665267 0.012665575 1.30e-07 20000 COMDE [27] 0.012665232 0.012667168 0.012676809 3.09e-06 24000 MVDE [28] 0.012665273 0.012667324 0.012719055 2.45e-06 10000 rank-iMDDE [29] 0.012665233 0.012665264 0.01266765 2.45e-07 19565 εDE-NNC 0.012665232788 0.012665232792 0.012665232816 5.09e-12 20000 (6630) 0.0126652359810 0.012665280131 0.012665508356 5.58e-08 10000 4.2.3. Pressure vessel design problem Experimental results on the problem with relative accuracy of 1e-4 are shown in Fig. 3 and Tab. 1. Clearly, εDE-NNC requires less evaluations of constraints and objective function than εDE. Comparing with εDE, εDE-NNC can omit 45.78% evaluations and has higher success evaluation rate. The wrong judgment is also low (6.82% of actual skipped evaluations as shown in Tab. 1). 0 1000 2000 3000 4000 5000 6000 0 0.5 1 1.5 2 2.5 x 10 4 Number of function evaluations O b je c ti v e f u n c ti o n v a lu e DE DE-NNC (a) 0 500 1000 1500 2000 0 0.1 0.2 0.3 0.4 0.5 Number of success evaluations S u c c e s s r a te DE DE-NNC (b) Fig. 3. Optimization of pressure vessel problem with accuracy of 1e-4 For this problem, there are five algorithms (i.e., MDDE, DELC, COMDE, rank- iMDDE, and εDE-NNC) that can obtain the optimal solution. According to the results given in Tab. 5, the proposed εDE-NNC is superior to MDDE, DELC, and COMDE with respect to the number of evaluations and has smaller standard deviation than that of rank-iMDDE. 98 Pham Hoang Anh Table 5. Comparison on the results of pressure vessel design Algorithm Best Mean Worst Std MaxNFEs MDDE [9] 6059.702 6059.702 6059.702 1.00e-12 24000 DELC [26] 6059.7143 6059.7143 6059.7143 2.10e-11 30000 COMDE [27] 6059.714335 6059.714335 6059.714335 3.62e-10 30000 MVDE [28] 6059.714387 6059.997236 6090.533528 2.91e+00 15000 rank-iMDDE [29] 6059.714335 6059.714335 6059.714335 7.47e-07 15000 εDE-NNC 6059.714335048 6059.714335049 6059.714335051 9.08e-10 15000 (8708) 4.2.4. Speed reducer design problem For this problem, εDE-NNC is also faster and requires less function evaluations than εDE (Fig. 4). There are 26.76% evaluations reduced with εDE-NNC, comparing with εDE (Tab. 1). The wrong judgment is less than ten percent (9.4%) of actual omitted eval- uations. Tab. 6 shows the results with 20000 evaluations. In this problem, εDE-NNC can ob- tain the optimal solution. However, it is slightly worse than rank-iMDDE and COMDE. 0 1000 2000 3000 4000 5000 2950 3000 3050 3100 3150 Number of function evaluations O b je c ti v e f u n c ti o n v a lu e DE DE-NNC (a) 0 500 1000 1500 2000 0 0.2 0.4 0.6 0.8 Number of success evaluations S u c c e s s r a te DE DE-NNC (b) Fig. 4. Optimization of speed reducer problem with accuracy of 1e-4 4.2.5. 200-bar truss sizing problem The displacement and stress of the structure are calculated by finite-element anal- ysis. The optimization results with 20000 evaluations are compared with the results ob- tained by ADEA and JADE given in [25] (Tab. 7). We can observe that εDE-NNC can obtain better solution than ADEA and JADE. Engineering optimization by constrained differential evolution with nearest neighbor comparison 99 Table 6. Comparison on the results of speed reducer design Algorithm Best Mean Worst Std MaxNEs MDDE [9] 2996.357 2996.367 2996.369 8.20e-03 24000 DELC [26] 2994.471066 2994.471066 2994.471066 1.90e-12 30000 COMDE [27] 2994.471066 2994.471066 2994.471066 1.54e-12 21000 MVDE [28] 2994.471066 2994.471066 2994.471069 2.82e-07 30000 rank-iMDDE [29] 2994.471066 2994.471066 2994.471066 7.93e-13 19920 εDE-NNC 2994.471066247 2994.471069502 2994.471079142 2.73E-06 20000 (9052) Table 7. Comparison on the results of 200-bar truss sizing problem Algorithm Best Mean Worst Std MaxNEs JADE [25] 25610.2086 25985.05665 - 177.03358 20000 ADEA [25] 25800.5708 26851.1460 - 1038.1452 20000 εDE-NNC 25267.904283635 25432.171192683 26298.638429529 224.06769 20000 (5618) 5. CONCLUSION This paper presented the combination of the nearest neighbor comparison, previ- ously used within unconstrained optimization, and the ε constrained method for han- dling constraints and proposed the εDE-NNC for constrained engineering design prob- lem. The performance of εDE-NNC was evaluated by five widely used engineering benchmark design problems. It was observed that εDE-NNC reduced the evaluations of the constraints and objective function about 26% to 49% compared to εDE. With low function evaluation requirement, εDE-NNC is also very competitive when comparing with other DE algorithms. Therefore, the εDE-NNC can solve constrained engineering optimization problems very effectively, especially for the problems with expensive objec- tive functions. ACKNOWLEDGMENTS This work is supported by National University of Civil Engineering, Vietnam (NUCE) under grant research number 98-2015/KHXD. REFERENCES [1] A. Ravindran, G. V. Reklaitis, and K. M. Ragsdell. Engineering optimization: methods and appli- cations. John Wiley & Sons, (2006). 100 Pham Hoang Anh [2] Z. Michalewicz and M. Schoenauer. Evolutionary algorithms for constrained parameter op- timization problems. Evolutionary Computation, 4, (1), (1996), pp. 1–32. [3] C. A. C. Coello. Theoretical and numerical constraint-handling techniques used with evolu- tionary algorithms: a survey of the state of the art. Computer Methods in Applied Mechanics and Engineering, 191, (11), (2002), pp. 1245–1287. [4] E. Mezura-Montes and C. A. C. Coello. Constraint-handling in nature-inspired numerical optimization: past, present and future. Swarm and Evolutionary Computation, 1, (4), (2011), pp. 173–194. [5] R. Storn and K. Price. Differential evolution-a simple and efficient heuristic for global opti- mization over continuous spaces. Journal of Global Optimization, 11, (4), (1997), pp. 341–359. [6] S. Das and P. N. Suganthan. Differential evolution: A survey of the state-of-the-art. IEEE Transactions on Evolutionary Computation, 15, (1), (2011), pp. 4–31. [7] R. Storn. System design by constraint adaptation and differential evolution. IEEE Transactions on Evolutionary Computation, 3, (1), (1999), pp. 22–34. [8] T. Takahama, S. Sakai, and N. Iwane. Solving nonlinear constrained optimization problems by the ε constrained differential evolution. In Proc. of the 2006 IEEE Conference on Systems, Man, and Cybernetics, Vol. 3, (2006), pp. 2322–2327. [9] E. Mezura-Montes, C. A. C. Coello, J. Vela´zquez-Reyes, and L. Mun˜oz-Da´vila. Multiple trial vectors in differential evolution for engineering design. Engineering Optimization, 39, (5), (2007), pp. 567–589. [10] Y. Wang and Z. Cai. Constrained evolutionary optimization by means of (µµ+λλ)-differential evolution and improved adaptive trade-off model. Evolutionary Computation, 19, (2), (2011), pp. 249–285. [11] Y. Wang and Z. Cai. Combining multiobjective optimization with differential evolution to solve constrained optimization problems. Evolutionary Computation, IEEE Transactions on, 16, (1), (2012), pp. 117–134. [12] E. K. da Silva, H. J. C. Barbosa, and A. C. C. Lemonge. An adaptive constraint handling tech- nique for differential evolution with dynamic use of variants in engineering optimization. Optimization and Engineering, 12, (1-2), (2011), pp. 31–54. [13] H. A. Pham. Reduction of function evaluation in differential evolution using nearest neigh- bor comparison. Vietnam Journal of Computer Science, 2, (2), (2015), pp. 121–131. [14] H. A. Pham, X. T. Nguyen, and V. H. Nguyen. Fuzzy structural analysis using improved differential evolutionary optimization. In Proceedings of the International Conference on Engi- neering Mechanics and Automation (ICEMA 3), Hanoi, (2014). pp. 492–498. [15] T. Takahama and S. Sakai. Constrained optimization by ε constrained particle swarm opti- mizer with ε-level control. In Proceedings of the 4th IEEE International Workshop on Soft Com- puting as Transdisciplinary Science and Technology (WSTST05). Springer, (2005), pp. 1019–1029. [16] T. Takahama and S. Sakai. Constrained optimization by the ε constrained differential evolu- tion with an archive and gradient-based mutation. In Proceedings of the 2006 IEEE Congress on Evolutionary Computation, (2006), pp. 308–315. [17] T. Takahama and S. Sakai. Fast and stable constrained optimization by the ε constrained differential evolution. Pacific Journal of Optimization, 5, (2), (2009), pp. 261–282. [18] T. Takahama and S. Sakai. Constrained optimization by the ε constrained differential evolu- tion with an archive and gradient-based mutation. In Proceedings of the 2010 IEEE Congress on Evolutionary Computation, (2010), pp. 1680–1688. Engineering optimization by constrained differential evolution with nearest neighbor comparison 101 [19] T. Takahama and S. Sakai. Efficient constrained optimization by the ε constrained adaptive differential evolution. In Proceedings of the 2010 IEEE Congress on Evolutionary Computation, (2010), pp. 2052–2059. [20] R. Storn and K. Price. Differential evolution-a simple and efficient adaptive scheme for global opti- mization over continuous spaces. International Computer Science Institute, Berkeley, (1995). [21] S. S. Rao. Engineering optimization. New York: Wiley, 3rd edition, (1996). [22] J. Arora. Introduction to optimum design. McGrawHill, (1989). [23] E. Sandgren. Nonlinear integer and discrete programming in mechanical design optimiza- tion. Journal of Mechanical Design, 112, (2), (1990), pp. 223–229. [24] J. Golinski. An adaptive optimization system applied to machine synthesis. Mechanism and Machine Theory, 8, (4), (1974), pp. 419–436. [25] S. Bureerat and N. Pholdee. Optimal truss sizing using an adaptive differential evolution algorithm. Journal of Computing in Civil Engineering, (2015). Doi:10.1061/(ASCE)CP.1943- 5487.0000487. [26] L. Wang and L.-P. Li. An effective differential evolution with level comparison for con- strained engineering design. Structural and Multidisciplinary Optimization, 41, (6), (2010), pp. 947–963. [27] A. W. Mohamed and H. Z. Sabry. Constrained optimization based on modified differential evolution algorithm. Information Sciences, 194, (2012), pp. 171–208. [28] V. V. De Melo and G. L. C. Carosio. Investigating multi-view differential evolution for solving constrained engineering design problems. Expert Systems with Applications, 40, (9), (2013), pp. 3370–3377. [29] W. Gong, Z. Cai, and D. Liang. Engineering optimization by means of an improved con- strained differential evolution. Computer Methods in Applied Mechanics and Engineering, 268, (2014), pp. 884–904. [30] J. Zhang and A. C. Sanderson. JADE: adaptive differential evolution with optional external archive. IEEE Transactions on Evolutionary Computation, 13, (5), (2009), pp. 945–958.