Improved Differential Evolution with Shrinking Space Technique for Constrained Optimization

To balance the convergence rate and accuracy of solution, an improved DE is used as the search engine for the proposed ATMDE algorithm. The set of offspring individuals Q _g is generated by the following three different mutant strategies (i.e. DE/rand/best/1, DE/current-to-rand/1, and DE/rand/2) which combine the advantages of different mutant strategies to generate individuals in order to increase the maximum probability of generating better offspring into evolutionary population. The strategies DE/current-to-rand/1 and DE/rand/2 have the strong explorative ability to generate new promising individuals to add into the evolutionary population, and the strategy DE/rand/best/1 has good exploitative ability to reproduce better individuals around the best individual. A good tradeoff between the global and local search performance can be achieved by combing the three different mutant strategies. The framework of generating offspring is shown in Fig. 2. Each individual in the parent population is employed to generate three different offspring with three different mutant strategies and binomial crossover, and then the better individuals retaining into the next generation are chosen from the new offspring and parent population by ATM strategy.

Generally, a constraint-handling technique to address constraints experiences three different situations in the whole evolutionary process: (1) the infeasible situation only includes the infeasible solutions; (2) the semi-feasible situation includes the feasible and infeasible individuals simultaneously; and (3) the infeasible situation only includes the infeasible individuals. The ATM strategy aims to construct an effective tradeoff scheme to address constraints for each situation according to their corresponding characteristics.

The shrinking space technique is one of the most pivotal ingredients of IS-PAES [27] and AATM [28]. This technique aims to reduce the search region to focus the computational effort on the specific promising feasible. The main procedure of the shrinking space technique is carried out as Algorithm 2, where T denotes that the technique is performed at every T generations, α _i is a threshold number, β is a reduced factor, and \(\bar{x}_{pob,i}\) and \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x}_{pob,i}\) denote the upper and lower bounds of the i-th variable in the selected offspring population, respectively. Afterward, the following specific operations are performed to shrink the search space around the promising individuals to determine the new boundaries for design variables.

ATMDE algorithm including an improved differential evolution and adaptive trade-off model and shrinking space technique is constructed to deal with COPs, and the main procedure of ATMDE is shown in Fig. 3. Firstly, it randomly generates NP individuals from the search domain [x _min, x _max] and then calculates the constraint value G(x), the function value f(x) and the feasibility proportion of current population φ. Secondly, it generates 3NP offspring individuals from the parent population P _g by the improved DE operator. Thirdly, the better NP individuals are selected from the combined population Q _g into the next generation by ATM strategy, and then the shrinking space technique is employed to reduce the search region to focus the search effort on the promising feasible region when it is satisfied the given condition. Finally, the procedures repeat until meeting the stopping criterion (the maximum generation or the maximum function fitness evaluations, max_FFEs).

For the numerical simulations, the following parameter settings are utilized unless some changes are mentioned: population size NP = 50, tolerance of equality δ = 1 × 10⁻⁴, crossover rate CR = 0.9, scaling factor F = 0.8, maximum generation g = 600. Meanwhile, the parameters of the shrinking space technique are set as:

To compare the robustness of different algorithms, benchmark functions are optimized at 30 independent runs. Then their statistical performances of the optimal solutions such as mean, standard deviation criteria are utilized to compare.

The numerical results of the eighteen-benchmark functions obtained by the ATMDE algorithm are summarized in Table 2. This table includes the known “optimal” solution for each benchmark function and the “best”, “median”, “mean”, “worst”, and “standard deviation” of each test function solved by the ATMDE algorithm. From Table 2, it shows that ATMDE has the strong ability to converge to the global optima for all test functions expect for functions g02 and g19. However, the benchmark functions g02 and g19 solved by ATMDE are extremely close to the known “optimal” values with small standard deviations. The rest sixteen test functions (g01, g03, g04, g05, g06, g07, g08, g09, g10, g11, g12, g14, g15, g16, g18, g24) can be found consistently to achieve “optimal” values in terms of the “best”, “median”, “mean”, “worst”, and “standard deviation” criteria. Especially, the functions g01 and g12 can both converge to “optimal” values and their standard deviations are zero, which means that 30 independent runs can obtain their corresponding optimal values.

The principal aim of this part verifies that the improved DE as the search optimizer is very effective and can be utilized to further enhance the performance of AATM. To make a fair comparison, the results of benchmark functions by AATM are obtained from the original literature [28]. The comparison results between AATM and ATMDE are listed in Table 3. w/t/l denotes that the proposed ATMDE wins in w functions, equals to t functions, and loses in l functions, compared with AATM algorithm. The results by ATMDE is significantly better than those solved by AATM in 8, 11, 11, and 15 functions in terms of the “best”, “mean”, “worst”, and “standard deviation” criteria, respectively. It equals to its corresponding results in 10, 7, 7, and 1 functions from the above criteria. For the standard deviations, the results only lose in functions g08 and g24, but their differences are extremely close and they both can achieve the “optimal” results with exceedingly small standard deviation. Based on the above comparison, it is clear that ATMDE achieves the competitive better performance than AATM in terms of the quality of the results by these benchmark functions.

Furthermore, the computational cost of ATMDE and AATM both are relatively low compared with the IS-PAES algorithm [27], but the performance of ATMDE is better than those solved by AATM in terms of quality of results. It should be noted that comparison results between AATM and IS-PAES are shown in the reference [28] in which AATM with smaller fitness function evaluations (FFEs) has better performance than IS-PAES. Hence, ATMDE is an effective and efficient algorithm with limited FFEs for solving COPs.

In order to verify the effectiveness of the proposed “DE/rand/best/1” strategy, 18 test functions are also employed to perform another numerical simulation (i.e. ATMDE1) which only “DE/rand/1” without “DE/best/1” strategy is used to generate the first offspring y ₁ in the whole search process. For each function, 30 independent runs are also conducted without changing any parameter settings. The comparing results of ATMDE and ATMDE1 summarizes in Table 4. Eleven functions (i.e. g04, g05, g06, g08, g09, g11, g12, g14, g15, g18, g24) can consistently converge to the global optima solved by both ATMDE and ATMDE1. However, the results of the seven functions (i.e. g01, g02, g03, g07, g10, g16, g19) solved by ATMDE can achieve the global optima but the ATMDE1 cannot consistently obtain ones especially for the functions g02, g03 and g10. More specifically, the results achieved by ATMDE are better than those solved by ATMDE1 in 6, 6, 7, 7 and 14 functions in terms of the “best”, “median”, “mean”, “worst”, and “standard deviation” criteria, respectively. It ties its corresponding results in 12, 12, 11, 11 and 3 functions from the above criteria. For the standard deviation criterion, the function g11 solved by ATMDE1 is smaller than one by ATMDE, but they both can obtain the optimal result with an exceedingly small difference. Based on the above analysis, the proposed “DE/rand/best/1” strategy is a very important part of the proposed ATMDE algorithm.

The four mechanical benchmark engineering problems are used by many researchers to demonstrate the performance of different algorithms. Different characteristics of objective functions and constraints are illustrated as shown in Table 5. The four mechanical engineering designs [28] are the minimum cost of a weld-beam design, the minimum weight of a spring design, the minimum weight of a speed reducer design and the minimum volume of a three-bar truss design, respectively. Table 6 summarizes the comparative results of these problems solved by ATMDE and AATM in terms of the “best”, “mean”, “worst”, and “standard deviation” metrics. It shows that the ATMDE has the better statistically quality and robustness than AATM under the same number of function fitness evaluations in terms of the selected performance criteria. Moreover, the four standard deviations obtained by ATMDE are relatively small, which is a crucial feature for application of the approach for solving the practical world problems. Table 7 lists the best design variables obtained by ATMDE and AATM for four engineering design problems associated with their corresponding optimal results, which show the ATMDE algorithm can obtain better solution than the AATM.

In the automotive industry, structural optimization design for vehicle crashworthiness has become a paramount research field. In this paper, different characteristics of low- and high-speed crashworthiness are considered simultaneously [30]. For the frontal impact, the most crucial energy absorbing components including rail, collision beam and stiffener can directly affect the performance of vehicle crashworthiness and safety. Therefore, the total mass M(x) of selected parts including collision beam, stiffener, front rail and front rail cover as shown in Fig. 4 is considered as our optimization objective and it is also subjected to acceleration, energy-absorbing and maximum intrusion constraints. Then, the crashworthiness problem could be formulated aswhere x = [x ₁,x ₂,x ₃,x ₄,x ₅], 2 mm ≤ x ₁ ≤ 3 mm, 1 mm ≤ x ₂, x ₃ ≤ 2.5 mm, 1.5 mm ≤ x ₄,x ₅ ≤ 3 mm, \(\bar{a}({\varvec{x}})\) is the mean value of integral acceleration, E(x) is the energy-absorbing of inner and outer front rail, I _up(x) and I _down(x) are the intrusion of the two points at the engine as shown in Fig. 5, respectively.

The finite element model (FEM) of the vehicle including 755 parts and 977 742 elements is established for the above objective and constraints. To improve efficiency, the response surfaces are established based on the samples by Latin hypercube sampling method. Then, the minimum mass M(x) solved by ATMDE algorithm is 10.53 kg and its corresponding five design variables are 2.00 mm, 2.50 mm, 2.50 mm, 2.76 mm and 1.68 mm, respectively. Under this circumstance, values of constraints are 0 g, −4208.33 J, − 60.74 mm, −0.0002 mm, respectively. Specifically, the mean value of integral acceleration \(\bar{a}({\varvec{x}})\) is 35 g, which can effectively protect passengers in the automobile when the collision inevitably occurs. Meanwhile, the inner and outer front rail can absorb 3908.33 J. In addition, the intrusions of upper and lower point at the engine are 289.26 mm and 200 mm, which can effectively reduce the occupants’ injuries to protect the passengers’ safety.

Currently, Tablet computer is one of typical popular consumer electronic devices, which have high-integrated density and large power dissipation. It is inevitably for its structural design to consider various aspects of design requirements, such as appearance, portability, operating safety, etc. Hence, structural optimization design of tablet computer should be guaranteed to work well under different conditions. This subsection considers the structural optimization design of a 7-inches tablet computer, as illustrated in Fig. 6 which mainly includes the touch screen, the display, the battery, the mainboard, the inner bracket, the front shell and the back shell [31]. Our optimization objective mainly considers the minimization of tablet’s thickness D(x). The design problem should be satisfied four practical work conditions including high-temperature constraint g ₁(x), room-temperature constraint g ₂(x), and alternating temperature constraint g ₃(x) and free fall constraint g ₄(x). Hence, this optimization problem could be formulated aswhere T ^CH(x) denotes the temperature of the chip on the main board under the high-temperature (45 °C); T ^SH(x) is the shell surface temperature with a full load under the room temperature (25 °C) for an hour continuously work; Γ ^BA(x) is the thermal stress of the battery and Γ ^TS(x) is the maximal stress of the touch screen under the collision of the 0.5 m-height free fall. The design variables are the thickness of the front shell x ₁, the thickness of the display x ₂, the thickness of the inner bracket x ₃, the thickness of the back shell x ₄, respectively. And their design values should be restricted to the domains 4 mm ≤ x ₁ ≤ 6 mm, 0.5 mm ≤ x ₂,x ₃,x ₄ ≤ 2 mm.

Four finite element models (FEM) are constructed for the above four performance constraints. To improve efficiency, the four corresponding response surfaces are established based on the given samples. Furthermore, the accuracy of the response surfaces is verified. Then the ATMDE algorithm is utilized to solve the tablet computer optimization problem. The structural thickness of optimized tablet computer is 6.42 mm which is a 31.7% reduction in compared with that of the original design (6.00 mm, 1.20 mm, 1.20 mm, 1.00 mm), and its design variables are 4.00 mm, 0.51 mm, 1.41 mm and 0.50 mm, respectively. Under this circumstance, the temperature of the chip is 62.05 °C and the temperature of shell surface is 37.66 °C which can ensure consumer daily-using comfortably. The thermal stress of battery is about 24 MPa, which can make sure the operating safety in daily. The maximal stress of touch screen is about 100 MPa, which can avoid the device broken during the collision of 0.5 m free fall. This optimized structural design is meaningful because the consumers are satisfied with the final design with better the appearance and portability for the tablet.