Research Article | Open Access
Lei Han, Handing Wang, Shuo Wang, "A Surrogate-Assisted Evolutionary Algorithm for Space Component Thermal Layout Optimization", Space: Science & Technology, vol. 2022, Article ID 9856362, 15 pages, 2022. https://doi.org/10.34133/2022/9856362
A Surrogate-Assisted Evolutionary Algorithm for Space Component Thermal Layout Optimization
In space engineering, the electronic component layout has a very important impact on the centroid stability and heat dissipation of devices. However, the expensive thermodynamic simulations in the component thermal layout optimization problems bring great challenges to the current optimization algorithms. To reduce the cost, a surrogate-assisted evolutionary algorithm with restart strategy is proposed in this work. The algorithm is consisted of the local search and global search. A restart strategy is designed to make the local search jump out of the local optimum promptly and speed up the population convergence. The proposed algorithm is compared with two state-of-the-art algorithms on the CEC2006, CEC2010, and CEC2017 benchmark problems. The experiment results show that the proposed algorithm has a high convergence speed and excellent ability to find the optimum in the expensive constrained optimization problems under the very limited computation budget. The proposed algorithm is also applied to solve an electronic component layout optimization problem. The final results demonstrate the good performance of the proposed algorithm, which is of great significance to the component layout optimization.
In space engineering, the layout of electronic components on circuit board has an important impact on the working state of components. However, the diversity of components, a variety of spatial location constraints, and thermal performance constraints have brought great challenges to the layout design. Moreover, the thermodynamic simulation is very expensive and time-consuming. For convenience, the component layout optimization problems (CLOPs) can be formulated as the expensive constrained optimization problems (ECOPs) as follows: where is the decision vector, . When the function evaluations and are time-consuming or expensive experiments [1, 2] which generally cannot be described by the analytic formula, they are expensive optimization problems. For example, the thermal layout simulation of electronic components in and is time-consuming, so it is expensive to evaluate the performance of a solution. Therefore, it is unrealistic to use a mount of real function evaluations in the optimization process. The ECOPs should be solved under the limited function evaluations. In addition, the ECOPs need the specific constraint handling strategies.
Evolutionary algorithms (EAs) have outstanding advantages in solving black-box optimization problems . EAs only rely on the input and output to keep approaching the optimum through its iterations, which do not need the mathematical formulation of the problems. EAs are the biological heuristic algorithms which cannot guarantee to get the global optimum but an approximated optimum. The representative EAs are the genetic algorithm (GA)  and the differential evolution (DE) .
Although EAs can deal with the black-box problems, they need a lot of evaluations. The high cost still hinders solving the problems. Thus, the surrogates are used for modeling the evaluation process to replace some real evaluations with the model prediction in the optimization process, which is called the surrogate-assisted evolutionary optimization [6–8]. Many machine learning techniques have been applied to construct the surrogate models, such as the Gaussian process regression (GP) , the radial basis function network (RBFN) , the polynomial regression (PR) , the support vector machine (SVM) , and the artificial neural network (ANN) . For finding the optimum and improving the surrogates, there have many strategies which are generally called the model management strategies designed for selecting the candidate solutions for real evaluations [14, 15].
Another core issue of ECOPs is how to handle constraints. There are mainly three kinds of constraint handling methods. The first is the penalty function method [16, 17] to impose penalties on the constraints. The second is the feasibility rule proposed in . The feasible solutions and infeasible solutions are compared by the specific rules. The stochastic ranking proposed in  is also a kind of feasibility rule method. The third is the multiobjective method [20, 21] which regards the constraints as the objective to transform the origin constrained optimization problems into the multiobjective optimization problems.
There are some representative algorithms proposed to address the ECOPs. For example, a global and local surrogate-assisted differential evolution algorithm (GLoSADE) is proposed in . In its global search stage, the algorithm adopts two DE variants for prescreening search and the generalized regression neural network for modeling the objective functions and the constraint functions. In its local search stage, the interior point method is applied to find the best solution in the local area and the RBFN models are established for the objectives and the constraints. The constraint handling adapts the feasibility rule for comparing the solutions. An evolutionary algorithm with multiple penalty functions and multiple local surrogates (MPMLS) is proposed in . The algorithm decomposes the search space into multiple subregions and builds one local surrogate model for each subregion. The penalty function method is applied to deal with the constraints.
In this work, we aim to design a surrogate-assisted evolutionary algorithm for solving the ECOPs efficiently. The local search can converge rapidly but easily fall into local optimum while the global search is on the contrary. Thus, we propose a restart strategy for combining the advantages of both strategies. Meanwhile, a local search method is proposed for enhancing local exploitation, and a new constraint violation calculation method is developed for handling the constraints. In addition, the modeling efforts are reduced by the proposed model management strategy. We term the proposed algorithm surrogate-assisted evolutionary algorithm with restart strategy (SAEA-RS).
The rest of this paper is organized as follows. Section 2 gives some preliminaries to establish a basis for the following discussion. Section 3 introduces the framework and some main parts in the proposed algorithm. Section 4 presents ablation experiments and comparative experiments with other algorithms. Section 5 presents the experimental results of the proposed algorithm on a circuit component thermal layout optimization problem. Section 6 concludes the paper and discusses the future research directions for the expensive constrained optimization.
2.1. Radial Basis Function Network
RBFN is a three-layer forward network which are the input layer, the hidden layer, and the output layer. It has excellent fitting ability, simple structure, and high training speed. Thus, it has been widely used as the surrogate model in surrogate-assisted evolutionary optimization. Given that , RBFN can be described in the function expression as follows: where is the weight and is the basis function. The can be calculated using the following expression: where and . The uncertainty of the solution can well represent the confidence of the current solution, which can guide the generation and retention of the solutions and further improve the accuracy of the model. The uncertainty can be calculated through the formula below: where .
2.2. Differential Evolution
DE is a classical EA which has a fast convergent speed. Meanwhile, it has very excellent robustness. Also, it is inherently parallelized which is very convenient for parallel computing. Mutation, crossover, and selection are three main operators of DE.
The mutation operator has many variants with different characteristics. The most representative variant is described below:
For each , the , , and are three different integers randomly selected from . Thus, , , and are the individuals in the population. is the scaling factor, and is the population size.
The crossover operator can be described as follows. The is the combination of and through the crossover. The is the crossover probability, and is a random location. If a random number between and is less than or equal to or , the should be the . Otherwise, the should be the .
The selection operator has many approaches, such as the roulette wheel selection , the tournament selection , feasibility ranking, and stochastic ranking. They have different selection intensity that lead to different convergent speed.
2.3. Constraint Handling
For constrained optimization problems, the core issue is how to handle their constraints. The common approaches can be concluded in three categories.
The first method is the penalty function. Through the penalty function, the objective and the constraint function are integrated into a fitness function, so the constrained problems are transformed into an unconstrained problems. A prominent problem in this approach is how to set the proper penalty coefficients, which greatly affects the optimization performance.
The second method is the feasibility rule, which provides a comparison rule among the feasible solutions and the infeasible solutions. The feasibility rule has three basic rules. (i)Between two feasible solutions, the solution with a better objective value is the better one(ii)Between a feasible solution and a infeasible solution, the feasible solution is the better one(iii)Between two infeasible solutions, the solution with a less constraint violation value is the better one
The third method is the multiobjective method. The constraints or constraint violation are considered as the objectives. Therefore, the constrained optimization problems are transformed into the multiobjective optimization problems. Hence, plenty of multiobjective optimization algorithms are available for solving the constrained optimization problems.
3. Proposed Algorithm
The flowchart of the proposed algorithm is shown in Figure 1. The algorithm has two coordinate optimization processes. One is a global search process, and another one is a local search process. The local search can converge rapidly but easily fall into the local optimum. In order to solve this problem, a restart strategy is introduced to restart the population when the search falls into the local optimum or falls behind the global search. A local search method is proposed to exploit the local region efficiently. The global search adopts the DE as the optimizer. To reduce the number of the real function evaluations, the RBFN models are established for each real objective and constraint functions to evaluate the newly generated individuals. The RBFN models are updated during the optimization process by the model management strategy proposed in this paper. For adapting to the different characteristics of the two optimization processes, the feasibility ranking is applied to the local search, and the stochastic ranking is applied to the global search. The feasibility ranking focuses on the convergence which is more suitable for the local search, while the stochastic ranking does not only retain the more feasible solutions but also preserves some infeasible solutions closed to the feasible region to maintain the population diversity. So, the stochastic ranking is adopted in the global search.
3.2. Proposed Constraint Violation
In the feasibility rule, the degree of constraint violation is generally defined as the sum of all constraint violations, which can be described as follows:
We proposed a new constraint violation calculation method as below: where chooses the maximum of all the constraint violations as the constraint violation of . For two infeasible solutions, the one closer to the feasible region should be the better one. The infeasible solution closer to the feasible region is the one with smaller maximum of all the constraint violations, which is similar to the “cask effect.” However, an infeasible solution with a smaller sum of constraint violation may not be the one closer to the feasible region. The reason is that an infeasible solution with a smaller sum of all constraint violations may have a large maximum of all constraint violations. Therefore, the maximum of all constraint violations can better compare the infeasible solutions.
3.3. Restart Strategy
The local search has a fast convergence ability, but it also easily falls into the local optimum. Therefore, a restart strategy is proposed in order to make the local search jump out of the current local optimum. The pseudocode of the restart strategy is shown in Algorithm 1. Firstly, the restart condition of local search population is defined as that the historical optimal solution obtained by local search is worse than that obtained by global search, or the local population diversity is lower than the diversity threshold. The diversity threshold is calculated by the following formula: where is a threshold factor. It should be between 0 and 1. The smaller it is, the higher the tolerance of losing population diversity is. The local population diversity is calculated by the formulae below:
It should be noted that the local population has a protection period which has been defined within 5 iterations after each restart. Therefore, the local population will not be restarted during the protection period. The restart population is generated by Equation (11), but is always the historical best individual in this part. Then, the local population will be set to the new restarted population.
3.4. Local Search
The global search adopts the mutation and crossover operator in DE to generate the new individuals of global population. Then, the new population and parent population are combined and evaluated by the RBFN models. Finally, the stochastic ranking is used to select individuals as the offspring population. In order to enhance the local exploitation, we propose a new local search method. After the restart strategy, the local population uses Equation (11) to generate new individuals, and the newly obtained population are combined with the parent population. The combined population are sorted by the feasibility rule, and the best individuals are selected as the offspring population. where , , and are three different integers randomly sampled from and is a random variable with a mean value of and a variance of . The operator is very similar to the mutation operator of DE, but it can generate more diverse local solutions for enhancing the local exploitation ability.
3.5. Model Management
For improving the models’ prediction ability, a model management strategy is proposed as shown in Algorithm 2. The sampling frequency is set to once every generations. Therefore, the models and will be updated every generations. The prediction ability of the model can be divided into the local prediction ability and the global prediction ability. In order to improve the local prediction ability, the best individual which has not been evaluated by the real function in the local search population is selected as the sampling point. In order to improve the global prediction accuracy, the most uncertain individual calculated by Equation (4) in the global search population is selected as the sampling point. Similar to the local search, if the individual has been evaluated by the real function, the second uncertain individual is selected, and so on. In the above two sampling methods, if all individuals of the population have been evaluated by the real function, a random individual in the search space is chosen as the sampling point. The sampling points are evaluated by real function and added to the training dataset for training the surrogate models and again.
4. Comparative Experiment
4.1. Ablation Experiment
In order to verify the performance of each part of the proposed algorithm, the ablation experiments are carried out on the restart strategy and the local search. Therefore, there are three different algorithms which are a basic SAEA without the restart strategy and the local search and a SAEA with the local search(SAEA-LS) and the SAEA-RS. In this experiment, the surrogate model of all algorithms adopts the RBFN model, and the kernel function adopts cubic kernel function. The model management strategy adopts the method in Algorithm 2, where the sampling frequency is set to . Since the initial sampling takes real function evaluations, there remain real function evaluations that can be used during the iterations. Since the RBFN models are updated every generations and each update costs real function evaluations, the number of evolutionary generations is set to . The constraint violation calculation method is the proposed .
Firstly, the basic algorithm is the SAEA without the restart strategy and the local search. The initial population whose size is is generated by the Latin hypercube sampling method . The main optimizer is the DE whose scaling factor is set to , and the crossover probability is set to . The selection adopts the stochastic ranking whose probability is set to . The model management adopts the method proposed in the Algorithm 2. It should be noted that the and are both sampled in the since there is no .
The second algorithm is the SAEA-LS which has the and the . The initial and are the same population generated by the Latin hypercube sampling. The local search adopts the proposed local search method in the Equation (11), and selection operator adopts the feasibility ranking. In addition, the global search uses the DE, and selection operator uses the stochastic ranking. For fair comparison, the other parameter settings are same as the SAEA.
The third algorithm is the SAEA-RS. The only difference between SAEA-RS and SAEA-LS is that SAEA-RS adds the restart strategy proposed in Algorithm 1. The in the is set to 1-10. The other parts including the parameter settings are same as SAEA-LS.
The ablation experiments adopts the 30D CEC2010  as the benchmark problems. The equality constraint problems are not considered in this work, so the c01, c07, c08, c13, c14, and c15 are chosen as the test problems. To avoid the instability of the algorithms, the experiments are conducted independently for times. The mean obtained optimum of independent experiments are shown in Table 1. The best results among three algorithms are highlighted in bold. The average convergence curve on the c01, c07, c08, and c13 are shown in Figure 2.
From Table 1, SAEA-RS has achieved the best results on the c01, c07, c08, and c13 problems. The three algorithms have failed in finding the feasible solutions in some experiments on the c14 and c15 problems. Hence, the rate of finding the feasible solutions in experiments are used as the comparison target. The SAEA-LS achieves the best results on c14 and c15 problems. In Figure 2, the SAEA-RS has the fastest convergence speed and best final results in c01, c07, c08, and c13 problems. Due to the restart strategy, the SAEA-RS has the faster convergence speed than the SAEA and SAEA-LS. Since the is sampled in the , the diversity of SAEA-LS is not as good as that of SAEA, and the performance of SAEA-LS is worse than the SAEA. The initial convergence curve has some fluctuations owing to not finding the infeasible solutions. In a word, the SAEA-RS performs best in the three algorithms.
For investigating the effects of the proposed , the SAEA-RS with is compared with the SAEA-RS with the general on the c01, c07, c08, and c13 problems of the 30D CEC2010. The other parameter settings are same as those in the previous experiments. The comparative experiments are carried out times independently. The average convergence curves of the two algorithms are shown in Figure 3.
From Figure 3, the has a higher convergence speed than the on the c01, c08, and c13 problems. The has the lower convergence speed but similar final results on the c07 problems. Therefore, the has the different effects on the different problems. On the whole, the is benefit for improving the performance of the proposed algorithm, which can increase the convergence speed on some problems.
4.2. Comparative Experiment
In order to compare the performance of the proposed algorithm with other existing algorithms, the comparative experiment is carried out in this part. The SAEA-RS is compared with two state-of-the-art algorithms which are the GLoSADE and the MPMLS. For a fair comparison, the total real function evaluations of all algorithms are set to . The other parameter settings of the SAEA-RS are same as the ablation experiment. The rest of the parameter settings of GLoSADE and MPMLS are consistent with that in [22, 23]. The comparative experiments are, respectively, conducted on three sets of benchmark problems which are the CEC2006 , the CEC2010, and the CEC2017 . To avoid the experimental instability, the comparative experiments are repeated 30 times independently. The average optimal objective values of experiments are as the performance indicator. If an algorithm fails to find a feasible solution in some experiments, the percentage of the algorithm finding a feasible solution in experiments is used as the performance indicator.
The experimental results on 13 benchmark problems of CEC2006 are shown in Table 2, and the best results among all the algorithms are italicized. From Table 2, SAEA-RS and GLoSADE have acquired the very close results on the g01, g04, and g24 problems. However, SAEA-RS converges faster and achieves more accurate results than GLoSADE. On g06, g07, g10, g16, g18, and g19 problems, SAEA-RS achieves a significant lead. Especially, SAEA-RS obtains feasible solutions in all experiments on g18 problems, but GLoSADE and MPMLS do not obtain the feasible solutions in some experiments. On the whole, SAEA-RS gets a lead on of the benchmark problems. In order to explore the convergence performance of the algorithms, the convergence curve of SAEA-RS, GLoSADE, and MPMLS on g01, g04, g19, and g24 problems are shown in Figure 4. The vertical axis is the objective values of the current best solution, and the horizontal axis is the number of real evaluations. Since the first real evaluations are applied in the initial sampling, the horizontal axis is set from to . The objective values of GLoSADE and MPMLS increase in the first real evaluations owing to not finding the feasible solutions, while SAEA-RS converges rapidly and even reaches near the optimum. The GLoSADE converges to the optimum rapidly at the real evaluations since GLoSADE enters the local search stage, which have the similar situations in other problems. Similarly, SAEA-RS has the fastest convergence speed among the three algorithms on g04, g19, and g24 problems. Therefore, when the number of real evaluations is small, the advantages of SAEA-RS are more prominent and more suitable for expensive problems than GLoSADE and MPMLS.
The experiment results on D CEC2010 are shown in Table 3. SAEA-RS acquires the best results on c01 and c13 problems, and GLoSADE acquires best results of the remaining four problems. However, the convergence curves in Figure 5 show that SAEA-RS has the best performance on the convergence speed. SAEA-RS has the best results on c01, c07, c08, and c13 problems before the real evaluations. Owing to the local search, GLoSADE converges quickly after real evaluations on c07 and c08 problems, but the final results are very close to SAEA-RS. In particular, the convergence curve fluctuates on c13 problems since SAEA-RS and GLoSADE do not find the feasible solutions before real evaluations and MPMLS do not find the feasible solutions in real evaluations.
The average results of SAEA-RS, GLoSADE, and MPMLS on benchmark problems of D CEC2017 are reported in Table 4. SAEA-RS achieves the best results on c01, c02, c04, and c05 problems, and GLoSADE acquires the best solutions on c20 problems. In addition, all the algorithms fail to find a feasible solution in experiments on the c13, c19, c22, and c28 problems. Therefore, SAEA-RS has an outstanding performance on the D CEC2017 problems. The convergence curve on c01 and c04 problems are shown in Figure 6. SAEA-RS has the fastest convergence speed on c01 and c04 problems which reflects the advantage of SAEA-RS when the real evaluations is small.
In summary, SAEA-RS has the highest convergence speed and finds the best solutions on most benchmark problems. For the constrained expensive optimization problems, the advantages of SAEA-RS can help to find a better solution within a few real evaluations.
5. Application to Electronic Component Layout Optimization
In this section, the proposed algorithm is applied to solve a CLOP on a space circuit board. As shown in Figure 7, the red border indicates the circuit board whose length is , and the width is . The four green rectangles represent the heat pipes on the circuit board, and the blue rectangles represent different electronic components. The heat pipes can evacuate the heat generated by the components. The length and width of component are and , respectively, which can be recorded as (15, 18). Similarly, the size of component to can be denoted as (20, 10), (10, 15), (14, 16), (8, 13), and (15, 12). The first constraint is that the components should not overlap each other or exceed the layout domain. The second constraint is that the deviation between the centroid of the system and the expected centroid should not exceed a specific value. The third constraint is that the heat load on each heat pipe should not exceed its maximum capacity. The last constraint is that each component should overlap with the heat pipes to dissipate heat. For uniform heat distribution, the objective is to minimize the maximum thermal load on all heat pipes. The problem can be described as follows: where denotes the maximum thermal load, denotes the component layout overlap constraint, denotes the centroid deviation constraint, denotes the heat load constraint, denotes the heat pipe overlap constraint, and , and denotes the center coordinate of component to , , and .
The and require the thermodynamic simulation which is expensive and time-consuming, but , , and can be easily calculated by the cheap function evaluations. In order to take advantage of this characteristic, a population initialization strategy shown in Algorithm 3 is designed for replacing the original Latin hypercube sampling in the SAEA-RS. Through Algorithm 3, there are more approximately feasible individuals selected as the initial and , which can increase the convergence process to some extent.
In this section, the is set to . The other parameter settings of SAEA-RS are same as those in Section 4. Therefore, there are real thermodynamic simulations used in the whole optimization process, which are simulations for evaluating the