solutions of low-fidelity

Proceedings of the 2016 Winter Simulation Conference T. M. K. Roeder, P. I. Frazier, R. Szechtman, E. Zhou, T. Huschka, and S. E. Chick, eds.

IMPROVING THE EFFICIENCY OF EVOLUTIONARY ALGORITHMS FOR LARGE-SCALE OPTIMIZATION WITH MULTI-FIDELITY MODELS Chun-Chih Chiu James T. Lin

Si Zhang Lu Zhen

Industrial Engineering & Engineering Management National Tsing Hua University No. 101, Section 2, Kuang-Fu Road Hsinchu, Taiwan 30013, R.O.C.

School of Management Shanghai University No. 99, Shang-Da Road Shanghai, 200444, CHINA

Edward Huang Systems Engineering & Operations Research George Mason University 4400 University Drive, MS 4A6 Fairfax, VA 22030, USA

ABSTRACT Large scale optimization problems are often with complex systems and large solution space, which significantly increase their computing cost. The idea of ordinal transformation (OT) is proposed in the method MO2TOS which can improve the efficiency of solving optimization problems with limited scale solution space by using multi-fidelity models. In this paper, we integrate OT with evolutionary algorithms to speed up the solving of large-scale problems. Evolutionary algorithms are employed to search the solutions of low-fidelity model from a large solution space and provide a good direction to the OT procedure. Meanwhile, evolutionary algorithms need to determine how to select solutions from multifidelity models after the OT procedure to update the next generation. We theoretically show the improvement by using multi-fidelity models and employ genetic algorithm (GA) as an example to exhibit the detailed implementation procedure. The numerical experiments demonstrate that the new method can lead to significant improvement. 1

INTRODUCTION

As a popular research area, optimization problem often arises in varied applications and has been well studied in many years. A lot of methods have been developed to solve optimization problems, such as Linear Programming, Non-linear Programming, and Dynamic Programming. These conventional computational methods are analytically well developed but have certain requirements on the problem configuration. However, the real world problems are usually large-scale optimization problems. These problems have large solution space, the noisy or incomplete data, or multimodality. These significantly increase the difficulty for the conventional methods solving the real optimization problems due to these approaches’ rigorous requirements on the problem structure. Some evolutionary techniques are then proposed or applied in optimization area for tackling the

978-1-5090-4486-3/16/$31.00 ©2016 IEEE

815

Chiu, Zhang, Huang, Lin and Zhen large scale problems with complex configuration because of their few or no assumptions about the optimization problem, such as Genetic Algorithms (GAs) (Holland 1975), Cross-Entropy Method (CE) (Rubinstein 1999), Particle Swarm Optimization (PSO) (Kennedy and Eberhart 1995, Bratton and Kennedy 2007), and so on. Although the evolutionary algorithms provide useful tools for large-scale optimization problems, the computational efficiency is still one challenge point. One reason is the large and nonlinear solution spaces of large-scale optimization problems. The other reason is the high computing cost to evaluate the solutions due to the very complex systems. One way to tackle the computational efficiency issue of large-scale problems is the multiple-fidelity optimization. For the same system, it is often possible to build multiple evaluating models with different fidelity levels. High-fidelity models have high accuracy on predicting the performance of a solution but they may take a long time to run, which leads to the high computing cost. Low-fidelity models usually have biased estimation of the objective value of solutions but their computational cost is very low and can provide certain useful information to help us find the true best solution(s). There are some literatures related to multi-fidelity optimization for complex systems optimization problems. Huang et al. (2006) proposed a Multi-Fidelity Sequential Kriging Optimization (MFSKO) procedure and use the expected improvement (EI) as a measurement to determine the next solution to evaluate and the level of fidelity for that solution. Moore (2012) used the value of information (VOI) instead of EI as the maximized objective to determine the next solution and the fidelity of model to evaluate. Both of these two methods use Kriging to predict the bias of the low-fidelity model. However, kriging may require a large number of design points to perform well when the response surface is highly nonlinear and multi-modal, and/or the dimension of the solution space is high. Xu et al. (2014, 2016) and Huang et al. (2015) then proposed a novel ordinal transformation (OT) framework, named as MO2TOS to effectively use the information of multi-fidelity models to solve deterministic optimization problems with a finite number of solutions. In this paper, we integrate the idea of OT with evolutionary algorithms (GA) to improve their efficiency for large-scale deterministic optimization problems with multi-fidelity models. Because the low-fidelity model has low computing cost, much more solutions in the low-fidelity model can be generated and evaluated at each iteration of evolutionary algorithms. OT helps find good solutions to be evaluated by high-fidelity model and improve the search direction of evolutionary algorithms. Therefore, by applying OT with multi-fidelity models, the evolutionary algorithms can increase its capability in exploration and exploitation. The rest of the paper is organized as follows. In Section 2, we propose a way to use low-fidelity models for evolutionary algorithms and theoretically analyze the benefits of using low-fidelity models. Section 3 provides an algorithm integrating with GA. The numerical results are shown in Section 4. Section 5 concludes the whole paper. 2 2.1

GENERAL GUIDELINES MULTIPLE FIDELITY MODELS IN RANDOM SAMPLING OPTIMIZATION ALGORITHMS Language Introduction of Ordinal Transformation

The Ordinal Transformation is the main part of MO2TOS. Firstly, all solutions are evaluated by the lowfidelity model. The Ordinal Transformation (OT) methodology then transforms the original solution space into a one-dimensional ordinal space based on the ranking of solutions using the low-fidelity model. Compared to the original solution space, which can be highly nonlinear, multi-modal, high-dimensional, and include a mix of discrete and categorical decision variables, the ordinal space is one-dimensional and often owns good structure. Xu et al. (2014, 2016) apply OT to a machine allocation problem in a flexible manufacturing system with an objective to maximize system resilience as measured by steady-state cycle time (smaller is better) under demand disruptions and machine failures. There are 37 machines and 5 workstations. The problem

816

Chiu, Zhang, Huang, Lin and Zhen is to determine the number of machines allocated to each workstation such that the average cycle time can be minimized. The constraints is the number of machines in each work station must be between 5 and 10. There are total 780 feasible solutions for the optimization problem. Figure 1 shows the performance of each solution evaluated by high-fidelity model. A low-fidelity model for the problem is built by using Jackson network analysis. Figure 2 shows the performance of each solution in low-/high-fidelity models after OT. From Figures 1 and 2, we can see that using low-fidelity model by OT can help to re-organize the structure of high-fidelity model in the original solution space and show a global trend. By this trend, the efficiency of finding good solutions can be significantly improved.

Figure 1: Cycle times in the original decision space.

Figure 2: Cycle times in the transformed decision space. 2.2

Theoretical Analysis About The Benefits of Using Low-Fidelity Model

In this section, we analyze the application of OT in evolutionary algorithms. The following is the list of notations. X: a solution, of dimension K; g(X)/f(X): the performance of solution X according to the low/high-fidelity deterministic model; Nh: the total number of solutions evaluated using high-fidelity model at each iteration; sNh: the total number of solutions evaluated using low-fidelity model at each iteration; For each solution 𝑋𝑋, we assume that g(X) and f(X) follow bi-variate normal distribution as below. µ σ 2 ρσ f σ g f f N , µ g ρσ f σ g σ g2

Therefore, f ( X ) and g ( X ) can be expressed as

817

Chiu, Zhang, Huang, Lin and Zhen = f ( X ) ag ( X ) + δ ( X )

in which a = σ f / σ g and δ ( x) = µ f − ρµ gσ f / σ g − 1 − ρ 2 Z ( Z is a standard normal random variable). In the formula, we also can see that δ (x) follows normal distribution and is independent to g ( X ) . Hence, the best Nh solutions from the sNh solutions evaluated by low-fidelity model are selected to run the highfidelity model. Based on the above assumptions and parameter settings, we have the following lemmas. Lemma 1. If we only use high-fidelity model during the implementation of evolutionary algorithms, the best solution of each iteration has the following expected performance. 1 σ f E [ f ( X b )] ≈ µ f + Φ −1 Nh + 1

Lemma 2. If we use both low- and high- fidelity models during the implementation of evolutionary algorithms, the best solution of each iteration has the following upper bound of the expected performance. 1 ρσ f E [ f ( X sb )] ≤ aE [g ( X 1 )] + E [δ ( X 1 )] ≈ µ f + Φ −1 sN h + 1

Based on Lemma 1 and 2, we can see there always exists a large enough s such that the best solutions found by using two fidelity levels models is better that that found by only using high fidelity model, which theoretically shows the benefits of using low-fidelity models in evolutionary algorithms. 3

PROCEDURES TO IMPLEMENT GAOT

In this section, we proposed an algorithm, GAOT, in a large design space with multiple-fidelity models. In the large design space, even though the low-fidelity model cannot evaluate all solutions. The quality of solutions selected based on low-fidelity model performance will impact searching efficiency in highfidelity model. We thus use GA to explore good low-fidelity solutions to OT stage. GA is a stochastic search algorithm which was proposed by Holland 1975 and it has been successfully applied to a wide variety optimization problems (Lin and Chen 2015). The GAOT implements the GA and the OT alternatively in the total evolution process. The GA works on and updates the population of the lowfidelity model, while the OT is implemented to determine the evaluation of high-fidelity model and feedback evaluated chromosomes to GA for updating the next generation. The schematic structure of the GAOT is depicted in Figure 3. The main components of GAOT are the chromosome encoding, the evaluation of ﬁtness function, rank, selection, crossover and mutation. In the chromosome encoding, we use the binary encoding. The evaluation fitness function of GA is performed using the low-fidelity model. When GA finish evaluating all solutions of the low-fidelity model, these results will input the OT procedure. The first step of OT, ranks chromosome by using low-fidelity performance. We then evaluate high-fidelity model from the transformed space until all computing budget of each generation has been used. Now there are some solutions are evaluated and the other solutions are not evaluated. The selection process in GA determines which of the chromosomes from the current population will crossover to create new chromosomes. Based on the chromosomes that better fitness values have better chances of selection, we first select evaluated solutions of the high-fidelity model according to their rank to the mating pool. This means evaluated solutions have higher priority to be selected in the crossover process. For the rest candidates, we use the non-evaluated solutions based on their rank of low-fidelity to fill the mating pool. The mating pool is performed roulette wheel method, where individuals are given a probability of being selected which is directly proportionate to their rank. The crossover operator in this study is two cut-point exchange from two selected parent chromosomes to produce one child chromosome. In the mutation strategy, two genes are chosen randomly and their bit in the chromosome are inverted. The detailed pseudocode of the GAOT is described in Table 1.

818

Chiu, Zhang, Huang, Lin and Zhen

Figure 3: Schematic structure of GAOT. Table 1: Procedure of GAOT. Initialization

Set the following parameters: Gmax: maximum generation; Nh: the population size of high-fidelity (computing budget); Nl: the population size of low-fidelity; Cr: crossover rate; Mr: mutation rate. NL chromosomes are randomly generated.

Loop while generation < Gmax do Evaluation

All chromosomes are evaluated by using low-fidelity model.

Loop while computing budget < Nh do: OT Rank N solutions using low-fidelity performance. Perform ordinal transformation using the ranking of all solutions Select solutions from the transformed space as design points and evaluate them with high-fidelity models. End of Loop Updating (GA)

For i=1 to Nh Select evaluated solutions to mating pool according to their rank of high-fidelity model End for

819

Chiu, Zhang, Huang, Lin and Zhen For i= Nh to Nl Select non-evaluated solutions to mating pool according to their rank of low-fidelity model. End for

End of Loop

The optimal solution and its performance are output.

Stopping 4

Perform roulette wheel method in mating pool. For i=1 to Nl Randuniform(0,1) If Rand

IMPROVING THE EFFICIENCY OF EVOLUTIONARY ALGORITHMS FOR LARGE-SCALE OPTIMIZATION WITH MULTI-FIDELITY MODELS Chun-Chih Chiu James T. Lin

Si Zhang Lu Zhen

Industrial Engineering & Engineering Management National Tsing Hua University No. 101, Section 2, Kuang-Fu Road Hsinchu, Taiwan 30013, R.O.C.

School of Management Shanghai University No. 99, Shang-Da Road Shanghai, 200444, CHINA

Edward Huang Systems Engineering & Operations Research George Mason University 4400 University Drive, MS 4A6 Fairfax, VA 22030, USA

ABSTRACT Large scale optimization problems are often with complex systems and large solution space, which significantly increase their computing cost. The idea of ordinal transformation (OT) is proposed in the method MO2TOS which can improve the efficiency of solving optimization problems with limited scale solution space by using multi-fidelity models. In this paper, we integrate OT with evolutionary algorithms to speed up the solving of large-scale problems. Evolutionary algorithms are employed to search the solutions of low-fidelity model from a large solution space and provide a good direction to the OT procedure. Meanwhile, evolutionary algorithms need to determine how to select solutions from multifidelity models after the OT procedure to update the next generation. We theoretically show the improvement by using multi-fidelity models and employ genetic algorithm (GA) as an example to exhibit the detailed implementation procedure. The numerical experiments demonstrate that the new method can lead to significant improvement. 1

INTRODUCTION

As a popular research area, optimization problem often arises in varied applications and has been well studied in many years. A lot of methods have been developed to solve optimization problems, such as Linear Programming, Non-linear Programming, and Dynamic Programming. These conventional computational methods are analytically well developed but have certain requirements on the problem configuration. However, the real world problems are usually large-scale optimization problems. These problems have large solution space, the noisy or incomplete data, or multimodality. These significantly increase the difficulty for the conventional methods solving the real optimization problems due to these approaches’ rigorous requirements on the problem structure. Some evolutionary techniques are then proposed or applied in optimization area for tackling the

978-1-5090-4486-3/16/$31.00 ©2016 IEEE

815

Chiu, Zhang, Huang, Lin and Zhen large scale problems with complex configuration because of their few or no assumptions about the optimization problem, such as Genetic Algorithms (GAs) (Holland 1975), Cross-Entropy Method (CE) (Rubinstein 1999), Particle Swarm Optimization (PSO) (Kennedy and Eberhart 1995, Bratton and Kennedy 2007), and so on. Although the evolutionary algorithms provide useful tools for large-scale optimization problems, the computational efficiency is still one challenge point. One reason is the large and nonlinear solution spaces of large-scale optimization problems. The other reason is the high computing cost to evaluate the solutions due to the very complex systems. One way to tackle the computational efficiency issue of large-scale problems is the multiple-fidelity optimization. For the same system, it is often possible to build multiple evaluating models with different fidelity levels. High-fidelity models have high accuracy on predicting the performance of a solution but they may take a long time to run, which leads to the high computing cost. Low-fidelity models usually have biased estimation of the objective value of solutions but their computational cost is very low and can provide certain useful information to help us find the true best solution(s). There are some literatures related to multi-fidelity optimization for complex systems optimization problems. Huang et al. (2006) proposed a Multi-Fidelity Sequential Kriging Optimization (MFSKO) procedure and use the expected improvement (EI) as a measurement to determine the next solution to evaluate and the level of fidelity for that solution. Moore (2012) used the value of information (VOI) instead of EI as the maximized objective to determine the next solution and the fidelity of model to evaluate. Both of these two methods use Kriging to predict the bias of the low-fidelity model. However, kriging may require a large number of design points to perform well when the response surface is highly nonlinear and multi-modal, and/or the dimension of the solution space is high. Xu et al. (2014, 2016) and Huang et al. (2015) then proposed a novel ordinal transformation (OT) framework, named as MO2TOS to effectively use the information of multi-fidelity models to solve deterministic optimization problems with a finite number of solutions. In this paper, we integrate the idea of OT with evolutionary algorithms (GA) to improve their efficiency for large-scale deterministic optimization problems with multi-fidelity models. Because the low-fidelity model has low computing cost, much more solutions in the low-fidelity model can be generated and evaluated at each iteration of evolutionary algorithms. OT helps find good solutions to be evaluated by high-fidelity model and improve the search direction of evolutionary algorithms. Therefore, by applying OT with multi-fidelity models, the evolutionary algorithms can increase its capability in exploration and exploitation. The rest of the paper is organized as follows. In Section 2, we propose a way to use low-fidelity models for evolutionary algorithms and theoretically analyze the benefits of using low-fidelity models. Section 3 provides an algorithm integrating with GA. The numerical results are shown in Section 4. Section 5 concludes the whole paper. 2 2.1

GENERAL GUIDELINES MULTIPLE FIDELITY MODELS IN RANDOM SAMPLING OPTIMIZATION ALGORITHMS Language Introduction of Ordinal Transformation

The Ordinal Transformation is the main part of MO2TOS. Firstly, all solutions are evaluated by the lowfidelity model. The Ordinal Transformation (OT) methodology then transforms the original solution space into a one-dimensional ordinal space based on the ranking of solutions using the low-fidelity model. Compared to the original solution space, which can be highly nonlinear, multi-modal, high-dimensional, and include a mix of discrete and categorical decision variables, the ordinal space is one-dimensional and often owns good structure. Xu et al. (2014, 2016) apply OT to a machine allocation problem in a flexible manufacturing system with an objective to maximize system resilience as measured by steady-state cycle time (smaller is better) under demand disruptions and machine failures. There are 37 machines and 5 workstations. The problem

816

Chiu, Zhang, Huang, Lin and Zhen is to determine the number of machines allocated to each workstation such that the average cycle time can be minimized. The constraints is the number of machines in each work station must be between 5 and 10. There are total 780 feasible solutions for the optimization problem. Figure 1 shows the performance of each solution evaluated by high-fidelity model. A low-fidelity model for the problem is built by using Jackson network analysis. Figure 2 shows the performance of each solution in low-/high-fidelity models after OT. From Figures 1 and 2, we can see that using low-fidelity model by OT can help to re-organize the structure of high-fidelity model in the original solution space and show a global trend. By this trend, the efficiency of finding good solutions can be significantly improved.

Figure 1: Cycle times in the original decision space.

Figure 2: Cycle times in the transformed decision space. 2.2

Theoretical Analysis About The Benefits of Using Low-Fidelity Model

In this section, we analyze the application of OT in evolutionary algorithms. The following is the list of notations. X: a solution, of dimension K; g(X)/f(X): the performance of solution X according to the low/high-fidelity deterministic model; Nh: the total number of solutions evaluated using high-fidelity model at each iteration; sNh: the total number of solutions evaluated using low-fidelity model at each iteration; For each solution 𝑋𝑋, we assume that g(X) and f(X) follow bi-variate normal distribution as below. µ σ 2 ρσ f σ g f f N , µ g ρσ f σ g σ g2

Therefore, f ( X ) and g ( X ) can be expressed as

817

Chiu, Zhang, Huang, Lin and Zhen = f ( X ) ag ( X ) + δ ( X )

in which a = σ f / σ g and δ ( x) = µ f − ρµ gσ f / σ g − 1 − ρ 2 Z ( Z is a standard normal random variable). In the formula, we also can see that δ (x) follows normal distribution and is independent to g ( X ) . Hence, the best Nh solutions from the sNh solutions evaluated by low-fidelity model are selected to run the highfidelity model. Based on the above assumptions and parameter settings, we have the following lemmas. Lemma 1. If we only use high-fidelity model during the implementation of evolutionary algorithms, the best solution of each iteration has the following expected performance. 1 σ f E [ f ( X b )] ≈ µ f + Φ −1 Nh + 1

Lemma 2. If we use both low- and high- fidelity models during the implementation of evolutionary algorithms, the best solution of each iteration has the following upper bound of the expected performance. 1 ρσ f E [ f ( X sb )] ≤ aE [g ( X 1 )] + E [δ ( X 1 )] ≈ µ f + Φ −1 sN h + 1

Based on Lemma 1 and 2, we can see there always exists a large enough s such that the best solutions found by using two fidelity levels models is better that that found by only using high fidelity model, which theoretically shows the benefits of using low-fidelity models in evolutionary algorithms. 3

PROCEDURES TO IMPLEMENT GAOT

In this section, we proposed an algorithm, GAOT, in a large design space with multiple-fidelity models. In the large design space, even though the low-fidelity model cannot evaluate all solutions. The quality of solutions selected based on low-fidelity model performance will impact searching efficiency in highfidelity model. We thus use GA to explore good low-fidelity solutions to OT stage. GA is a stochastic search algorithm which was proposed by Holland 1975 and it has been successfully applied to a wide variety optimization problems (Lin and Chen 2015). The GAOT implements the GA and the OT alternatively in the total evolution process. The GA works on and updates the population of the lowfidelity model, while the OT is implemented to determine the evaluation of high-fidelity model and feedback evaluated chromosomes to GA for updating the next generation. The schematic structure of the GAOT is depicted in Figure 3. The main components of GAOT are the chromosome encoding, the evaluation of ﬁtness function, rank, selection, crossover and mutation. In the chromosome encoding, we use the binary encoding. The evaluation fitness function of GA is performed using the low-fidelity model. When GA finish evaluating all solutions of the low-fidelity model, these results will input the OT procedure. The first step of OT, ranks chromosome by using low-fidelity performance. We then evaluate high-fidelity model from the transformed space until all computing budget of each generation has been used. Now there are some solutions are evaluated and the other solutions are not evaluated. The selection process in GA determines which of the chromosomes from the current population will crossover to create new chromosomes. Based on the chromosomes that better fitness values have better chances of selection, we first select evaluated solutions of the high-fidelity model according to their rank to the mating pool. This means evaluated solutions have higher priority to be selected in the crossover process. For the rest candidates, we use the non-evaluated solutions based on their rank of low-fidelity to fill the mating pool. The mating pool is performed roulette wheel method, where individuals are given a probability of being selected which is directly proportionate to their rank. The crossover operator in this study is two cut-point exchange from two selected parent chromosomes to produce one child chromosome. In the mutation strategy, two genes are chosen randomly and their bit in the chromosome are inverted. The detailed pseudocode of the GAOT is described in Table 1.

818

Chiu, Zhang, Huang, Lin and Zhen

Figure 3: Schematic structure of GAOT. Table 1: Procedure of GAOT. Initialization

Set the following parameters: Gmax: maximum generation; Nh: the population size of high-fidelity (computing budget); Nl: the population size of low-fidelity; Cr: crossover rate; Mr: mutation rate. NL chromosomes are randomly generated.

Loop while generation < Gmax do Evaluation

All chromosomes are evaluated by using low-fidelity model.

Loop while computing budget < Nh do: OT Rank N solutions using low-fidelity performance. Perform ordinal transformation using the ranking of all solutions Select solutions from the transformed space as design points and evaluate them with high-fidelity models. End of Loop Updating (GA)

For i=1 to Nh Select evaluated solutions to mating pool according to their rank of high-fidelity model End for

819

Chiu, Zhang, Huang, Lin and Zhen For i= Nh to Nl Select non-evaluated solutions to mating pool according to their rank of low-fidelity model. End for

End of Loop

The optimal solution and its performance are output.

Stopping 4

Perform roulette wheel method in mating pool. For i=1 to Nl Randuniform(0,1) If Rand