Evolutionary optimization with hierarchical surrogates
Introduction
Computationally expensive problems (CEPs) widely exist in the engineering filed due to the use of increasingly high-fidelity analysis codes such as computational structural mechanics, computational electro-magnetics and computational fluid dynamics. For a CEP, evaluating the quality of a candidate solution may take from minutes to hours of supercomputer time. For example, in aerodynamic wing design, one function evaluation involving the solution of the Navier-Stokes equations can take many hours of computer time [1]. As evolutionary algorithms (EAs) usually need a lot of fitness evaluations to achieve a satisfying solution, to reduce the number of fitness evaluations, the most popular method is to use surrogate models in lieu of the real fitness function in the evolutionary process [[2], [3], [4]].
Surrogate models are computationally efficient models, which can be interpolation or regression models that are built to approximate the real fitness function based on some input-output pairs that are evaluated with the real fitness function. In the literature, various modeling techniques exist and have been applied in surrogate-assisted evolutionary optimization, including polynomial regression method (PR), radial basis function (RBF) network, Kriging/Gaussian process (GP) and so on [2]. It has been shown that the performance of surrogate-assisted evolution highly depends on the used modeling technique [5]. However, one modeling technique usually models differently on different problem landscapes [6]. As a result, it is almost impossible to know which modeling technique is best beforehand without any knowledge about the fitness landscape of the underlying problem.
To address this issue, some researchers proposed building multiple surrogate models by using different modeling techniques and then selecting the one with the minimum training error or building a more accurate ensemble model based on these models [[7], [8], [9], [10]]. However, the most accurate model might not lead to the best performance [6]. The authors in Refs. [11,12] have shown that smoothing models can help find a better solution on multimodal problems than an exact evaluation model. Thus, the performance metric like training error might not be a good metric for modeling technique selection in surrogate-assisted evolutionary optimization.
Considering this, researchers in Ref. [6] proposed GSM method in which multiple surrogate models are used separately for local optimization in the framework of memetic algorithm (MA) and only the model that provides the best solution is selected. Specifically, in GSM, two surrogate models are built using different modeling techniques. Each of the two models undergoes a local search on it to generate an improved solution. The two generated solutions are then evaluated with the real fitness function, and only the one with the better fitness is kept and compared to the individuals in the population. As this method selects a modeling technique according to the true quality of solutions generated on the built models, it can select the best modeling technique each time. However, it costs more than one fitness evaluations in the selection, which is cost ineffective especially when many modeling techniques are considered.
In another work [5], the authors proposed the evolvability learning of surrogates (EvoLS) method in which a novel performance metric was introduced for selecting modeling techniques in surrogate-assisted MA. The performance metric is called the evolvability of modeling technique which indicates the expected fitness improvement that a modeling technique can bring in the local search. The modeling technique that has the best evolvability is selected to build a model for local search each time. To calculate the evolvability of a modeling technique, the quality of the solutions provided by the modeling technique in previous generations are used. Experimental studies in Ref. [5] have shown the superiority of this method over only using one modeling technique. Compared to the GSM method, the advantage of EvoLS is that it only needs one fitness evaluation in the selection of modeling technique. However, the evolvability calculation process in EvoLS has limitations to the type of used evolutionary operators, and thus it can not be applied to any EA. Thus, good modeling techniques are still in demand in the field of surrogate-assisted evolutionary optimization.
Motivated by this, in this paper, we propose a novel modeling technique selection strategy in the framework of MA, which has a two-level hierarchical structure. In this structure, multiple modeling techniques are used to build multiple low-level local models. Then, local search is conducted on each low-level local model to obtain an improved individual. After this, a high-level model is built to select the best one from these individuals. Only the selected individual will be evaluated with the real fitness function and allowed to compete in the evolutionary optimization. The corresponding optimization method is called EHS (evolution with hierarchical surrogates) in this paper. Compared to the selection method employed in GSM, the proposed hierarchical method only needs one fitness evaluation in the modeling technique selection, and thus is more cost effective. Compared to EvoLS, EHS does not have any assumption about the type of evolutionary operators, and thus can be applied to any EA.
The remaining parts of this paper are organized as follows. Section 2 will give a brief introduction to GSM and EvoLS. In Section 3, the proposed hierarchical selection method and EHS along with a mathematic analysis will be detailed. In Section 4, experimental studies will be presented to show the efficacy of EHS. Finally, Section 5 will conclude this paper.
Section snippets
Memetic algorithm
Memetic algorithm (MA) was proposed by Moscato et al., in 1989 as an optimization framework which aims to combine population-based global search and individual-based local search [13,14]. Algorithm 1 gives the pseudo-code for MA. MA firstly generates an initial population P0 randomly from the search space. At each generation, MA selects a parent population Ppar from the current population PG, and then generates an offspring population PG+1. After evaluating each offspring individual xi,G+1 in PG
The proposed EHS algorithm
The proposed EHS uses a novel modeling technique selection method in the framework of MA. Inspired by hierarchical mixture of expert [16], a two-level hierarchical surrogate structure is introduced in EHS to do modeling technique selection. In the two-level hierarchical structure, the lower level is composed of multiple local models, each built using a different modeling technique. EHS does local search on each low-level model to obtain a new individual, and then builds a high-level model to
Experimental studies
To evaluate the efficacy of EHS, experimental studies were conducted to make comparisons between EHS and each of GSM and EvoLS as well as 3 state-of-the-art optimization algorithms. The details of the experimental studies will be presented in this section.
Conclusions and future work
When using EAs to solve computationally expensive optimization problems, surrogate models are usually applied to reduce the number of true fitness evaluations. In surrogate-assisted evolutionary search, the choice of surrogate modeling technique can highly affect the performance of the search. However, it is not easy to decide which modeling technique to use. To address this issue, we proposed a novel strategy in this paper to do surrogate modeling technique selection in the memetic
Acknowledgment
This work was supported in part by the National Key R&D Program of China (Grant No. 2017YFC0804003), the Program for Guangdong Introducing Innovative and Enterpreneurial Teams (Grant No. 2017ZT07X386), the Science and Technology Innovation Committee Foundation of Shenzhen (Grant No. ZDSYS201703031748284), the Shenzhen Peacock Plan (Grant No. KQTD2016112514355531) and the Program for University Key Laboratory of Guangdong Province (Grant No. 2017KSYS008).
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