An on-line variable-fidelity surrogate-assisted harmony search algorithm with multi-level screening strategy for expensive engineering design optimization

Abstract

This paper presents an on-line variable-fidelity surrogate-assisted harmony search algorithm (VFS-HS) for expensive engineering design optimization problems. VFS-HS employs a novel model-management strategy that uses a multi-level screening mechanism based on non-dominated sorting to strictly control the numbers of low-fidelity and high-fidelity evaluations and to keep a balance between exploration and exploitation. The performance of VFS-HS is validated through comparison not only to those of four single-fidelity surrogate-assisted optimization methods (i.e. the particle swarm optimization algorithm with radial basis function-based surrogate (OPUS-RBF), the two-layer surrogate-assisted particle swarm optimization algorithm (TLSAPSO), the surrogate-assisted hierarchical particle swarm optimization (SHPSO) and the hybrid surrogate-based optimization using space reduction (HSOSR)) but also to that a multi-fidelity surrogate-assisted optimization method (the multi-fidelity Gaussian process and radial basis function-model-assisted memetic differential evolution (MGPMDE)) on the CEC2014 expensive optimization test suite. A real-world problem of the optimal design for a long cylindrical gas-pressure vessel is also investigated. The results show that VFS-HS outperforms all the compared methods.

Introduction

Computer simulation (e.g., finite element analysis (FEA) and computational fluid dynamics (CFD)) have gradually developed into an indispensable and important technology in the design optimization of modern complex engineering products. Due to the long running time of the computer analysis code, these simulations are often expensive and time-consuming [1]. For example, it may take 36–160 h for the Ford Motor Company to run a single car-crash simulation [2]. However, for most optimization methods, the need for large number of function evaluations prior to the convergence to the global optimum makes them unsuitable for solving such expensive design optimization problems. In addition, given the rapid development in computer technologies, the finite element size of simulation and analysis models grows progressively smaller, whereas the amount of the finite elements grows progressively larger. Thus, improvement in simulation accuracy is accompanied by increased computational expenses, which present a challenge to the engineering practice of complex product design optimization.

A currently-popular approach is to build a surrogate model (or metamodel) directly from the selected high-fidelity (HF) analysis model, and use this surrogate model to replace the expensive evaluations in the optimization. Familiar surrogate modeling methods include the polynomial response surface method, radial basis functions (RBF), kriging, and support vector regression (SVR). Such optimization methods with the help of surrogate models are thus called surrogate-assisted optimization methods. A more detailed overview of surrogate-assisted evolutionary optimization algorithms has been presented by Jin et al. [3]. It is noteworthy that the quality of the surrogate models critically affects the computational cost and convergence characteristics of the surrogate-assisted optimization. The quality of the surrogate models directly depends on the sample points at which the computer simulation or physical experiments are conducted. Generally, more sample points offer more information of the system but with a higher computational cost [4]. Conversely, fewer sample points incur a lower expense, but may lead to inaccurate or even distorted surrogate models. Hence, the trade-off between high accuracy and low expense seems to be inevitable in building surrogate models.

In addition to directly approximating the HF function with a single-fidelity surrogate model, another approach uses an existing, low-fidelity (LF) analysis model (which is much easier and computationally-cheaper to obtain than HF models) and then build a surrogate model that corrects the LF model to the HF model. This kind of modeling method is called variable-fidelity modeling [5]. Because of the use of computationally-cheap LF information, variable-fidelity modeling can achieve accurate approximation even if the size of the HF sample is limited. Given its superiority to single-fidelity surrogates, the research in variable-fidelity modeling has intensified over recent years [6], [7], [8], [9]. A more detailed review of the variable-fidelity models can be found in Godino et al. [10]. However, compared to variable-fidelity modeling, the research of variable-fidelity surrogate-assisted optimization methods is relatively rare. Mehmani et al. [11] proposed a particle swarm optimization (PSO) method based on a variable-fidelity model that adaptively switches models with different levels of fidelity for evaluation during the optimization. Liu et al. [12] developed a multi-fidelity surrogate model-assisted memetic differential evolution (DE) method. Zhu et al. [13] developed a variable-fidelity surrogate-assisted genetic algorithm for multi-objective optimization problems. They further improved this method through the K-means cluster algorithm to divide model data into several local surrogate models [14].

The harmony search (HS) algorithm is a meta-heuristic algorithm which was first proposed by Geem et al. [15]. It imitates music improvisation in which the musicians continuously adjust the pitch of their instruments to attain better harmony. The search process of global optimization problems is similar to the music improvisation, in that each decision variable continuously changes its value during the search process to converge to the global optimum. Since its introduction in 2001, HS has caught much scholarly attention and found practical utility in many areas, such as robot path planning [16], [17], tool path planing for five-axis flank milling [18], designing renewable energy systems [19], [20], structure designs [21], [22], scheduling [23], [24], [25], experiment designs [26], wireless sensor networks [27], [28], data clustering [29], [30], feature selection [31], [32], [33], parameters fine-tuning for deep belief networks [34] and self-driving cars [35]. The successful applications of HS algorithm in various disciplines demonstrate its outstanding global search capability.

However, because HS needs a large number of function evaluations before convergence, it is impractical to directly apply the HS algorithm for solving expensive engineering optimization problems. In this paper, an improved harmony search algorithm based on a variable-fidelity surrogate (VFS-HS) is developed to tackle such problems. Different from previous methods, VFS-HS employs a novel model management strategy that uses a multi-level screening mechanism based on non-dominated sorting to strictly control the numbers of low-fidelity and high-fidelity evaluations and to keep a balance between exploration and exploitation. To validate the performance of VFS-HS, 16 numerical benchmark problems on CEC2014 expensive optimization test suite (8 well-known problems with 10 and 30 dimensions) are tested, through which comparison is done with both single-fidelity surrogate-assisted optimization methods (i.e. the particle swarm optimization algorithm with radial basis function-based surrogate (OPUS-RBF), the two-layer surrogate-assisted particle swarm optimization algorithm (TLSAPSO), the surrogate-assisted hierarchical particle swarm optimization (SHPSO) and the hybrid surrogate-based optimization using space reduction (HSOSR)) and a multi-fidelity surrogate-assisted optimization method (the multi-fidelity Gaussian process and radial basis function-model-assisted memetic differential evolution (MGPMDE)). In addition, a real-world problem of the optimal design for a long cylindrical gas-pressure vessel is also investigated. Both results of the numerical benchmark problems and the real-world problem show that VFS-HS outperforms all the compared methods. The main contributions of this paper can be summarized as follows:

(1)

A improved HS algorithm based on variable fidelity surrogate is proposed. It can be considered as a promising method for expensive engineering design optimization problems when both the HF model and LF model are available.

(2)

A novel model management strategy based on multi-level screening mechanism has been developed in VFS-HS. There are two major advantages of this strategy: firstly, the multi-level screening mechanism can strictly control the number of real LF and HF evaluations and ensure judicious expenditure of the computational budget at the right place; secondly, the separation of HF evaluations at each iteration can supervise the search direction throughout the optimization.

(3)

An evaluation-point selection strategy based on non-dominated sorting is proposed. In most of previous literature, exploitation and exploration are discussed on only concept level, which lack of direct measures. However, in this paper, the exploration and exploitation are formulated and considered as two conflict objectives, and a balance is kept between them through the non-dominated sorting method.

The rest of the paper is organized as follows. Section 2 introduces the RBF model, variable-fidelity surrogate modeling, and HS algorithm. In Section 3, the proposed VFS-HS is characterized. Comparison of VFS-HS to four single-fidelity surrogate-assisted optimization methods and a multi-fidelity surrogate-assisted optimization method are discussed in Section 4. An engineering application of designing a long cylinder gas-pressure vessel is given in Section 5. Finally, Section 6 gives the concluding remarks.