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عنوان فارسی مقاله:
روش بهینه سازی الحاقی چند جانشینه دو سطحی برای مسائل غیر خطی تحلیل ابعادی بالا
عنوان انگلیسی مقاله:
Two-level Multi-surrogate Assisted Optimization method for high dimensional nonlinear problems
سال انتشار : 2016
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مقدمه انگلیسی مقاله:
1. Introduction
In the past 30 years, Surrogate Assisted Optimization (SAO) methods have been extensively used in multidiscipline. The SAO is invoked as a substitution for physical models or simulation-based evaluations, improving the efficiency of optimization procedure significantly. However, with the development of complexity of practical engineering design, it is difficult for the SAO to handle such complicated problems, especially for high dimensional problems. Generally, the essential of SAOs is approximation. To improve the performance of surrogate in terms of accuracy and reliability, various surrogate modeling methods have been developed and applied in various disciplines in the past 30 years, such as Polynomial Regression (PR) [1], Radial Basis Function (RBF) [2–4], Moving Least Square (MLS) [5,6], Kriging [7,8] and SVR [9,10]. The most influential SAO might be Efficient Global Optimization (EGO) developed by Jones et al. [11], which can be utilized to find global optimization assisted by Kriging. Chen et al. used heuristics methodto lead the surface refinementto a smaller design space [12]. Wujek and Renaud compared a number of move-limit strategies which focused on controlling function approximation in a more “meaningful” design space [13,14]. Alexandrov et al. advocated the use of a sequential modeling approach using Trust Region Method (TRM) [15]. Rodríguez employed trust region augment Lagrangian method for Sequential Response Surface Method (SRSM) [16]. Wang et al. developed an Adaptive RSM (ARSM), which systematically reduced the size of design space [17]. Wang et al. proposed a Boundary and Best Neighbor Sampling (BBNS)-based ARSM, which is integrated with the MLS approximation for nonlinear problems [18]. Another sub-branch of SAOs is Surrogate Assisted Evolutionary Optimization (SAEO). Studies on SAEOs began over two decades ago and have received considerably increasing interest in recent years. For the SAEOs, surrogates are commonly integrated with Evolutionary Algorithms (EAs), such as Genetic Algorithm (GA), Particle SwarmOptimization(PSO) andTeaching Learning-Based Optimization (TLBO) [19–21]. Theoretically, the surrogate can be applied to almost all operations of the EA, such as population initialization, cross-over, mutation and local search and fitness evaluations [22]. The SAEO commonly uses surrogates in the local search for both single and multi-objective optimization methods [23,24]. In such cases, sophisticated model management methods developed in traditional design optimization, such as TRM can be employed directly [25]. The surrogate has also been used in stochastic search methods other than EAs, such as surrogate assisted SimulatedAnnealing (SA) [26] or Artificial Immune (AI) systems [27]. Although SAOs have achieved good results for various disciplines, most of high dimensional nonlinear problems still cannot be handled, especially for costly simulation evaluations. According to Wang and Shan [28] and Chen et al.s’ [29] suggestions, 10 or more dimensional problems can be considered high ifthe corresponding evaluation is time consuming, and such problems widely exist in various disciplines. For a practical industrial design application, if all input variables are independent, each parameter can be designed individually. Ideally, under such circumstance, a complicated case can be decomposed easily. Practically, for most of the problems, design parameters are commonly correlated. However, according to application experiences, some of them are weak correlated. If the correlations among input variables can be identified, a high dimensional problem might be decomposed into a combination of some low or medium problems and can be solved easier than the original one. The HDMR is a particular family of representations where each term reflects the independent and cooperative contributions ofinputs upon the output. The HDMR was elaborated by Rabitz et al. [30]. Similar to Taylor expansion, a HDMR expansion expresses a high dimensional function as a finite hierarchical correlated function expansion in terms of inputs, which can efficiently reduce sampling effort for learning the behavior of a high dimensional system and automatically identify the correlated relationship among input variables. Recently, Shorter et al. utilized the HDMR to build an efficient chemical kinetics solver [31]. Li et al. proposed a HDMRbased random sampling method and developed an approach to approximate its different component functions [32]. Shan and Wang integrated RBF with cut-HDMR for high dimensional expensive black-box problems [33]. Wang et al. utilized the MLS as a basis function for cut-HDMR and applied to high dimensional problems [34]. However, the HDMR is still rather a modeling technique than an optimizer. If a practical case is optimized by HDMRs, the computational cost is still expensive.
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