Sequential Approximate Optimization using Radial Basis Function network for engineering optimization

金沢大学理工研究域機械工学系This paper presents a Sequential Approximate Optimization (SAO) procedure that uses the Radial Basis Function (RBF) network. If the objective and constraints are not known explicitly but can be evaluated through a computationally intensive numerical simulation, the response surface, which is often called meta-modeling, is an attractive method for finding an approximate global minimum with a small number of function evaluations. An RBF network is used to construct the response surface. The Gaussian function is employed as the basis function in this paper. In order to obtain the response surface with good approximation, the width of this Gaussian function should be adjusted. Therefore, we first examine the width. Through this examination, some sufficient conditions are introduced. Then, a simple method to determine the width of the Gaussian function is proposed. In addition, a new technique called the adaptive scaling technique is also proposed. The sufficient conditions for the width are satisfied by introducing this scaling technique. Second, the SAO algorithm is developed. The optimum of the response surface is taken as a new sampling point for local approximation. In addition, it is necessary to add new sampling points in the sparse region for global approximation. Thus, an important issue for SAO is to determine the sparse region among the sampling points. To achieve this, a new function called the density function is constructed using the RBF network. The global minimum of the density function is taken as the new sampling point. Through the sampling strategy proposed in this paper, the approximate global minimum can be found with a small number of function evaluations. Through numerical examples, the validities of the width and sampling strategy are examined in this paper. © 2010 Springer Science+Business Media, LLC

Simple estimate of the width in Gaussian kernel with adaptive scaling technique

This paper presents a simple method to estimate the width of Gaussian kernel based on an adaptive scaling technique. The Gaussian kernel is widely employed in radial basis function (RBF) network, support vector machine (SVM), least squares support vector machine (LS-SVM), Kriging models, and so on. It is widely known that the width of the Gaussian kernel in these machine learning techniques plays an important role. Determination of the optimal width is a time-consuming task. Therefore, it is preferable to determine the width with a simple manner. In this paper, we first examine a simple estimate of the width proposed by Nakayama et al. Through the examination, four sufficient conditions for the simple estimate of the width are described. Then, a new simple estimate for the width is proposed. In order to obtain the proposed estimate of the width, all dimensions are equally scaled. A simple technique called the adaptive scaling technique is also developed. It is expected that the proposed simple method to estimate the width is applicable to wide range of machine learning techniques employing the Gaussian kernel. Through examples, the validity of the proposed simple method to estimate the width is examined. © 2011 Elsevier B.V. All rights reserved

Bayesian approach to support vector machines

Ph.DDOCTOR OF PHILOSOPH