1896 IEEE TRANSACTIONS ON MAGNETICS, VOL. 33, NO. 2, MARCH 1997 zy ent” and Statistical Approach to ethod in the Optimisation of ~~ltimin~ Piergiorgio Alotto, Marina Gaggero, Giorgio Molinari, Mario Nervi Dipartimento di Ingegneria Elettrica, Universitl di Genova 1 la, Via Opera Pia, zyxwvu 1-16145 Genova, Italy Abstract - The present paper discusses the use of different, “Design of Experiment” and statistical theory based methods to improve and speed up the “Generalised Response Surface Method”, using multiquadrics to describe the objective function in parameter space, for the optimization of multiminima design problems in magnetics. The effectiveness of the proposed approach to enhance the convergence of stochastic optimisation algorithms is tested on a couple of test problems and results are discussed. I. INTRODUCTION The “Response Surface Method” (RSM) has already been used successfully in conjunction with deterministic minimisation procedures for the optimisation of electromagnetic devices [ 11. Recently, a more general, global “Generalised Response Surface Method” (GRSM) capable of describing multiminima functions has been also presented [2]. In this paper the use of “Design of Experiment” (DOE) and statistical theory techniques [3-41 to provide a theoretical and operational framework to increase the robustness and efficiency of GRSM is proposed. DOE can be used in this context as a tool to give qualitative evaluations on the relevance of optimisation parameters on the behaviour of the overall objective function, providing the user with some insight on how to define the optimisation problem in the most efficient way. Statistical theory has proven useful to improve the quality of some important parameters of the multiquadric approximation In the following sections the procedures applied are briefly described, and some test results to evaluate the effectiveness of the methods axe displayed and discussed. zyxwvu 11. THE GENERALISED RESPONSE SURFACE METHOD The main difference between RSM and GRSM is that the second one is a global approximation method, and therefore can be applied zyxwvutsrqp to the solution of multiminima problems, as opposed to single-minimum problems where local, deterministic methods can be successfully used to locate the minimum. The global interpolation strategy is based on multiquadric radial basis functions (MQ), capable of providing an analytical, multiminima approximation of the objective function with a limited number of sampling points Manuscript received March 19, 1996. Work supported by the Italian Ministry of University and Research (MURST) under the MURST National Projects scheme. zyxwvutsr [5]. In its original form, the method relies on a series of successive zooming steps in parameter space, each of which consists in the placement of a regular grid of points centred on a previously found “quasi-optimum” which is updated at each iteration. GRSM uses approximations, g, of the objective function, f, of the form: j=1 where h is the so called shift parameter. The coefficients c, are determined by inverting the system of equations: The value of h, that controls the curvature of the multiquadric approximating functions, is generally chosen as smaller than the average spacing of the sampling points. 111. DESIGN OF EXPERIMENT METHODS Design of Experiments is a well known technique that has been developed to help planning large experimental campaigns, trying to devise strategies to optimise the efficiency of experimental measurements [3]. DOE methods can serve in our case for the analysis of the relative importance of degrees of freedom and to get indications on their interactions. However, a significant difference between the present context and standard applications of DOE theory is that in our case the problem is lacking “measurement errors” in the evaluation of the objective function, so that we may consider the “experiment” (i.e. a sampling of parameter space, generally requiring a computer aided analysis and consequently expensive or very expensive) to be a so called “single replicate factorial design”. Due to possible constraints which exclude some internal parts of the parameter space (see for example [6]) the factorial design must be of the incomplete type, also to allow the handling of optimisation problems with a fairly large number of parameters (i.e. greater than five or six) where the complete factorial approach becomes unfeasible. Main effects and higher order effects can then be used to guide the optimisation algorithm, in our case a stochastic 0018-9464/97$10.00 zyxwvu 0 1997 IEEE