Effective Two-Point Function Approximation for Design Optimization

Abstract

Anew,two-pointapproximationofafunctionthatisverystraightforwardtobuildandthatprovidesaccurateand stable approximation is presented. Earlier developments of the two-point approximations had either incomplete matches at two data points or needed the solution of additional equations to get all of the parameters. The present two-point approximation is an incomplete second-order Taylor-series expansion in terms of intervening variables; theHessianmatrixhasonlydiagonalelements,anditdependsondesignvariables.Theexponentofeachintervening design variable and the unknown constant of the second-order terms are evaluated by matching the derivatives and the value of the approximation with the previous data point gradients and the value of the original function, respectively.All of theunknownsareidentie ed in aclosedform.Both thefunctionandthegradientof thetwo-point approximation are equal to those of the original function at two data points. Several examples are given to show the accuracy and efe ciency of this method.