Effective two-point function approximation for design optimization

A new two-point approximation of a function that is very straightforward to build and that provides accurate and 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 ail of the parameters. The present two-point approximation is an incomplete second-order Taylor-series expansion in terms of intervening variables; the Hessian matrix has only diagonal elements, and it depends on design variables. The exponent of each intervening 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 the unknowns are identified in a closed form. Both the function and the gradient of the two-point approximation are equal to those of the original function at two data points. Several examples are given to show the accuracy and efficiency of this method.