Least $H^2$ Norm Updating Quadratic Interpolation Model Function for Derivative-free Trust-region Algorithms

Derivative-free optimization methods are numerical methods for optimization problems in which no derivative information is used. Such optimization problems are widely seen in many real applications. One particular class of derivative-free optimization algorithms is trust-region algorithms based on quadratic models given by under-determined interpolation. Different techniques in updating the quadratic model from iteration to iteration will give different interpolation models. In this paper, we propose a new way to update the quadratic model by minimizing the $H^2$ norm of the difference between neighbouring quadratic models. Motivation for applying the $H^2$ norm is given, and theoretical properties of our new updating technique are also presented. Projection in the sense of $H^2$ norm and interpolation error analysis of our model function are proposed. We obtain the coefficients of the quadratic model function by using the KKT conditions. Numerical results show advantages of our model, and the derivative-free algorithms based on our least $H^2$ norm updating quadratic model functions can solve test problems with fewer function evaluations than algorithms based on least Frobenius norm updating.