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Derivative-free optimization methods are numerical methods for optimizationproblems in which no derivative information is used. Such optimization problemsare widely seen in many real applications. One particular class ofderivative-free optimization algorithms is trust-region algorithms based onquadratic models given by under-determined interpolation. Different techniquesin updating the quadratic model from iteration to iteration will give differentinterpolation models. In this paper, we propose a new way to update thequadratic model by minimizing the $H^2$ norm of the difference betweenneighbouring 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 ourmodel function are proposed. We obtain the coefficients of the quadratic modelfunction by using the KKT conditions. Numerical results show advantages of ourmodel, and the derivative-free algorithms based on our least $H^2$ normupdating quadratic model functions can solve test problems with fewer functionevaluations than algorithms based on least Frobenius norm updating.

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