  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.
