Efficient Estimators Using the Second Order Interval Lipschitz Condition

In this paper, we introduce smooth continuously differentiable upper and lower estimators for a univariate function with the first derivative satisfying an interval Lipschitz property being a generalization of the Lipschitz condition. The constructed estimators are piecewise quadratic and can be readily used for bounding ranges of a function, for reducing the search intervals in global optimization or for finding roots of non-linear equations. The new estimators are derived analytically and studied experimentally in the framework of branch-and-bound global optimization algorithms. The first experiments are very promising and show clearly a notable superiority of the proposed technique over the traditional approaches.