A GUARANTEED, ADAPTIVE, AUTOMATIC ALGORITHM FOR UNIVARIATE FUNCTION MINIMIZATION

This thesis proposes a guaranteed, adaptive, automatic algorithm for solving univariate function minimization problem on the unit interval. The key to this adaptive algorithm is performing the analysis for cones of input functions that are twice differentiable. This cone is defined in terms of two semi-norms, a stronger one and a weaker one. Three fixed-cost algorithms based on linear splines are used to find the bounds for an input function and its minimum value. The estimated minimum value and possible optimal solution set are given by those bounds. This algorithm is guaranteed to provide either a minimum value within a user-specified tolerance or a possible optimal solution set whose volume is less than another user-specified tolerance.M.S. in Applied Mathematics, July 201