Optimal centers in branch-and-prune algorithms for univariate global optimization

We present an interval branch-and-prune algorithm for computing verified enclosures for the global minimum and all global minimizers of univariate functions subject to bound constraints. The algorithm works within the branch-and-bound framework and uses first order information of the objective function. In this context, we investigate valuable properties of the optimal center of a mean value form and prove optimality. We also establish an inclusion function selection criterion between natural interval extension and an optimal mean value form for the bounding process. Based on optimal centers, we introduce linear (inner and outer) pruning steps that are responsible for the branching process. The proposed algorithm incorporates the above techniques in order to accelerate the search process. Our algorithm has been implemented and tested on a test set and compared with three other methods. The method suggested shows a significant improvement on previous methods for the numerical examples solved.