Box-Splitting strategies for the interval Gauss-Seidel step in a global optimization method

Abstract

We consider an algorithm for computing verified enclosures for all global minimizersx * and for the global minimum valuef *=f(x *) of a twice continuously differentiable functionf:ℝ n →→ within a box [x]∈I→. Our algorithm incorporates the interval Gauss-Seidel step applied to the problem of finding the zeros of the gradient off. Here, we have to deal with the gaps produced by the extended interval division. It is possible to use different box-splitting strategies for handling these gaps, producing different numbers of subboxes. We present results concerning the impact of these strategies on the interval Gauss-Seidel step and therefore on our global optimization method. First, we give an overview of some of the techniques used in our algorithm, and we describe the modifications improving the efficiency of the interval Gauss-Seidel step by applying a special box-splitting strategy. Then, we have a look on special preconditioners for the Gauss-Seidel step, and we investigate the corresponding results for different splitting strategies. Test results for standard global optimization problems are discussed for different variants of our method in its portable PASCAL-XSC implementation. These results demonstrate that there are many cases in which the splitting strategy is more important for the efficiency of the algorithm than the use of preconditioners.