Use of a Real-Valued Local Minimum in Parallel Interval Global Optimization
Ole Caprani, Brian Godthaab, Kaj Madsen · 1994
We consider a parallel method for finding the global minimum (and all of the global minimizers) of a continuous non-linear function f: D → R, where D is an n-dimensional interval. The method combines one of the well known branch-and-bound interval search methods of Skelboe, Moore and Hansen with a real-valued optimization method. Initially we use a standard real-valued optimization method to find a local minimizer xp (or rather: a prediction of a local minimizer). Then the interval Newton method is applied to an interval Ip containing xp as its midpoint. Ip is chosen as large as possible under the restriction that the Newton interval method must converge when Ip is used as starting interval. In this way the original problem has been reduced to the problem of searching a domain D \\Ip which does not contain the local (and perhaps global) minimizer. The remaining domain is searched by the branch-and-bound interval method, starting by splitting the remaining domain into 2n intervals and hence avoiding Ip. This branch-and-bound search then either verifies that the point xp is the global minimizer, or the opposite is detected and it finds the global minimum (and the global minimizers) in the usual way. The combined method parallizes well. On one test case the combined method is faster than the branch-and-bound method itself. However, for another test case we get the opposite result. This is explained. Использование вещественнозначного локального минимума для параллельной интервальной глобальной оптимизации