Interval Analysis: Unconstrained and Constrained Optimization

Interval optimization methods (Interval analysis: Unconstrained and Constrained Optimization) have the guarantee not to loose global optimizer points. To achieve this, a de-terministic branch-and-bound framework is applied. Still heuristic algorithmic improvements may increase the convergence speed while keeping the guaranteed reliability. The indicator parameter called RejectIndex pf * (X) = f * − F (X) F (X) − F (X). was suggested by L. G. Casado as a measure of the closeness of the interval X to a global mini-mizer point [1]. First it was applied to improve the work load balance of global optimization algorithms. A subinterval X of the search space with the minimal value of the inclusion function F (X) is usually considered as the best candidate to contain a global minimum. However, the larger the interval X, the larger the overestimation of the range f (X) on X compared to F (X). Therefore a box could be considered as a good candidate to contain a global minimum just because it is larger than the others. In order to compare subintervals of different size we normalize the distance between the global minimum value f * and F (X). The idea behind pf * is that in general we expect the overestimation to be symmetric, i.e., the overestimation above f (X) is closely equal to the overestimation below f (X) for small subintervals containing a global minimizer point. Hence, for such intervals X the relative place of the global optimum value inside the F (X) interval should be high, while for intervals far from global minimizer points pf * must be small. Obviously, there are exceptions, and there exists no theoretical proof that pf * would be a reliable indicator of nearby global mini-mizer points. The value of the global minimum is not available in most cases. A generalized expression for a wider class of indicators is: p(ˆ f , X) = ˆ f − F (X) F (X) − F (X) , where thê f value is a kind of approximation of the global minimum. We assume thatˆf ∈ F (X), i.e., this estimation is realistic in the sense thatˆf is within the known bounds of the objective function on the search region. According to the numerical experience collected, we need a good approximation of the f * value to improve the efficiency of the algorithm. Subinterval selection. I. Among the possible applications of these indicators the …