A New Super-Convergent Inclusion Function Form and its Use in Global Optimization

Recently, Lin and Rokne [10] introduced the so-called Taylor-Bernstein form as an inclusion function form for multidimensional functions. This form was theoretically shown to have the super-convergence property. Here, we present an improvement of Lin and Rokne’s Taylor-Bernstein form to make it more effective in practice. We test and compare the super-convergence behavior of the proposed form with that of Lin and Rokne’s Taylor-Bernstein form and also with that of the Taylor model of Berz et al. [3]. We obtain super-convergence of orders up to 9 with the proposed form. Moreover, with the proposed form we quite easily obtain such high orders of super-convergence for up to 5− dim problems. We also investigate the use of higher order inclusion functions in the Moore-Skelboe (MS) algorithm of interval analysis (IA) for unconstrained global optimization. We use the improved TB form as an inclusion function in a prototype MS algorithm and also modify the cut-off test and termination condition in the algorithm. We test and compare on several examples the performances of the proposed algorithm, the MS algorithm, and the MS algorithm with the Taylor model of Berz et al. [3] as inclusion function. The results of these (preliminary) tests indicate that the proposed algorithm with the improved TB form as inclusion function is quite effective for low to medium dimension problems studied.