Abstract. The role of the interval subdivision selection rule is investigated in branch-and-bound algorithms for global optimization. The class of rules that allow convergence for the model algorithm is characterized, and it is shown that the four rules investigated satisfy the conditions of convergence. A numerical study with a wide spectrum of test problems indicates that there are substantial di erences between the rules in terms of the required CPU time, the number of function and derivative e v aluations and space complexity, and two rules can provide substantial improvements in e ciency.
Abstract. This paper investigates the in uence of the interval subdivision selection rule on the convergenceof interval branch-and-bound algorithmsfor global optimization. F or the class of rules that allows convergence, we study the e ects of the rules on a model algorithm with special list ordering. Four di erent rules are investigated in theory and in practice. A wide spectrum of test problems is used for numerical tests indicating that there are substantial di erences between the rules with respect to the required CPU time, the number of function and derivative e v aluations, and the necessary storage space. Two rules can provide considerable improvements in e ciency for our model algorithm.
| ZusammenfassungBox-Splitting Strategies for the Interval Gauss-Seidel Step in a Global Optimization Method. W e consider an algorithm for computing veri ed enclosures for all global minimizers x and for the global minimum value f = f x o f a t wice continuously di erentiable function f : I R n ! I R within a b o x x 2 I I R n . Our algorithm incorporates the interval Gauss-Seidel step applied to the problem of nding the zeros of the gradient o f f . Here, we h a v e to deal with the gaps produced by the extended interval division. It is possible to use di erent b o x-splitting strategies for handling these gaps, producing di erent n umbers 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 a n o v erview of some of the techniques used in our algorithm, and we describe the modi cations improving the e ciency of the interval Gauss-Seidel step by applying a special box-splitting strategy. Then, we h a v e a look on special preconditioners for the Gauss-Seidel step, and we i n v estigate the corresponding results for di erent splitting strategies. Test results for standard global optimization problems are discussed for di erent v ariants 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 e ciency of the algorithm than the use of preconditioners.
We consider the problem of finding interval enclosures ot all zeros of a nonlinear system of polynomial equations. We presen~ a method which combines th e meth(g! of Grfibner: bases (used as a preprocessing step), some techniques frmn interval analysis, and a special version of the algorithm of E. Hansen for s~lving nonlinear equations in i~ne variable. The latter is applied to a tri.qngular fi)rm of the system ~d" eqnations, which is generated by. the preprocessing step: ~ Our" method is able to check if the given system has a finite mnnher of zeros and to r verifg'd enclosures for all these zeros. Several test resnhs detn~mstrate that ,our tneth~ is mpch faster than the application of Hansen's multidimensional algorithm (or similar methods) to the original nonlinear systems of polyt~nmial eqnations.
The new programming language PASCAL XSC is presented with an emphasis on the new concepts for scienti c computation and numerical data processing of the PASCAL XSC compiler. PASCAL XSC is a universal PASCAL extension with extensive standard modules for scienti c computation. It is available for personal computers, workstations, mainframes and supercomputers by means of an implementation in C. By using the mathematical modules of PASCAL XSC, numerical algorithms which deliver highly accurate and automatically veri ed results can be programmed easily. PASCAL XSC simpli es the design of programs in engineering scienti c computation by modular program structure, user-de ned operators, overloading of functions, procedures, and operators, functions and operators with arbitrary result type, dynamic arrays, arithmetic standard modules for additional numerical data types with operators of highest accuracy, standard functions of high accuracy and exact evaluation of expressions. The most important advantage of the new language is that programs written in PASCAL XSC are easily readable. This is due to the fact that all operations, even those in the higher mathematical spaces, have been realized as operators and can be used in conventional mathematical notation. In addition to PASCAL XSC a large number of numerical problem-solving routines with automatic result veri cation are available. The language supports the development of such routines.
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