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59
Decision Making Under Interval Probabilities
, 1998
"... If we know the probabilities p1 ; : : : ; pn of different situations s1 ; : : : ; sn , then we can choose a decision A i for which the expected benefit C i = p1 \Delta c i1 + : : : + pn \Delta cin takes the largest possible value, where c ij denotes the benefit of decision A i in situation s j . In ..."
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Cited by 10 (3 self)
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If we know the probabilities p1 ; : : : ; pn of different situations s1 ; : : : ; sn , then we can choose a decision A i for which the expected benefit C i = p1 \Delta c i1 + : : : + pn \Delta cin takes the largest possible value, where c ij denotes the benefit of decision A i in situation s j . In many real life situations, however, we do not know the exact values of the probabilities p j ; we only know the intervals p j = [p \Gamma j ; p + j ] of possible values of these probabilities. In order to make decisions under such interval probabilities, we would like to generalize the notion of expected benefits to interval probabilities. In this paper, we show that natural requirements lead to a unique (and easily computable) generalization. Thus, we have a natural way of decision making under interval probabilities. 1 Introduction to the Problem Decision making: case of exactly known consequences. One of the main problems in decision making is the problem of choosing one of (finitel...
Theoretical Justification of a Heuristic Subbox Selection Criterion for Interval Global Optimization
, 2001
"... The most widely used guaranteed methods for global optimization are probably the intervalbased branchandbound techniques. In these techniques, we start with a single box  the entire function domain  as a possible location of the global minimizer points, and then, in each step, subdivide some ..."
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Cited by 8 (4 self)
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The most widely used guaranteed methods for global optimization are probably the intervalbased branchandbound techniques. In these techniques, we start with a single box  the entire function domain  as a possible location of the global minimizer points, and then, in each step, subdivide some of the boxes, use interval computations to compute the enclosure [F (X); F (X)] f(X) of the range f(X) of the objective function f(x) on each new subbox X , and, based on these computations, eliminate the boxes which cannot contain a global minimizer point. The computational eciency of these methods strongly depends on which boxes we select for subdivision. Traditionally, for subdivision, the algorithms selected a box with the smallest value of F (X). Recently, it was shown that the algorithm converges much faster if we select, instead, a box with the largest possible value of the ratio e f F (X) F (X) F (X) ; where e f is a current upper bound on the actual global minimum. In this paper, we give a theoretical justication for this empirical criterion. Namely, we show that:  first, this criterion is the only one that is invariant w.r.t. some reasonable symmetries; and  second, that this criterion is optimal in some reasonable sense.
On Hardware Support For Interval Computations And For Soft Computing: Theorems
, 1994
"... This paper provides a rationale for providing hardware supported functions of more than two variables for processing incomplete knowledge and fuzzy knowledge. The result is in contrast to Kolmogorov's theorem in numerical (nonfuzzy) case. ..."
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Cited by 7 (4 self)
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This paper provides a rationale for providing hardware supported functions of more than two variables for processing incomplete knowledge and fuzzy knowledge. The result is in contrast to Kolmogorov's theorem in numerical (nonfuzzy) case.
Improving the efficiency of a nonlinear–system–solver using a componentwise newton method
 Institut für Angewandte Mathematik, Universität Karlsruhe (TH
, 1997
"... ..."
Interval rational = algebraic
 ACM SIGNUM Newsletter
, 1995
"... Abstract. Rational functions can be defined as compositions of arithmetic operations (+, −, ·,:). What class of functions will be get if we add to this list the “interval ” operation (that transforms a function f of n variables and given intervals X1,..., Xn into the bounds for the range f(X1,..., X ..."
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Cited by 7 (7 self)
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Abstract. Rational functions can be defined as compositions of arithmetic operations (+, −, ·,:). What class of functions will be get if we add to this list the “interval ” operation (that transforms a function f of n variables and given intervals X1,..., Xn into the bounds for the range f(X1,..., Xn))? In this paper, we prove that adding this “interval ” operation to rational functions leads exactly to the set of all (locally) algebraic functions. In other words, algebraic functions can be described as compositions of arithmetic operations and the “interval ” operation. This result provides an additional explanation of why naive interval computations sometimes overshoot: • the desired dependency is (locally) a general algebraic function; • naive interval methods results in a (locally) rational function; • not all algebraic functions are rational. 1. FORMULATION OF THE PROBLEM 1.1. Estimating accuracy of the results of data processing, and interval mathematics
W.: The interval library filib++ 2.0  design, features and sample programs. Preprint 2001/4, Universität
, 2001
"... InternetZugriff Die Berichte sind in elektronischer Form erhältlich über die World Wide Web Seiten ..."
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Cited by 5 (0 self)
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InternetZugriff Die Berichte sind in elektronischer Form erhältlich über die World Wide Web Seiten
MACHINE INTERVAL EXPERIMENTS
, 2005
"... A statistical experiment is a mathematical object that provides a framework for statistical inference, including hypothesis testing and parameter estimation, from observations of an empirical phenomenon. When observations in the continuum of real numbers are not empirically measurable to infinite pr ..."
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Cited by 5 (1 self)
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A statistical experiment is a mathematical object that provides a framework for statistical inference, including hypothesis testing and parameter estimation, from observations of an empirical phenomenon. When observations in the continuum of real numbers are not empirically measurable to infinite precision and when conventional floatingpoint computations used in the inference procedure are not exact, the statistical experiment can become epistemologically invalid. The family of measures of the conventional statistical experiment indexed by a compact finitedimensional continuum is extended to the complete metric space of all compact subsets (of a certain form) of the index set. This is accomplished by the natural interval extension of the likelihood function. The extended experiment allows a statistical decision made with the aid of a computer to be equivalent to a numerical proof of its global optimality. Three open problems in computational statistics were solved using the extended experiment: (1) parametric bootstraps of likelihood ratio test statistics for finite mixture models, (2) rigorous maximum likelihood estimates of the branch lengths of a phylogenetic tree with a fixed topology or shape and
Enclosing the maximum likelihood of the simplest DNA model evolving on fixed topologies: towards a rigorous framework for phylogenetic inference
, 2004
"... An interval extension of the recursive formulation for the likelihood function of the simplest Markov model of DNA evolution on unrooted phylogenetic trees with a fixed topology is used to obtain rigorous enclosure(s) of the global maximum likelihood. Validated global maximizer(s) inside any compact ..."
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Cited by 5 (3 self)
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An interval extension of the recursive formulation for the likelihood function of the simplest Markov model of DNA evolution on unrooted phylogenetic trees with a fixed topology is used to obtain rigorous enclosure(s) of the global maximum likelihood. Validated global maximizer(s) inside any compact set of the parameter space which is the set of all branch lengths of the tree are thus obtained. The algorithm is an adaptation of a widely applied global optimization method using interval analysis for the phylogenetic context. The method is applied to enclose the maximizer(s) and the global maximum for the simplest DNA model evolving on trees with 2, 3, and 4 taxa. The method is also applicable to a wide class of inclusion isotonic likelihood functions.
On Branching Rules in SecondOrder BranchandBound Methods for Global Optimization
 Scienti Computing and Validated Numerics, AkademieVerlag
, 1996
"... This paper investigates different branching rules, i.e. rules for selecting the subdivision direction, in interval branchandbound algorithms for global optimization. Earlier studies ([2],[9]) dealt with the subdivision direction selection in methods which do not use secondorder information about ..."
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Cited by 5 (2 self)
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This paper investigates different branching rules, i.e. rules for selecting the subdivision direction, in interval branchandbound algorithms for global optimization. Earlier studies ([2],[9]) dealt with the subdivision direction selection in methods which do not use secondorder information about the objective function. The investigated model algorithm (similar to that in [3]) now uses the enclosure of the Hessian matrix to incorporate a concavity test for boxdiscarding and an interval Newton GaussSeidel step to reduce the widths of the boxes resulting from the underlying generalized bisection method. Four different branching rules are investigated, and a wide spectrum of test problems is used for numerical tests. The results indicate that there are substantial differences between the rules with respect to the performance of the model algorithm. This is clarified by comparing the required CPU times, the numbers of function and derivative evaluations, and the necessary amounts of st...