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16
Complete search in continuous global optimization and constraint satisfaction
 ACTA NUMERICA 13
, 2004
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A Global Optimization Method, αBB, for General TwiceDifferentiable Constrained NLPs: I  Theoretical Advances
, 1997
"... In this paper, the deterministic global optimization algorithm, αBB, (αbased Branch and Bound) is presented. This algorithm offers mathematical guarantees for convergence to a point arbitrarily close to the global minimum for the large class of twicedifferentiable NLPs. The key idea is the constru ..."
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Cited by 81 (4 self)
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In this paper, the deterministic global optimization algorithm, αBB, (αbased Branch and Bound) is presented. This algorithm offers mathematical guarantees for convergence to a point arbitrarily close to the global minimum for the large class of twicedifferentiable NLPs. The key idea is the construction of a converging sequence of upper and lower bounds on the global minimum through the convex relaxation of the original problem. This relaxation is obtained by (i) replacing all nonconvex terms of special structure (i.e., bilinear, trilinear, fractional, fractional trilinear, univariate concave) with customized tight convex lower bounding functions and (ii) by utilizing some α parameters as defined by Maranas and Floudas (1994b) to generate valid convex underestimators for nonconvex terms of generic structure. In most cases, the calculation of appropriate values for the α parameters is a challenging task. A number of approaches are proposed, which rigorously generate a set of α par...
Qualitative and Quantitative Simulation: Bridging the Gap
 Artificial Intelligence
, 1997
"... Shortcomings of qualitative simulation and of quantitative simulation motivate combining them to do simulations exhibiting strengths of both. The resulting class of techniques is called semiquantitative simulation. One approach to semiquantitative simulation is to use numeric intervals to represe ..."
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Cited by 52 (1 self)
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Shortcomings of qualitative simulation and of quantitative simulation motivate combining them to do simulations exhibiting strengths of both. The resulting class of techniques is called semiquantitative simulation. One approach to semiquantitative simulation is to use numeric intervals to represent incomplete quantitative information. In this research we demonstrate semiquantitative simulation using intervals in an implemented semiquantitative simulator called Q3. Q3 progressively refines a qualitative simulation, providing increasingly specific quantitative predictions which can converge to a numerical simulation in the limit while retaining important correctness guarantees from qualitative and interval simulation techniques. Q3's simulations are based on a technique we call step size refinement. While a pure qualitative simulation has a very coarse step size, representing the state of a system trajectory at relatively few qualitatively distinct states, Q3 interpolates newly expl...
Automatically Verified Reasoning with Both Intervals and Probability Density
 Interval Computations
, 1993
"... Information about a value is frequently best expressed with an interval. Frequently also, information is best expressed with a probability density function. We extend automatically verified numerical inference to include combining operands when both are intervals, both are probability density funct ..."
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Cited by 41 (14 self)
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Information about a value is frequently best expressed with an interval. Frequently also, information is best expressed with a probability density function. We extend automatically verified numerical inference to include combining operands when both are intervals, both are probability density functions, or one is an interval and the other a probability density function. This technique, termed the automatically verified histogram method, uses interval techniques and forms a sharp contrast with traditional Monte Carlo methods, in which operands are all intervals or all density functions, and which are not automatically verifying. Автоматически проверяемые рассуждения с использованием интервалов и функций плотности вероятности Д. Берлеант Информация о значении величины часто лучше всего может быть выражена с помощью интервала, а также и с помощью функции плотности вероятности. Мы обобщаем автоматически проверяемый численный вывод таким образом, чтобы включить случай комбинированных операндов, то есть случай, когда оба операнда являются интервалами, или оба функциями плотности вероятности, или когда один является интервалом, а другой — функцией плотности вероятности. Этот метод, называемый методом гистограмм с автоматической проверкой, использует интервальную технику и резко отличается от традиционного метода МонтеКарло, в котором все операнды являются либо интервалами, либо функциями плотности вероятности, и в котором отсутствует автоматическая верификация.
Taylor models and other validated functional inclusion methods
 Int. J. Pure Appl. Math
"... Abstract: A detailed comparison between Taylor model methods and other tools for validated computations is provided. Basic elements of the Taylor model (TM) methods are reviewed, beginning with the arithmetic for elementary operations and intrinsic functions. We discuss some of the fundamental prope ..."
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Cited by 37 (3 self)
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Abstract: A detailed comparison between Taylor model methods and other tools for validated computations is provided. Basic elements of the Taylor model (TM) methods are reviewed, beginning with the arithmetic for elementary operations and intrinsic functions. We discuss some of the fundamental properties, including high approximation order and the ability to control the dependency problem, and pointers to many of the more advanced TM tools are provided. Aspects of the current implementation, and in particular the issue of floating point error control, are discussed. For the purpose of providing range enclosures, we compare with modern versions of centered forms and mean value forms, as well as the direct computation of remainder bounds by highorder interval automatic differentiation and show the advantages of the TM methods. We also compare with the socalled boundary arithmetic (BA) of Lanford, Eckmann, Wittwer, Koch et al., which was developed to prove existence of fixed points in several comparatively small systems, and the ultraarithmetic (UA) developed by Kaucher, Miranker et al. which
The Cluster Problem In Multivariate Global Optimization
 Journal of Global Optimization
, 1994
"... . We consider branch and bound methods for enclosing all unconstrained global minimizers of a nonconvex nonlinear twicecontinuously differentiable objective function. In particular, we consider bounds obtained with interval arithmetic, with the "midpoint test," but no acceleration procedu ..."
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Cited by 24 (4 self)
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. We consider branch and bound methods for enclosing all unconstrained global minimizers of a nonconvex nonlinear twicecontinuously differentiable objective function. In particular, we consider bounds obtained with interval arithmetic, with the "midpoint test," but no acceleration procedures. Unless the lower bound is exact, the algorithm without acceleration procedures in general gives an undesirable cluster of boxes around each minimizer. In a previous paper, we analyzed this problem for univariate objective functions. In this paper, we generalize that analysis to multidimensional objective functions. As in the univariate case, the results show that the problem is highly related to the behavior of the objective function near the global minimizers and to the order of the corresponding interval extension. 1. Introduction and Basic Concepts Our underlying problem is: (1) find all global minimizers to f(x) subject to x 2 X; where X ae R m is a compact right parallelepiped with face...
Taylor Forms  Use and Limits
 Reliable Computing
, 2002
"... This review is a response to recent discussions on the reliable computing mailing list, and to continuing uncertainties about the properties and merits of Taylor forms, multivariate higher degree generalizations of centered forms. They were invented around 1980 by Lanford, documented in detail in 19 ..."
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Cited by 8 (2 self)
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This review is a response to recent discussions on the reliable computing mailing list, and to continuing uncertainties about the properties and merits of Taylor forms, multivariate higher degree generalizations of centered forms. They were invented around 1980 by Lanford, documented in detail in 1984 by Eckmann, Koch and Wittwer, and independently studied and popularized since 1996 by Berz, Makino and Hoefkens. A highlight is their application to the verified integration of asteroid dynamics in the solar system in 2001, although the details given are not sufficient to check the validity of their claims.
The Cluster Problem in Global Optimization  The Univariate Case
, 1993
"... We consider a branch and bound method for enclosing all global minimizers of a nonlinear C² or C¹ objective function. In particular, we consider bounds obtained with interval arithmetic, along with the "midpoint test," but no acceleration procedures. Unless the lower bound is exact, the al ..."
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Cited by 4 (3 self)
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We consider a branch and bound method for enclosing all global minimizers of a nonlinear C² or C¹ objective function. In particular, we consider bounds obtained with interval arithmetic, along with the "midpoint test," but no acceleration procedures. Unless the lower bound is exact, the algorithm without acceleration procedures in general gives an undesirable cluster of intervals around each minimizer. In this article, we analyze this problem in the one dimensional case. Theoretical results are given which show that the problem is highly related to the behavior of the objective function near the global minimizers and to the order of the corresponding interval extension.
A Lower Bound for Range Enclosure in Interval Arithmetic
 Theor. Comp. Sci
, 1998
"... Including the range of a rational function over an interval is an important problem in numerical computation. A direct interval arithmetic evaluation of a formula for the function yields in general a superset with an error linear in the width of the interval. Special formulas like the centered f ..."
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Cited by 2 (0 self)
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Including the range of a rational function over an interval is an important problem in numerical computation. A direct interval arithmetic evaluation of a formula for the function yields in general a superset with an error linear in the width of the interval. Special formulas like the centered forms yield a better approximation with a quadratic error. Alefeld posed the question whether in general there exists a formula whose interval arithmetic evaluation gives an approximation of better than quadratic order. In this paper we show that the answer to this question is negative if in the interval arithmetic evaluation of a formula only the basic four interval operations +; #; #;= are used. Keywords : Interval arithmetic, range enclosure, approximation of quadratic order, centered form. 1 Introduction In numerical computations one often wishes to compute the interval f#I# for a given continuous or rational function f and a given closed interval I such that f is de#ned at all po...
The Extrapolated Taylor Model
"... The Taylor model [8] is one of the inclusion functions available to compute the range enclosures. It has the property of (m + 1) th convergence order, where, m is the order of the Taylor model used. It computes a high order polynomial approximation to a multivariate Taylor expansion, with a remainde ..."
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The Taylor model [8] is one of the inclusion functions available to compute the range enclosures. It has the property of (m + 1) th convergence order, where, m is the order of the Taylor model used. It computes a high order polynomial approximation to a multivariate Taylor expansion, with a remainder term that rigorously bound the approximation error. The sharper bounds on the enclosures computed using the Taylor model can be obtained either by successively partitioning the domain x using suitable subdivision factors, or by increasing the convergence rate of the Taylor model using higher order Taylor models. However, higher order Taylor forms require higher degrees of the polynomial part, which in turn require more computational effort and more memory. This is the major drawback of increasing the order m of Taylor models for obtaining range enclosures with higher order convergence rates. In this paper, we attempt to overcome these drawbacks by using a lower order Taylor model, and then using extrapolation to accelerate the convergence process of the sequences generated with the lower order Taylor model. The effectiveness of all the proposed algorithms is tested on various multivariate examples and compared with the conventional methods. The test results show that the proposed extrapolationbased methods offer considerable speed improvements over the conventional methods.