Results 1 
8 of
8
Complete search in continuous global optimization and constraint satisfaction, Acta Numerica 13
, 2004
"... A chapter for ..."
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 ..."
Abstract

Cited by 52 (3 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 43 (1 self)
 Add to MetaCart
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...
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 procedures. Unles ..."
Abstract

Cited by 15 (4 self)
 Add to MetaCart
. 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 ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
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 C 2 or C 1 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 algo ..."
Abstract

Cited by 4 (3 self)
 Add to MetaCart
. We consider a branch and bound method for enclosing all global minimizers of a nonlinear C 2 or C 1 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. Das Cluster Problem bei globalen Optimierungsaufgaben Wir betrachten ein Bisektionsverfahren zur Einschließung aller globalen Minimalpunkte fur stetig differenzierbare bzw. zweimal stetig differenzierbare Zielfunktionen. Das Verfahren basiert auf den Grundlagen der Intervallarithmetik und benutzt den Mittelpunkt...
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 ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
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 ..."
Abstract
 Add to MetaCart
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.