## Empirical Evaluation Of Innovations In Interval Branch And Bound Algorithms For Nonlinear Systems (1994)

Venue: | SIAM J. Sci. Comput |

Citations: | 20 - 10 self |

### BibTeX

@ARTICLE{Kearfott94empiricalevaluation,

author = {R. Baker Kearfott},

title = {Empirical Evaluation Of Innovations In Interval Branch And Bound Algorithms For Nonlinear Systems},

journal = {SIAM J. Sci. Comput},

year = {1994},

volume = {18},

pages = {574--594}

}

### OpenURL

### Abstract

. Interval branch and bound algorithms for finding all roots use a combination of a computational existence / uniqueness procedure and a tesselation process (generalized bisection). Such algorithms identify, with mathematical rigor, a set of boxes that contains unique roots and a second set within which all remaining roots must lie. Though each root is contained in a box in one of the sets, the second set may have several boxes in clusters near a single root. Thus, the output is of higher quality if there are relatively more boxes in the first set. In contrast to previously implemented similar techniques, a box expansion technique in this paper, based on using an approximate root finder, ffl-inflation and exact set complementation, decreases the size of the second set, increases the size of the first set, and never loses roots. In addition to the expansion technique, use of second-order extensions to eliminate small boxes that do not contain roots, and interval slopes versus interval d...

### Citations

552 |
Interval Methods for Systems of Equations
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(Show Context)
Citation Context ...awczyk method (from [20] and explained in [24] and [26]) and the interval Gauss--Seidel method (such as in [9] and [17]), although preconditioned interval Gaussian elimination could also be used (see =-=[29]-=-). Theory and practice indicate that the interval Gauss--Seidel method is somewhat better than Krawczyk's method for most purposes; see [29]. Here we use the interval Gauss--Seidel method. Regardless ... |

488 |
Introduction to Interval Computations
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Citation Context ... in x6, while experimental results appear in x7. We summarize in x9. We assume prior familiarity with interval analysis, and will not repeat elementary details. Introductions to the field are [25] or =-=[1]-=-, while a treatise on the theory of interval methods for nonlinear systems is [29]. A well-prepared up-to-date general review of advanced aspects of the subject is [6]. 2. Notation. Throughout, we wil... |

487 |
Global Optimization using Interval Analysis
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Citation Context ...IF there was no progress in steps 2, 3, 5, or 6 THEN bisect X along a coordinate direction i, placing the resulting two regions on S. END DO Algorithms with similar basic steps have appeared in [25], =-=[5]-=- and the references therein, and elsewhere. An abstract version of such an algorithm along with an analysis appears in [8]. Commonly the regions X are taken to be boxes, i.e. interval vectors or, geom... |

154 |
Numerical continuation methods - an introduction
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(Show Context)
Citation Context ...ry of the graph and interval dependencies. In particular, the function decreases rapidly from x = \Gamma12, then is relatively flat in the interval [\Gamma11; 8]; it is extremely flat in the interval =-=[1; 2]-=-; graphs are available from the author. Step 4 7 from G. V. Balaji and J. D. Seader, private communication 12 R. B. Kearfott of Algorithm 5, particularly step 4(c), as well as use of second order exte... |

72 |
Computer Methods for the Range of Functions
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(Show Context)
Citation Context ...uires more work to evaluate than F 1 . The following algorithm combines the two. 4 including, possibly, loops and subroutines 5 Various interval extensions, of first and second order, are detailed in =-=[30]-=-. Branch and Bound Algorithms for Nonlinear Systems 7 X W 1 W 4 W 3 W 2 W 1 X (a) W 1 W 2 X (b) (c) Fig. 1. Complementation of a box X in a box W Algorithm 4. (Try a first-order, then second-order fun... |

63 |
Newton-Algorithmen zur Bestimmung von Nullstellen mit Fehlerschranken
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- 1969
(Show Context)
Citation Context ...eds. Along these lines, implementations generally involve some form of interval Newton method for steps 2, 3 and 5. The most common such interval Newton methods appear to be the Krawczyk method (from =-=[20]-=- and explained in [24] and [26]) and the interval Gauss--Seidel method (such as in [9] and [17]), although preconditioned interval Gaussian elimination could also be used (see [29]). Theory and practi... |

59 |
Algorithm 737: INTLIB : a portable Fortran 77 interval standard function
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- 1994
(Show Context)
Citation Context ...plementation Environment. The algorithm was implemented within the research and prototyping environment described in [13]. This environment has an interval data type that uses the routines in INTLIB (=-=[14]-=-), as well as dynamically allocated linked lists of boxes. The problems are input as expressions, loops and subroutines in Fortran syntax, and an internal representation is then generated. This single... |

52 | Preconditioners for the interval Gauss-Seidel method
- Kearfott
- 1990
(Show Context)
Citation Context ...while we have found preconditioners satisfying various types of optimality conditions can give better performance when solving nonlinear systems problems in small to moderate dimensions; see [11] and =-=[16]-=-. To date, we have preferred interval Jacobi matrices to slopes, since it is simple to incorporate them in a computational uniqueness test, such as in [24] or in [29, Theorem 5.1.7]. Interval slopes p... |

41 |
III, INTBIS: A portable interval Newton/ bisection package
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- 1990
(Show Context)
Citation Context ...or steps 2, 3 and 5. The most common such interval Newton methods appear to be the Krawczyk method (from [20] and explained in [24] and [26]) and the interval Gauss--Seidel method (such as in [9] and =-=[17]-=-), although preconditioned interval Gaussian elimination could also be used (see [29]). Theory and practice indicate that the interval Gauss--Seidel method is somewhat better than Krawczyk's method fo... |

39 |
A Test for Existence of Solutions for Non-Linear Systems
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- 1977
(Show Context)
Citation Context ..., implementations generally involve some form of interval Newton method for steps 2, 3 and 5. The most common such interval Newton methods appear to be the Krawczyk method (from [20] and explained in =-=[24]-=- and [26]) and the interval Gauss--Seidel method (such as in [9] and [17]), although preconditioned interval Gaussian elimination could also be used (see [29]). Theory and practice indicate that the i... |

37 |
Interval Slopes for Rational Functions and Associated Centered Forms
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- 1985
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Citation Context ...ntries of slope matrices generally have smaller widths than those of corresponding Lipschitz matrices, and thus are more likely to lead to ~ X ae X when bounding solutions of Equation 1. In [32, x3], =-=[21]-=- and other works, we see Definition 1. Let F : D ` R n ! R n be a continuous function and let X ` D and X c ` D. An interval matrix S(F ,X,X c ) with X c ` X is called a slope matrix for F over X at X... |

35 | The optimization problem: An introduction - Dixon, Szego - 1978 |

33 |
Computing all solutions to polynomial systems using homotopy continuation
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- 1987
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Citation Context ... A less rigorous, "probability one" alternative for polynomial systems is continuation methods, described in [2]. Good theory and practice have been developed for such algorithms by Morgan e=-=t al, eg. [28]-=-. These algorithms, though particular to polynomial systems and not rigorous in the sense of Theorem 8.1, are useful in many applications. 9. Summary. We have implemented a rigorous algorithm to compu... |

31 |
Kleine Fehlerschranken bei Matrixproblemen
- Rump
- 1980
(Show Context)
Citation Context ...computation of an approximate solution) in the second. Verification of approximate solutions began with Krawczyk [20] and Moore [24], and continued with the introduction of fixed point theory by Rump =-=[31]-=-. In global search algorithms for nonlinear systems, besides in [10], it has been used in the univariate global optimization algorithm proposed by Caprani and Madsen in [3] and in the multivariate glo... |

29 |
Abstract generalized bisection and a costbound
- Kearfott
- 1987
(Show Context)
Citation Context ...gions on S. END DO Algorithms with similar basic steps have appeared in [25], [5] and the references therein, and elsewhere. An abstract version of such an algorithm along with an analysis appears in =-=[8]-=-. Commonly the regions X are taken to be boxes, i.e. interval vectors or, geometrically, rectangular parallelepipeds. Along these lines, implementations generally involve some form of interval Newton ... |

25 |
A Global Minimization Method: The Multi-dimensional case
- Jansson, Knüppel
- 1992
(Show Context)
Citation Context ...for nonlinear systems, besides in [10], it has been used in the univariate global optimization algorithm proposed by Caprani and Madsen in [3] and in the multivariate global optimization algorithm in =-=[7]-=-, and is discussed in [29, p. 211]. The goals of the present study are to: (i) examine the practical worth of the approximate root-finding / complementation process; (ii) examine the practicality of s... |

21 |
Safe starting regions for iterative methods
- Jones
- 1977
(Show Context)
Citation Context ...ntations generally involve some form of interval Newton method for steps 2, 3 and 5. The most common such interval Newton methods appear to be the Krawczyk method (from [20] and explained in [24] and =-=[26]-=-) and the interval Gauss--Seidel method (such as in [9] and [17]), although preconditioned interval Gaussian elimination could also be used (see [29]). Theory and practice indicate that the interval G... |

18 | The cluster problem in multivariate global optimization
- Kearfott, Du
- 1994
(Show Context)
Citation Context ...ot be any solutions of F within X. In the function evaluation, we obtain a region F(X) that contains the range of F over the region X; F has no roots in X if F(X) does not contain the zero vector. In =-=[15]-=-, theoretical and empirical analysis indicates that the order of the interval extension (reviewed in x4 below) of an objective function in global optimization greatly affects the ability of the algori... |

17 |
User Guide for MINPACK-1
- Morge, Garbow, et al.
- 1980
(Show Context)
Citation Context ...the box expansion idea from [17] to allow us to rigorously eliminate clusters, without discarding roots. Furthermore, use of a classical root-finder (such as a "globalized" quasi-Newton meth=-=od, as in [27]-=-) need not be confined to the search for roots where the Jacobi matrix is ill-conditioned or singular. Assuming that it is easier to find an approximate solution and then verify it using interval arit... |

10 |
Epsilon-inflation in verification algorithms
- Mayer
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Citation Context ...ifferentiation and evaluation with interval arithmetic 6 R. B. Kearfott Obtaining X a and W involves a process termed ffl-inflation, originated by Rump in [31] and further described by Mayer, e.g. in =-=[22]-=-. This process works by constructing a small box centered at X a , then expanding it until existence (or uniqueness) can be verified. To find just one root, this process potentially is much less costl... |

9 | Use of a Real-Valued Local Minimum in Parallel Global Optimization
- Caprani, Godthåb, et al.
- 1993
(Show Context)
Citation Context ...fixed point theory by Rump [31]. In global search algorithms for nonlinear systems, besides in [10], it has been used in the univariate global optimization algorithm proposed by Caprani and Madsen in =-=[3]-=- and in the multivariate global optimization algorithm in [7], and is discussed in [29, p. 211]. The goals of the present study are to: (i) examine the practical worth of the approximate root-finding ... |

3 | Rigorous computation of surface patch intersection curves
- Kearfott, Xing
- 1993
(Show Context)
Citation Context ...gorithms for taking the complement of a box in a box and for generating a list of boxes whose union is the complement of a box in the union of the boxes in an original list first appeared in [33] and =-=[18]-=-, but had not been tried in general nonlinear systems codes. Details are available from the author. The important observation here is that a list of at most 2n boxes can be produced by complementing a... |

2 |
for the interval gauss--seidel method
- Preconditioners
- 1990
(Show Context)
Citation Context ...ly used, while we have found preconditioners satisfying various types of optimality conditions can give better performance when solving nonlinear systems problems in small to moderate dimensions; see =-=[11]-=- and [16]. To date, we have preferred interval Jacobi matrices to slopes, since it is simple to incorporate them in a computational uniqueness test, such as in [24] or in [29, Theorem 5.1.7]. Interval... |

1 |
uppel, PROFIL/BIAS --- A fast interval library
- Kn
- 1994
(Show Context)
Citation Context ...nce is sacrificed in favor of extreme ease of programming new functions. The INTLIB package was used with only one small low-level modification: As suggested by Knuppel in his BIAS subroutine package =-=[19]-=-, we replaced the simulated directed rounding subroutine with an assembler language routine that allowed true directed roundings on our machine, and we restructured the subroutines for the four elemen... |

1 |
An improved interval method for solving nonlinear systems of monotone functions
- Mladenov
- 1993
(Show Context)
Citation Context ...onlinear electrical circuit analysis. It has nine roots within the initial box [\Gamma2; 1] \Theta [\Gamma5; 1] \Theta [\Gamma2; 1] \Theta [\Gamma5; 1] ae R 4 . V. Mladenov (private communication and =-=[23]-=-) developed special algorithms for handling the special structure in the nonlinearities in this and the following two problems. The function is labelled MLAD3. Mladenov-4 This relatively easy circuit-... |

1 |
Rigorous Step Control for Continuation
- Xing
- 1993
(Show Context)
Citation Context ... roots where the Jacobi matrix is singular, even though interval verification processes fail there; the removed box portions were constructed around approximate solutions so found. Later, in [18] and =-=[33]-=-, reminiscent of the trisection process, we developed a geometrical complementation algorithm, in which we could form a new list of boxes from an old list, such that the union of the elements of the n... |