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## (1) THE CLUSTER PROBLEM IN GLOBAL OPTIMIZATION THE

### Citations

722 |
Methods and Applications of Interval Analysis
- Moore, Bierbaum
- 1979
(Show Context)
Citation Context ...rs. We will occasionally use a lower case bold letter to denote a non-interval function value which has been bounded using interval arithmetic. For introductions to interval arithmetic, see e.g. [8], =-=[5]-=-, or [1]. Inclusion functions. Using interval arithmetic, we may extend an objective function f to interval values such that f(X (1) ) contains the entire range of f over interval X (1) . Let F(X (1) ... |

635 | Interval Methods for Systems of Equations - Neumaier - 1990 |

592 |
Introduction to Interval Computations
- Alefeld, Herzberger
- 1983
(Show Context)
Citation Context ...ill occasionally use a lower case bold letter to denote a non-interval function value which has been bounded using interval arithmetic. For introductions to interval arithmetic, see e.g. [8], [5], or =-=[1]-=-. Inclusion functions. Using interval arithmetic, we may extend an objective function f to interval values such that f(X (1) ) contains the entire range of f over interval X (1) . Let F(X (1) ) denote... |

154 |
New Computer Methods for Global Optimization
- Ratschek, Rokne
- 1988
(Show Context)
Citation Context ...e set of global minimizers as X ∗ . In this paper, we study the branch and bound principle to enclose the solution set X ∗ of (1). Our analysis deals with algorithms similar to Algorithm 3, p. 111 of =-=[9]-=-. Also, as in [9], we will use interval arithmetic to obtain the bounds. Key words and phrases. branch and bound principle, inclusion function, interval extensions, midpoint test, global optimization,... |

82 |
Computer Methods for the Range of Functions
- Ratschek, Rokne
- 1984
(Show Context)
Citation Context ...letters. We will occasionally use a lower case bold letter to denote a non-interval function value which has been bounded using interval arithmetic. For introductions to interval arithmetic, see e.g. =-=[8]-=-, [5], or [1]. Inclusion functions. Using interval arithmetic, we may extend an objective function f to interval values such that f(X (1) ) contains the entire range of f over interval X (1) . Let F(X... |

41 |
Global Optimization Using Interval Analysis : The multidimensional Case
- Hansen
- 1980
(Show Context)
Citation Context ...ilities for improvement. 1. Use acceleration devices such as an interval Newton method. (Note that this is standard practice in interval arithmetic-based branch and bound algorithms; see, for example =-=[4]-=- for one of the earlier explanations of this process.) 2. Use higher order interval extensions. When f ′′ (x ∗ ) �= 0, an order 3 or higher extension should result in no cluster, even without accelera... |

17 |
Box-bisection for solving second-degree systems and the problem of clustering
- Morgan, Shapiro
- 1987
(Show Context)
Citation Context ...e algorithm cannot eliminate. We refer to this phenomenon as the cluster problem. Discussion of clustering in a branch and bound method in a particular context in the multidimensional case appears in =-=[6]-=-; however, the cause of the phenomenon studied there is different from the cause in the one-dimensional case studied here. In this paper, we consider the phenomenon, using interval extensions, in the ... |

15 |
Computing the range of values of real functions with accuracy higher than second order
- Cornelius, Lohner
- 1984
(Show Context)
Citation Context ... extension: F(X) = f(c) + F ′ (X)(X − c), which is also of order 2. The results occur in Table 2 ′ . initial ɛ ′ = 10 −2 ɛ ′ = 10 −3 ɛ ′ = 10 −4 ɛ ′ = 10 −5 ɛ ′ = 10 −6 ɛ ′ = 10 −7 [0, 2] 6 6 6 6 6 6 =-=[1, 3]-=- 4 4 4 4 4 4 [0.101, 5] 4 4 2 2 2 2 [1.01, 5] 6 5 6 5 5 5 Table 2 ′ . The behavior shown in Table 2 ′ is typical of an interval extension of order 2: the numbers of intervals in the final lists are al... |

12 |
Fortran-SC – A Study of a Fortran Extension for Engineering Scientific Computations with Access to
- BLEHER, RUMP, et al.
- 1987
(Show Context)
Citation Context ...dpoint test may not be able to reject all the intervals which do not contain global minimizers. This can be illustrated with the following example. and We know f(x) = (x − 1) 2 = x 2 − 2x + 1 for x ∈ =-=[0, 2]-=- F(X) = X 2 − 2X + 1. X ∗ = {1} and f ∗ = 0. The unique minimizer occurs at the endpoint of the adjacent intervals produced during the bisection process. Suppose at some stage, the length of each inte... |