## Subdivision Direction Selection In Interval Methods For Global Optimization (1997)

Venue: | SIAM J. Numer. Anal |

Citations: | 47 - 18 self |

### BibTeX

@ARTICLE{Csendes97subdivisiondirection,

author = {T. Csendes and D. Ratz},

title = {Subdivision Direction Selection In Interval Methods For Global Optimization},

journal = {SIAM J. Numer. Anal},

year = {1997},

volume = {34},

pages = {922--938}

}

### Years of Citing Articles

### OpenURL

### Abstract

. The role of the interval subdivision selection rule is investigated in branch-and-bound algorithms for global optimization. The class of rules that allow convergence for the model algorithm is characterized, and it is shown that the four rules investigated satisfy the conditions of convergence. A numerical study with a wide spectrum of test problems indicates that there are substantial differences between the rules in terms of the required CPU time, the number of function and derivative evaluations and space complexity, and two rules can provide substantial improvements in efficiency. Key words. global optimization, interval arithmetic, interval subdivision AMS subject classifications. 65K05, 90C30 Abbreviated title: Subdivision directions in interval methods. 1. Introduction. Interval subdivision methods for global optimization [7, 21] aim at providing reliable solutions to global optimization problems min x2X f(x) (1) where the objective function f : IR n ! IR is continuo...

### Citations

876 |
Interval Analysis
- MOORE
- 1966
(Show Context)
Citation Context ...lgorithm and subdivision direction selection rules. First we give a simple model algorithm that has the most important common features of the interval subdivision methods for global optimization (cf. =-=[2, 3, 5, 6, 7, 16, 21, 22]-=-). No local search procedure is included. The Newton-like steps are also not built in, since these would need the inclusion of the Hessian. On the other hand, the cut-off and monotonicity tests are ap... |

507 |
Interval methods for systems of equations
- Neumaier
- 1990
(Show Context)
Citation Context ...lied widths of the respective intervals (and these are in general not equal). After a short calculation, the right-hand side of (6) can be written as maxfj minF 0 i (X)j; j maxF 0 i (X)jgw(X i ) (cf. =-=[1, 17, 21]). This co-=-rresponds to the "maximum smear" (used as a direction selection merit function solving systems of nonlinear equations [13]) for the case f : IR n ! IR. It is easy to see that the Rules B and... |

465 |
Global optimization Using Interval Analysis
- HANSEN
- 1001
(Show Context)
Citation Context ...ic, interval subdivision AMS subject classifications. 65K05, 90C30 Abbreviated title: Subdivision directions in interval methods. 1. Introduction. Interval subdivision methods for global optimization =-=[7, 21]-=- aim at providing reliable solutions to global optimization problems min x2X f(x) (1) where the objective function f : IR n ! IR is continuously differentiable, and X ` IR n is an n-dimensional interv... |

454 |
Introduction to Interval Computation
- Alefeld, Herzberger
- 1983
(Show Context)
Citation Context ...l solutions are acceptable [4, 22], and the local minima are less important. No special problem structure is required: only inclusion functions of the objective function and its gradient are utilised =-=[1]-=-. Denote the set of compact intervals by II := f[a; b] j asb; a; b 2 IRg and the set of n-dimensional intervals (also called simply intervals or boxes) by II n . We call a function F : II n ! II to be... |

204 |
Lipschitzian Optimization without the Lipschitz Constant
- Jones, Perttunen, et al.
- 1993
(Show Context)
Citation Context ... is subdivided in a uniform way, then the width of the actual subintervals goes the quickest to zero. It has also been used for generating subdivision direction in other optimization procedures (e.g. =-=[11]-=-). 924 T. Csendes and D. Ratz The algorithm with Rule A is convergent both with and without the monotonicity test (e.g. in [5] and in [21]). This rule allows a relatively simple analysis of the conver... |

130 |
ROKNE New Computer Methods for Global optimization
- RATSCHEK, J
- 1988
(Show Context)
Citation Context ...ic, interval subdivision AMS subject classifications. 65K05, 90C30 Abbreviated title: Subdivision directions in interval methods. 1. Introduction. Interval subdivision methods for global optimization =-=[7, 21]-=- aim at providing reliable solutions to global optimization problems min x2X f(x) (1) where the objective function f : IR n ! IR is continuously differentiable, and X ` IR n is an n-dimensional interv... |

56 |
C++ Toolbox for Verified Computing
- Hammer, Hocks, et al.
- 1995
(Show Context)
Citation Context ...ction of the gradient of f(x) is denoted by F 0 (X). There are several ways to build an inclusion function for a given optimization problem (e.g. by using the Lipschitz constant). Interval arithmetic =-=[1, 6, 7, 21]-=- is a convenient tool for constructing the inclusion functions, and one can get those for almost all functions that can be calculated by a finite algorithm (i.e. not only for given expressions). It is... |

42 |
a portable interval Newton bisection package
- Kearfott, Novoa, et al.
- 1990
(Show Context)
Citation Context ...written as maxfj minF 0 i (X)j; j maxF 0 i (X)jgw(X i ) (cf. [1, 17, 21]). This corresponds to the "maximum smear" (used as a direction selection merit function solving systems of nonlinear =-=equations [13]-=-) for the case f : IR n ! IR. It is easy to see that the Rules B and C give the same merit function value if and only if either min F 0 i (X) = 0 or maxF 0 i (X) = 0. It is worth mentioning, that maxf... |

40 | A Fortran 90 environment for research and prototyping of enclosure algorithms for nonlinear equations and global optimization
- Kearfott
- 1995
(Show Context)
Citation Context ...thmetic package handling the outside rounding necessary for the inclusion functions. The authors thank R. B. Kearfott and W. V. Walter for their kind help in providing the interval arithmetic package =-=[12]-=- and the necessary modules. The inclusion functions were produced by the natural interval extension that fulfils the assumptions made in Section 1: the isotonicity and property (2). In this straightfo... |

30 |
Nonlinear parameter estimation by global optimizationâ€”efficiency and reliability
- Csendes
(Show Context)
Citation Context ...in x2X f(x) (1) where the objective function f : IR n ! IR is continuously differentiable, and X ` IR n is an n-dimensional interval. In many cases, only the globally optimal solutions are acceptable =-=[4, 22]-=-, and the local minima are less important. No special problem structure is required: only inclusion functions of the objective function and its gradient are utilised [1]. Denote the set of compact int... |

28 |
Automatische ergebnisverifikation bei globalen optimierungsproblemen. pHd dissertation, Universitat
- Ratz
- 1992
(Show Context)
Citation Context ... only one way how Rule B was applied in [7]. The subdivision was, e.g., carried out also for many directions in a single iteration step. Rule C. The next rule of our investigation was defined by Ratz =-=[23]-=-. The underlying idea was to minimize the width of the inclusion: w(F (X)) = w(F (X) \Gamma F (m(X)))sw(F 0 (X)(X \Gamma m(X))) = n X i=1 w \Gamma F 0 i (X)(X i \Gamma m(X i )) \Delta : Obviously, tha... |

24 |
Computation of rational interval functions
- Skelboe
- 1974
(Show Context)
Citation Context ...y using a merit function: k := min ae j j j 2 f1; 2; : : : ; ng and D(j) = n max i=1 D(i) oe (3) where D(i) is determined by the given rule. Rule A. First the interval width oriented rule was applied =-=[16, 21, 24]-=-, this chooses the coordinate direction with D(i) := w(X i ): (4) This rule was justified by the idea that if the original interval is subdivided in a uniform way, then the width of the actual subinte... |

22 |
A Global Minimization Method: the Multi-Dimensional Case
- Jansson, Knueppel
- 1992
(Show Context)
Citation Context ...er, quicker and less error prone than the earlier one in FORTRAN-77, when all the operations were transformed to function calls on new data structures. More sophisticated inclusion functions (like in =-=[10]-=- or [21]) would result in better efficiency figures at the cost of additional calculations or preliminary reformulations on the involved functions. The inclusions for the gradients were calculated com... |

13 |
An interval step control for continuation methods
- Kearfott, Xing
- 1994
(Show Context)
Citation Context ... time unit (STU, 1000 evaluations of the noninterval Shekel-5 function) was on the used workstation 0.0036 Sec. The large CPU times measured in STU are in part due to the interval implementation (cf. =-=[14]-=-) and the overhead of the list manipulation. The CPU values are in general proportional to the number of objective function (NFE) and derivative evaluations (NDE). The exceptions are the cases with hi... |

11 |
Stochastic global optimization
- Zhigliavsky, Zilinskas
- 2008
(Show Context)
Citation Context ...ms to be saved, but the size of the data structure can be decreased using the available information [11]. The numerical tests involved the set of standard global optimization problems (definitions in =-=[25]-=-, further numerical results in [4, 5, 10, 23]) and the set of test problems studied in Hansen's book (descriptions in [7], additional test results in [10, 23]). In some cases, where slight alterations... |

9 |
The impact of accelerating tools on the interval subdivision algorithm for global optimization
- Csendes, Pinter
- 1993
(Show Context)
Citation Context ...i-karlsruhe.de). 922 Subdivision direction selection in interval methods for global optimization 923 of the interval global optimization methods. After studying the effects of some accelerating tools =-=[5]-=-, the present paper investigates the role of the selection of the interval subdivision direction. 2. Model algorithm and subdivision direction selection rules. First we give a simple model algorithm t... |

8 |
Methodologies for tolerance intervals
- Kristinsdottir, Zabinsky, et al.
- 1993
(Show Context)
Citation Context ... with the number of derivative evaluations is more important than the required CPU time, because the computation of the involved functions are usually longer than those of the test problems (see e.g. =-=[15, 22]-=-). According to the test results, 7% improvement can be expected if Rules B or C are applied instead of Rule A, and Rule D causes 102% higher number of function evaluations. The sum of the numbers of ... |

5 |
Modeling of lowfrequency pulmonary impedance in dogs
- Hantos, Daroczy, et al.
- 1990
(Show Context)
Citation Context ... original problem with " = 10 \Gamma6 . These problems were completed by three new ones. The first one, called EX1 was defined in Section 2. EX2 is a simplified real life parameter estimation pro=-=blem [8, 18]-=-: f(x) = 6 X i=1 fi fi fi fi fi f i \Gamma / x 1 + x 2 ! x 3 i \Gamma -- / ! i x 4 \Gamma x 5 ! x 3 i !! fi fi fi fi fi 2 ; where the f i -s are 5:0 \Gamma 5:0--, 3:0 \Gamma 2:0--, 2:0 \Gamma --, 1:5 ... |

4 |
On the convergence of two branch-and-bound algorithms for nonconvex programming problems
- Benson
- 1982
(Show Context)
Citation Context ...lgorithm and subdivision direction selection rules. First we give a simple model algorithm that has the most important common features of the interval subdivision methods for global optimization (cf. =-=[2, 3, 5, 6, 7, 16, 21, 22]-=-). No local search procedure is included. The Newton-like steps are also not built in, since these would need the inclusion of the Hessian. On the other hand, the cut-off and monotonicity tests are ap... |

4 |
New LP-Bound in Multivariate Lipschitz Optimization: Theory and Applications
- Horst, V
- 1995
(Show Context)
Citation Context ... track of the three first list members. This implementation can be efficient for short lists, while problems of large memory complexity require data structures like the AVLtrees or other search trees =-=[9, 11]-=-. The implementation of the list can affect the required CPU time, but not the number of objective function and derivative evaluations. The memory complexity is invariant regarding the data structure ... |

1 |
Combining real and interval methods for global optimization
- Caprani, Hansen, et al.
- 1993
(Show Context)
Citation Context ...lgorithm and subdivision direction selection rules. First we give a simple model algorithm that has the most important common features of the interval subdivision methods for global optimization (cf. =-=[2, 3, 5, 6, 7, 16, 21, 22]-=-). No local search procedure is included. The Newton-like steps are also not built in, since these would need the inclusion of the Hessian. On the other hand, the cut-off and monotonicity tests are ap... |

1 |
oczy, Partitioning of pulmonary impedance: modeling vs. alveolar capsule approach
- ak, Hantos
- 1993
(Show Context)
Citation Context ... original problem with " = 10 \Gamma6 . These problems were completed by three new ones. The first one, called EX1 was defined in Section 2. EX2 is a simplified real life parameter estimation pro=-=blem [8, 18]-=-: f(x) = 6 X i=1 fi fi fi fi fi f i \Gamma / x 1 + x 2 ! x 3 i \Gamma -- / ! i x 4 \Gamma x 5 ! x 3 i !! fi fi fi fi fi 2 ; where the f i -s are 5:0 \Gamma 5:0--, 3:0 \Gamma 2:0--, 2:0 \Gamma --, 1:5 ... |

1 |
Extended univariate algorithms for n-dimensional global optimization
- er
- 1986
(Show Context)
Citation Context ...e available inclusion function information. This formulation shows how the model algorithm with the direction selection rule C can be related to Lipschitzian partition methods for global optimization =-=[19, 20]-=-. Rule D. The fourth rule is derivative-free like Rule A, and reflects the machine representation of the inclusion function F (X) (see [6]). It is again defined by (3) and by D(i) := ( w(X i ) if 0 2 ... |

1 |
using interval analysis for solving a circuit design problem
- Experiments
- 1993
(Show Context)
Citation Context ... with the number of derivative evaluations is more important than the required CPU time, because the computation of the involved functions are usually longer than those of the test problems (see e.g. =-=[15, 22]-=-). According to the test results, 7% improvement can be expected if Rules B or C are applied instead of Rule A, and Rule D causes 102% higher number of function evaluations. The sum of the numbers of ... |