## A Radial Basis Function Method for Global Optimization (1999)

Venue: | JOURNAL OF GLOBAL OPTIMIZATION |

Citations: | 49 - 1 self |

### BibTeX

@ARTICLE{Gutmann99aradial,

author = {H.-M. Gutmann},

title = {A Radial Basis Function Method for Global Optimization},

journal = {JOURNAL OF GLOBAL OPTIMIZATION},

year = {1999},

volume = {19},

pages = {2001}

}

### Years of Citing Articles

### OpenURL

### Abstract

We introduce a method that aims to find the global minimum of a continuous nonconvex function on a compact subset of R^d. It is assumed that function evaluations are expensive and that no additional information is available. Radial basis function interpolation is used to define a utility function. The maximizer of this function is the next point where the objective function is evaluated. We show that, for most types of radial basis functions that are considered in this paper, convergence can be achieved without further assumptions on the objective function. Besides, it turns out that our method is closely related to a statistical global optimization method, the P-algorithm. A general framework for both methods is presented. Finally, a few numerical examples show that on the set of Dixon-Szego test functions our method yields favourable results in comparison to other global optimization methods.

### Citations

388 |
Differential evolution - a simple and efficient heuristic for global optimization overcontinuous spaces
- Storn, Price
- 1997
(Show Context)
Citation Context ... are recent methods that, according to the results presented in those papers, are more efficient than most of their competitors on the Dixon-Szego testbed. DE (Differential Evolution, Storn and Price =-=[18]-=-) is an evolutionary method that operates only at the global level, which explains the large number of function evaluations. EGO (Efficient Global Optimization, Jones, Schonlau and Welch [10]) is the ... |

232 |
Efficient global optimization of expensive black-box functions
- Jones, Schonlau, et al.
- 1998
(Show Context)
Citation Context ...timization 25 no. of local no. of global function dimension minima minima domain Branin 2 3 3 [\Gamma5; 10] \Theta [0; 15] Goldstein-Price 2 4 1 [\Gamma2; 2] 2 Hartman 3 3 4 1 [0; 1] 3 Shekel 5 4 5 1 =-=[0; 10]-=- 4 Shekel 7 4 7 1 [0; 10] 4 Shekel 10 4 10 1 [0; 10] 4 Hartman 6 6 4 1 [0; 1] 6 Table 1: Dixon-Szego test functions and their dimension, the domain and the number of local and global minima. of h n li... |

207 |
Global Optimization
- Torn, Zilinskas
- 1989
(Show Context)
Citation Context ...fined on D. Under these assumptions, (GOP) is solvable, because f attains its minimum on D. Numerous methods to solve (GOP) have been developed (see e.g. Horst and Pardalos [4] and Torn and Zilinskas =-=[19]-=-). Stochastic methods like simulated annealing and genetic algorithms which use only function values are very popular among users, although their rate of convergence is usually rather slow. Determinis... |

204 |
Lipschitzian Optimization without the Lipschitz Constant
- Jones, Perttunen, et al.
- 1993
(Show Context)
Citation Context ...number of function evaluations needed to achieve a relative error less than 1% and 0:01%. RBF denotes our method using the target value strategy described above. DIRECT (Jones, Perttunen and Stuckman =-=[9]-=-) and MCS (Multilevel Coordinate Search, Huyer and Neumaier [5]) are recent methods that, according to the results presented in those papers, are more efficient than most of their competitors on the D... |

166 |
The Theory of Radial Basis Function Approximation
- Powell
- 1992
(Show Context)
Citation Context ...introduce the linear space Vm ae IR n containing alls2 IR n that satisfy n X i=1si q(x i ) = 0 8 q 2 \Pi m : (2.4) Formally, we set V \Gamma1 := IR n . Obviously, Vm+1 ae Vm for all ms\Gamma1. Powell =-=[15]-=- shows that, in the cubic and thin plate spline cases T \Phi ? 0 8s2 V 1 n f0g; (2.5) in the linear and multiquadric cases T \Phi ! 0 8s2 V 0 n f0g; (2.6) and in the Gaussian case T \Phi ? 0 8s2 IR n ... |

122 |
Approximation Theory and Methods
- Powell
- 1989
(Show Context)
Citation Context ... interpolant can be developed from the theory of natural cubic splines in one dimension. They can be written in the form (2.1), where OE(r) = r 3 ;s2 V 1 and p 2 \Pi 1 . It is well known (e.g. Powell =-=[14]-=-) that the interpolant s that is defined by the system (2.12) minimizes I(g) := R IR [g 00 (x)] 2 dx among all functions g : IR ! IR that satisfy the interpolation conditions g(x i ) = f i ; i = 1; : ... |

115 |
Analytic extensions of differentiable functions defined in closed sets
- Whitney
- 1934
(Show Context)
Citation Context ...unction method for global optimization 15 Proposition 10 Let OE, m andsbe defined as in Proposition 6, and let f 2 C (D), where D ae IR d is compact. Then f 2 N OE;m (D). Proof: By Whitney's theorem (=-=[20]-=-), f can be extended to a function F 2 C (IR d ) that is equal to f on D. Now D is contained in a closed ball of radius ffi, say, and there is an infinitely differentiable function g with g(x) = 1; kx... |

108 |
Handbook of Global Optimization
- Horst, Pardalos
- 1995
(Show Context)
Citation Context ...is a continuous function defined on D. Under these assumptions, (GOP) is solvable, because f attains its minimum on D. Numerous methods to solve (GOP) have been developed (see e.g. Horst and Pardalos =-=[4]-=- and Torn and Zilinskas [19]). Stochastic methods like simulated annealing and genetic algorithms which use only function values are very popular among users, although their rate of convergence is usu... |

73 | Global optimization by multilevel coordinate search
- Huyer, Neumaier
- 1999
(Show Context)
Citation Context ...or less than 1% and 0:01%. RBF denotes our method using the target value strategy described above. DIRECT (Jones, Perttunen and Stuckman [9]) and MCS (Multilevel Coordinate Search, Huyer and Neumaier =-=[5]-=-) are recent methods that, according to the results presented in those papers, are more efficient than most of their competitors on the Dixon-Szego testbed. DE (Differential Evolution, Storn and Price... |

34 |
P.: The Optimization Problem: An Introduction
- Dixon, Szego
- 1978
(Show Context)
Citation Context ...e inefficiencies are reduced if large function values are replaced by the median of all available function values. Some experiments were performed using the test functions proposed by Dixon and Szego =-=[2]-=-. Table 1 gives the name of each function, the dimension, the domain and the number of local and global minima in that domain. The maximization of (3.9) was carried out using a version of the tunnelin... |

34 |
The Tunneling Algorithm for the Global Minimization of Functions
- Levy, Montalvo
- 1985
(Show Context)
Citation Context ...each function, the dimension, the domain and the number of local and global minima in that domain. The maximization of (3.9) was carried out using a version of the tunneling method (Levy and Montalvo =-=[13]-=-). The target values f n are determined as follows. The idea is to perform cycles of N + 1 iterations for some N 2 IN , where each cycle employs a range of target values, starting with a low one (glob... |

25 | Comparison of radial basis function interpolantsâ€ť, in Multivariate Approximation: From CAGD to Wavelets
- Schaback
- 1993
(Show Context)
Citation Context ... msm 0 , i.e. s 2 A OE;m is not a polynomial in \Pi m . Thus (2.14) is a semi-inner product on A OE;m that induces the semi-norm hs; si with null space \Pi m (for details see Powell [16] and Schaback =-=[17]-=-). In analogy to the variational principle for cubic splines in one dimension, mentioned above, there is a theorem that states that the given interpolant is the solution to a minimization problem. 8 H... |

18 |
A new method of locating the maximum point of an arbitrary multipeak curve in the presence of noise
- Kushner
- 1964
(Show Context)
Citation Context ...tion, namely the P-algorithm ( Zilinskas [22]). Although being derived using a completely different approach, it is very similar to our method. One special case of a P-algorithm, developed by Kushner =-=[12]-=-, is even equivalent to a special case of our radial basis function method. Other global optimization methods based on radial basis functions have been developed. Alotto et. al. [1] use interpolation ... |

9 | Recent research at cambridge on radial basis functions. Report DAMTP 1998/NA05, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, 1998
- Powell
(Show Context)
Citation Context ... ifs2 Vm n f0g and msm 0 , i.e. s 2 A OE;m is not a polynomial in \Pi m . Thus (2.14) is a semi-inner product on A OE;m that induces the semi-norm hs; si with null space \Pi m (for details see Powell =-=[16]-=- and Schaback [17]). In analogy to the variational principle for cubic splines in one dimension, mentioned above, there is a theorem that states that the given interpolant is the solution to a minimiz... |

4 | On the semi-norm of radial basis function interpolants
- Gutmann
- 2000
(Show Context)
Citation Context ...e points has the property hs n ; s n isC: The characterization of N OE;m (D) is rather abstract. In the linear, cubic and thin plate spline cases, the following proposition that is taken from Gutmann =-=[3]-=- provides a useful criterion to check whether it is satisfied. In the multiquadric and Gaussian cases, however, no such criterion is known. Proposition 6 Let OE(r) = r, OE(r) = r 2 log r or OE(r) = r ... |

4 |
Axiomatic Characterization of a Global Optimization Algorithm and Investigation of its Search Strategy
- Zilinskas
- 1985
(Show Context)
Citation Context ...erpolation points, and a measure of bumpiness is also available. Close relations can be established between our method and one from statistical global optimization, namely the P-algorithm ( Zilinskas =-=[22]-=-). Although being derived using a completely different approach, it is very similar to our method. One special case of a P-algorithm, developed by Kushner [12], is even equivalent to a special case of... |

3 |
M.Repetto. A Multiquadrics-based Algorithm for the Acceleration of Simulated Annealing Optimization Procedures
- Alotto, Caiti, et al.
- 1996
(Show Context)
Citation Context ...eloped by Kushner [12], is even equivalent to a special case of our radial basis function method. Other global optimization methods based on radial basis functions have been developed. Alotto et. al. =-=[1]-=- use interpolation by multiquadrics to accelerate a simulated annealing method. Ishikawa et. al. [6], [7] employ radial basis functions to estimate the global minimizer and run an SQP algorithm to loc... |

3 |
An Optimization Method Based on Radial Basis Function
- Ishikawa, Matsunami
- 1997
(Show Context)
Citation Context ...her global optimization methods based on radial basis functions have been developed. Alotto et. al. [1] use interpolation by multiquadrics to accelerate a simulated annealing method. Ishikawa et. al. =-=[6]-=-, [7] employ radial basis functions to estimate the global minimizer and run an SQP algorithm to locate it. The properties of radial basis functions that are necessary for the description of our metho... |

2 |
Axiomatic Approach to Statistical Models and their Use
- Zilinskas
- 1982
(Show Context)
Citation Context ...e, the interpolant s n is identical to Mean. Further, except for a constant factor, Var(x) = \Gamma 1sn (x) : Therefore, Kushner's method and our method using linear splines are equivalent. Zilinskas =-=[21], [22] ext-=-ends this approach to Gaussian random processes in several dimensions. He uses the selection rule (5.1) and introduces the name "Palgorithm ". In addition, he gives an axiomatic description ... |

1 |
A Combines Method for the Global Optimization Using Radial Basis Function and Deterministic Approach
- Ishikawa, Tsukui, et al.
- 1999
(Show Context)
Citation Context ...lobal optimization methods based on radial basis functions have been developed. Alotto et. al. [1] use interpolation by multiquadrics to accelerate a simulated annealing method. Ishikawa et. al. [6], =-=[7]-=- employ radial basis functions to estimate the global minimizer and run an SQP algorithm to locate it. The properties of radial basis functions that are necessary for the description of our method are... |

1 |
Global optimization with response surfaces
- Jones
- 1996
(Show Context)
Citation Context ...e function evaluations, our goal is to require as few function evaluations as possible to find an adequate estimate of the global minimum. The method is based on a general technique proposed by Jones =-=[8]. Let A be-=- a linear space of functions, and assume that, for s 2 A, oe(s) is a measure of the "bumpiness" of s. Now assume that we have calculated x 1 ; : : : ; x n and the function values f(x 1 ); : ... |

1 |
A Versatile Model of a Function of Unknown and Time Varying Form
- Kushner
- 1962
(Show Context)
Citation Context ...ions to statistical global optimization In this section we consider the similarities between the given radial basis function method and the P-algorithm. The idea of that method is proposed by Kushner =-=[11]-=-, [12] for one-dimensional problems. Here the objective function is regarded as a realization of a Brownian motion stochastic process. If real numbers x 1 ! : : : !x n A radial basis function method f... |