Smithsonian Astrophysical Observatory, 60 Garden Street, MS 81, Cambridge, MA 02138 USA

As part of the Smithsonian Astrophysical Observatory AXAF
Science Center (ASC) Data Analysis Environment, ASCFIT
was designed as a generalized fitting engine to enable users to fit models to
the data that will be returned by AXAF, the Advanced X-ray
Astrophysical Facility (Doe et al. 1996). I present a
description of the use of a development version of
ASCFIT as a function optimizer, in which the optimization methods
available within ASCFIT are applied to seven benchmark functions of varying
complexity. These functions are the 2-dimensional Gaussian function previously
investigated by ASC, and six functions commonly used in optimization testing.
The optimization methods combining the Powell technique with
one of several other
techniques successfully converged for all but one of the
benchmark functions.

Simplex | Powell |

Grid | Grid + Powell |

Monte Carlo | Monte Carlo + Powell |

Simulated Annealing 1 | Simulated Annealing 1 + Powell |

Simulated Annealing 2 | Simulated Annealing 2 + Powell |

Function 2dg(x, y) is a two-dimensional Gaussian function:

,

,

The function has coefficients *s*_{i}, *t*_{i}, *d*_{i}, and *c*
defined such that the function is a
paraboloid with axes parallel to the coordinate axes, and a
set of holes that increase in depth near the origin. The
function is very difficult to
minimize, having over 10^{20} local minima in which the
optimization may become trapped:

if

otherwise,

*s*_{i}=0.2,*t*_{i}=0.05,*i*=1,....4,

*c*=0.15,

The function *f*_{1} is a simple sum of squares, with a
single minimum:

The function *f*_{2} is the classical Rosenbrock function:

The function *f*_{3} is discontinuous and has a single minimum:

The function *f*_{4} is a noisy function, as is a
uniform random variable bounded by
:

Function *f*_{5} has 25 local minima, and is optimized
with a value of approximately
0.001996 when
*x*_{1}=*x*_{2}=-32:

Table 1 summarizes the results for scenario B. For each
function and technique, the value shown is the normalized
final parameter distance, defined as the ratio of the
distance between the final parameter state and the global
minimum to the maximum distance between any
two states within the allowed parameter space. Distances of
less than 10^{-6} are shown as 10^{-6}. In general, the
results for scenario C (not shown) indicate that, as expected, a more
favorable initial state improved the
single-shot routines' optimizations of smoothly varying
functions. The final column is the execution time for the
gaussian, normalized to the time for the simplex method. The
result of one technique is plotted in Fig.
1, with the initial parameter distances for scenarios A, B
and C.

Using optimization scenario C, the global minimum was located for all functions except , in which the routines became trapped in local minima. For this function, the grid and simulated annealing techniques would presumably locate the minimum if strict enough criteria were used; the grid method showed improved optimization for scenario B compared to scenario A. Ingber & Rosen (1992) tested a similar function with 30 variables and obtained comparable results.

The single-shot routines successfully converged for the
smoothly varying functions (2dg, *f*_{1}, and *f*_{2}),
particularly when starting from a favorable parameter
state. For the discontinuous functions, these routines
sometimes became trapped in local minima. In these cases, the
optimizations were not improved substantially by using an
initial parameter state closer to the global minimum or
by using stricter convergence criteria.

The scatter-shot routines were less successful in optimizing the smooth functions, as they sample the entire parameter space without refining the search near the minimum. However, for both the smooth and the discontinuous functions, the addition of the Powell technique improved the optimization, at the expense of longer execution times (see Table 1). Such methods with refining steps following sampling of the parameter space are to be preferred.

This project is supported by NASA contract NAS8-39073. I would like to thank the following individuals for helpful discussions and their review: Mark Birkinshaw, Janet DePonte, Stephen Doe, Diab Jerius, Malin Ljungberg, Dan Nguyen, Aneta Siemiginowska, and Robert Zacher.

Birkinshaw, M. 1998, Some Advice on Minimization Routines, (AXAF Science Center Internal Memo)

Doe, S., Ljungberg, M., Siemiginowska, A., & Joye, W. 1996, in ASP Conf. Ser., Vol. 145, Astronomical Data Analysis Software and Systems VII, ed. R. Albrecht, R. N. Hook, & H. A. Bushouse (San Francisco: ASP), 157

Ingber, L. & Rosen, B. 1992, Mathematical and Computer Modelling, 16(11), 87

More, J., Garbow, B. S., Hillstrom, K. E. 1981, ACM Trans on Math. Software, 7, 17

© Copyright 1999 Astronomical Society of the Pacific, 390 Ashton Avenue, San Francisco, California 94112, USA

adass@ncsa.uiuc.edu