Authors

Citation

P. Tseng, Fortified-descent simplicial search method: A general approach, SIAM J OPTI, 10(1), 1999, pp. 269-288

Citations number

32

Language

INGLESE

art.tipo

Article

Categorie Soggetti

Mathematics

Journal title

SIAM JOURNAL ON OPTIMIZATION

ISSN journal

1052-6234
→ ACNP

Volume

10

Issue

1

Year of publication

1999

Pages

269 - 288

Database

ISI

SICI code

1052-6234(19991129)10:1<269:FSSMAG>2.0.ZU;2-9

Abstract

We propose a new simplex-based direct search method for unconstrained minim
ization of a real-valued function f of n variables. As in other methods of
this kind, the intent is to iteratively improve an n-dimensional simplex th
rough certain reflection/expansion/contraction steps. The method has three
novel features. First, a user-chosen integer (m) over bar(k) specifies the
number of "good" vertices to be retained in constructing the initial trial
simplices-reflected, then either expanded or contracted-at iteration k. Sec
ond, a trial simplex is accepted only when it satisfies the criteria of for
tified descent, which are stronger than the criterion of strict descent use
d in most direct search methods. Third, the number of additional function e
valuations needed to check a trial reflected/expanded simplex for fortified
descent can be controlled. If one of the initial trial simplices satisfies
the fortified-descent criteria, it is accepted as the new simplex; otherwi
se, the simplex is shrunk a fraction of the way toward a best vertex and th
e process is restarted, etc., until either a trial simplex is accepted or t
he simplex effectively has shrunk to a single point.
We prove several theoretical properties of the new method. If f is continuo
usly differentiable, bounded below, and uniformly continuous on its lower l
evel set and we choose (m) over bar(k) with the same value at all iteration
s k, then every cluster point of the generated sequence of iterates is a st
ationary point. The same conclusion holds if the function is continuously d
ifferentiable, bounded below, and we choose (m) over bar(k) = 1 at all iter
ations k.