Fortified-descent simplicial search method: A general approach

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.