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CATEGORIES:Applied and Computational Analysis
SUMMARY:Optimal Newton-type methods for nonconvex smooth o
ptimization - Coralia Cartis (School of Mathematic
s\, University of Edinburgh\, UK)
DTSTART;TZID=Europe/London:20120913T150000
DTEND;TZID=Europe/London:20120913T160000
UID:TALK38794AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/38794
DESCRIPTION:This talk addresses global rates of convergence an
d the worst-case evaluation complexity of methods
for nonconvex optimization problems. We show that
the classical steepest-descent and Newton's method
s for unconstrained nonconvex optimization under s
tandard assumptions may both require a number of i
terations and function evaluations arbitrarily clo
se to the steepest-descent's global worst-case com
plexity bound. This implies that the latter upper
bound is essentially tight for steepest descent an
d that Newton's method may be as slow as the steep
est-descent method in the worst case. Then the cub
ic regularization of Newton's method (Griewank (19
81)\, Nesterov & Polyak (2006)) is considered and
extended to large-scale problems\, while preservin
g the same order of its improved worst-case comple
xity (by comparison to that of steepest-descent)\;
this improved worst-case bound is also shown to b
e tight. We further show that the cubic regulariza
tion approach is\, in fact\, optimal from a worst-
case complexity point of view amongst a wide class
of second-order methods for nonconvex optimizatio
n. The worst-case problem-evaluation complexity of
constrained optimization will also be discussed.
This is joint work with Nick Gould (Rutherford App
leton Laboratory\, UK) and Philippe Toint (Univers
ity of Namur\, Belgium).
LOCATION:MR 14\, Centre for Mathematical Sciences
CONTACT:Carola-Bibiane Schoenlieb
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