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## Nonsmooth, Nonconvex Optimization: Theory, Algorithms and Applications

### Michael Overton

New York University

###
Thursday, May 4, 2006

L01 Carson Hall, 4 pm

Tea 3:30 pm, Math Lounge

**Abstract: ** Theory: there are two standard approaches to
generalizing derivatives to nonsmooth, nonconvex optimization: the
Clarke subdifferential (or generalized gradient), and the MIRW
subdifferential (or subgradient sets), as expounded in Rockafellar and
Wets (Springer, 1998). We briefly discuss these and mention their
advantages and disadvantages. They coincide for an important class of
functions: those that are locally Lipschitz and regular, which
includes continuously differentiable functions and convex
functions.

Algorithms: the usual approach is bundle methods, which
are complicated. We describe some alternatives: BFGS (a new look at
an old method), and Gradient Sampling (a simply stated method that,
although computationally intensive, has solved some previously
unsolved problems and has a nice convergence theory).

Applications: these abound in control, but surely in other areas too.
Of particular interest to me are applications involving eigenvalues
and singular values of nonsymmetric matrices. Sometimes even easily
stated problems in a few variables are hard. Our new code HIFOO
(H-Infinity Fixed-Order Optimization) is intended for use by
practicing control engineers and has solved some open problems in
control.

This is all joint work with James Burke and Adrian Lewis.
HIFOO is also joint with Didier Henrion.

This talk will be accessible to undergraduates.