**Convexity and Duality in Nonconvex
Optimization
**

R. Tyrrell Rockafellar

University of Washington, Emeritus

Already in the early days of optimization
theory in the 1950s, a new and fascinating phenomenon came to light. Problems of minimization, at least in some
frameworks, initially just in linear programming but later more widely, are
paired with problems of maximization with respect to completely different
variables. It is essentially impossible
to solve one of the problems without at the same time solving its hidden
partner.

This phenomenon of duality was
discovered to be fundamental to the convex analysis that, in optimization,
relaces classical applied analysis.
Later it was learned that augmented Lagrangian
functions could provide globally dual problems even in nonlinear programming
without convexity. Now the subject has
progressed much farther, and a localized form of duality has emerged as a key
to understanding local optimality in its very essence. This talk will explain
the progression of ideas that has led to this recent insight and what it can
mean for numerical methodology.