Convexity and Duality in Nonconvex
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.