The Paradigm Of The Implicit Function Theorem In Variational Analysis

par Asen L. Dontchev (University of Michigan)

mercredi 29 mai 2013, 16:30 → 17:30 Europe/Paris

Salle de conférence (XLIM)

The classical implicit function theorem revolves around solving an equation f(p,x)=0 for x in terms of a parameter p and tells us when the solution mapping associated with this equation is a differentiable function with respect to the parameter. In this talk we move into a much wider territory in replacing equation-solving problems by more complicated problems for "generalized equations'' that arise in constrained optimization, models of equilibrium, control theory, and many other areas. It turns out that if we put aside differentiability and focus on Lipschitz continuity only, or even more general metric regularity properties of mappings, we can cover a wider range of models and get estimates of the solution changes resulting from approximations of the model. After a review of metric regularity properties of set-valued mappings, we present an application to inexact Newton's method for solving variational inequalities. We also show an extension of the Dennis-Moré theorem to nonsmooth/generalized equations. An error estimate for the Euler-Newton method for tracking solution curves of variational inequalities will be also presented.