Exact augmented Lagrangians for constrained optimization problems in Hilbert spaces I: theory

Abstract

This two-part study is devoted to the analysis of the so-called exact augmented Lagrangians, introduced by Di Pillo and Grippo for finite dimensional optimization problems, in the case of optimization problems in Hilbert spaces. In the second part of our study we present applications of the general theory of exact augmented Lagrangians to several constrained variational problems and optimal control problems, including variational problems with additional constraints at the boundary, isoperimetric problems, problems with nonholonomic equality constraints (PDE constraints), and optimal control problems for linear evolution equations. We provide sufficient conditions for augmented Lagrangians for these problems to be globally/completely exact, that is, conditions under which a constrained variational problem/optimal control problem becomes equivalent to the problem of unconstrained minimization of the corresponding exact augmented Lagrangian in primal and dual variables simultaneously. PreliminariesLet us introduce notation and recall some auxiliary definitions from the first part of our study [14] and the theory of Sobolev spaces that will be utilised throughout this article.