  This work concerns the local convergence theory of Newton and quasi-Newton
methods for convex-composite optimization: minimize f(x):=h(c(x)), where h is
an infinite-valued proper convex function and c is C^2-smooth. We focus on the
case where h is infinite-valued piecewise linear-quadratic and convex. Such
problems include nonlinear programming, mini-max optimization, estimation of
nonlinear dynamics with non-Gaussian noise as well as many modern approaches to
large-scale data analysis and machine learning. Our approach embeds the
optimality conditions for convex-composite optimization problems into a
generalized equation. We establish conditions for strong metric subregularity
and strong metric regularity of the corresponding set-valued mappings. This
allows us to extend classical convergence of Newton and quasi-Newton methods to
the broader class of non-finite valued piecewise linear-quadratic
convex-composite optimization problems. In particular we establish local
quadratic convergence of the Newton method under conditions that parallel those
in nonlinear programming when h is non-finite valued piecewise linear.
