  The paper is devoted to a comprehensive study of composite models in
variational analysis and optimization the importance of which for numerous
theoretical, algorithmic, and applied issues of operations research is
difficult to overstate. The underlying theme of our study is a systematical
replacement of conventional metric regularity and related requirements by much
weaker metric subregulatity ones that lead us to significantly stronger and
completely new results of first-order and second-order variational analysis and
optimization. In this way we develop extended calculus rules for first-order
and second-order generalized differential constructions with paying the main
attention in second-order variational theory to the new and rather large class
of fully subamenable compositions. Applications to optimization include
deriving enhanced no-gap second order optimality conditions in constrained
composite models, complete characterizations of the uniqueness of Lagrange
multipliers and strong metric subregularity of KKT systems in parametric
optimization, etc.
