  This paper is devoted to developing and applications of a generalized
differential theory of variational analysis that allows us to work in
incomplete normed spaces, without employing conventional variational techniques
based on completeness and limiting procedures. The main attention is paid to
generalized derivatives and subdifferentials of the Dini-Hadamard type with the
usage of mild qualification conditions revolved around metric subregularity. In
this way we develop calculus rules of generalized differentiation in normed
spaces without imposing restrictive normal compactness assumptions and the like
and then apply them to general problems of constrained optimization. Most of
the obtained results are new even in finite dimensions. Finally, we derive
refined necessary optimality conditions for nonconvex problems of semi-infinite
and semidefinite programming.
