The paper is devoted to the development of a comprehensive calculus for directional limiting normal cones, subdifferentials and coderivatives in finite dimensions. This calculus encompasses the whole range of the standard generalized differential calculus for (non-directional) limiting notions and relies on very weak (non-restrictive) qualification conditions having also a directional character. The derived rules facilitate the application of tools exploiting the directional limiting notions to difficult problems of variational analysis including, for instance, various stability and sensitivity issues. This is illustrated by some selected applications in the last part of the paper.

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In this paper, we consider a sufficiently broad class of non-linear mathematical programs with disjunctive constraints, which, e.g. include mathematical programs with complemetarity/vanishing constraints. We present an extension of the concept of -stationarity which can be easily combined with the well-known notion of M-stationarity to obtain the stronger property of so-called -stationarity. We show how the property of -stationarity (and thus also of M-stationarity) can be efficiently verified for the considered problem class by computing -stationary solutions of a certain quadratic program. We consider further the situation that the point which is to be tested for -stationarity, is not known exactly, but is approximated by some convergent sequence, as it is usually the case when applying some numerical method.

Estimating the regular normal cone to constraint systems plays an important role for the derivation of sharp necessary optimality conditions. We present two novel approaches and introduce a new stationarity concept which is stronger than M-stationarity. We apply our theory to three classes of mathematical programs frequently arising in the literature.