Preprint A84/2001
The theory of 2-regularity for mappings with Lipschitzian derivatives and its applications to optimality conditions

Mikhail Solodov | Izmailov, Alexey

**Keywords: **
regularity | tangent cone | covering | optimality

We study local structure of a nonlinear mapping near points where standard
regularity and/or smoothness assumptions need not be satisfied.
We introduce a new concept of 2-regularity (a certain kind of
second-order regularity) for a once
differentiable mapping whose derivative is Lipschitz continuous.
Under this 2-regularity condition, we obtain
the representation theorem and the covering theorem (i.e., stability
with respect to ``right-hand side'' perturbations) under assumptions
which are weaker than those previously employed in the literature
for results of this type. These results are
further used to derive a constructive description of the tangent cone
to a set defined by (2-regular) equality constraints, and
optimality conditions for related optimization problems.
The class of mappings introduced and studied in the paper
appears to be a convenient tool for treating complementarity structures
by means of an appropriate equation-based reformulation.
Optimality conditions for mathematical programs with
(equivalently reformulated) complementarity constraints are also discussed.