The theory of 2-regularity for mappings with Lipschitzian derivatives and its applications to optimality conditions
Mikhail Solodov | Izmailov, Alexey
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.