On Metric Regularity for Weakly Almost Piecewise Smooth Functions and Some Applications in Nonlinear Semidefinite Programming

The one-to-one relation between the points fulfilling the KKT conditions of an optimization problem and the zeros of the corresponding Kojima function is well-known. In the present paper we study the interplay between metric regularity and strong regularity of this a priori nonsmooth function in the context of semidefinite programming. Having in mind the topological structure of the positive semidefinite cone we identify a class of locally Lipschitz functions which turn out to have coherently oriented Bsubdifferentials if metric regularity is assumed. This class is general enough to contain the Kojima function corresponding to the nonlinear semidefinite programming problem. Using a characterization of strong regularity for semismooth functions in terms of B-subdifferentials we arrive at an equivalence between metric regularity and strong regularity provided that an assumption involving the topological degree is fulfilled. Moreover, we shall show that metric regularity of the Kojima function implies constraint nondegeneracy.