On mathematical programs with sign constraints and their applications in regularization theory

We study a novel class of nonsmooth optimization problems, called mathematical programs with sign constraints (MPSiC).For MPSiC, the critical point theory is developed.It includes the introduction of topologically relevant T-stationary points along with their T-index.Two typical results within the scope of Morse theory are demonstrated.Outside the set of T-stationary points, the MPSiC lower level sets remain homotopy-equivalent.If surpassing a nondegenerate T-stationary point, a cell of dimension equal to its T-index has to be attached for this purpose.Further, we apply our findings on the MPSiC class to the sign-type regularization of mathematical programs with complementarity constraints (MPCC) from [A.Kadrani, J.P. Dussault, A. Benchakroun, A new regularization scheme for mathematical programs with complementarity constraints, SIAM J. Optim.20 (2009), 78-103].By doing so, the convergence analysis for the sign-type regularization of MPCC is refined.Among the other results, an index shift for limiting C-stationary points of MPCC can not be avoided in the generic sense.In particular, a sequence of saddle points of the sign-type regularization might converge to a minimizer of MPCC.This phenomenon turns out to be stable with respect to C 2 -perturbations of the MPCC defining functions.Moreover, due to the stably occurring bifurcations, the global structure of the sign-type regularization becomes much more involved than that of the underlying MPCC.