Optimality Conditions for Mathematical Programs with Orthogonality Type Constraints

We consider the class of mathematical programs with orthogonality type constraints. Orthogonality type constraints appear by reformulating the sparsity constraint via auxiliary binary variables and relaxing them afterwards. For mathematical programs with orthogonality type constraints a necessary optimality condition in terms of T-stationarity is stated. The justification of T-stationarity is threefold. First, it allows to capture the global structure of mathematical programs with orthogonality type constraints in terms of Morse theory, i. e. deformation and cell-attachment results are established. For that, nondegeneracy for the T-stationary points is introduced and shown to hold generically. Second, we prove that Karush-Kuhn-Tucker points of the Scholtes-type regularization converge to T-stationary points of mathematical programs with orthogonality type constraints. This is done under the tailored linear independence constraint qualification, which turns out to be a generic property too. Third, we show that T-stationarity applied to the relaxation of sparsity constrained nonlinear optimization naturally leads to its M-stationary points. Moreover, we argue that all T-stationary points of this relaxation become degenerate.