Bilevel Imaging Learning Problems as Mathematical Programs with Complementarity Constraints

We investigate a family of bilevel imaging learning problems where the lower-level instance corresponds to a convex variational model involving first- and second-order nonsmooth regularizers. By using geometric properties of the primal-dual reformulation of the lower-level problem and introducing suitable changes of variables, we are able to reformulate the original bilevel problems as Mathematical Programs with Complementarity Constraints (MPCC). For the latter, we prove tight constraint qualification conditions (MPCC-MFCQ and partial MPCC-LICQ) and derive Mordukovich (M-) and Strong (S-) stationarity conditions. The S-stationarity system for the MPCC turns also into S-stationarity conditions for the original formulation. Second-order sufficient optimality conditions are derived as well. The proposed reformulation may be extended to problems in function spaces, leading to MPCC's with additional constraints on the gradient of the state. Finally, we report on some numerical results obtained by using the proposed MPCC reformulations together with available large-scale nonlinear programming solvers.