Bilevel Imaging Learning Problems as Mathematical Programs with Complementarity Constraints: Reformulation and Theory

TL;DR

This paper reformulates bilevel imaging learning problems with nonsmooth regularizers as Mathematical Programs with Complementarity Constraints (MPCC), providing theoretical analysis and enabling efficient large-scale solver application.

Contribution

It introduces a novel MPCC reformulation for bilevel imaging problems with nonsmooth regularizers, along with constraint qualifications and stationarity conditions.

Findings

Established tight constraint qualification conditions for MPCC

Derived Mordukhovich and Strong stationarity conditions

Enabled application of large-scale nonlinear programming solvers

Abstract

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 sparsity-based regularizers. By using geometric properties of the primal-dual reformulation of the lower-level problem and introducing suitable auxiliar 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-RCPLD and partial MPCC-LICQ) and derive Mordukhovich (M-) and Strong (S-) stationarity conditions. The stationarity systems for the MPCC turn also into stationarity conditions for the original formulation. Second-order sufficient optimality conditions are derived as well, together with a local uniqueness result for stationary points. The proposed reformulation may be extended to problems in function spaces, leading to MPCC's with constraints on the gradient of the state. The MPCC reformulation also leads to the efficient use of available large-scale nonlinear programming solvers, as shown in a companion paper, where different imaging applications are studied.