Optimality conditions for bilevel imaging learning problems with total variation regularization. (English) Zbl 1504.94012
Summary: We address the problem of optimal scale-dependent parameter learning in total variation image denoising. Such problems are formulated as bilevel optimization instances with total variation denoising problems as lower-level constraints. For the bilevel problem, we are able to derive M-stationarity conditions, after characterizing the corresponding Mordukhovich generalized normal cone and verifying suitable constraint qualification conditions. We also derive B-stationarity conditions, after investigating the Lipschitz continuity and directional differentiability of the lower-level solution operator. A characterization of the Bouligand subdifferential of the solution mapping, by means of a properly defined linear system, is provided as well. Based on this characterization, we propose a two-phase nonsmooth trust-region algorithm for the numerical solution of the bilevel problem and test it computationally for two particular experimental settings.
MSC:
| 94A08 | Image processing (compression, reconstruction, etc.) in information and communication theory |
| 90C33 | Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming) |
| 68U10 | Computing methodologies for image processing |
| 68T05 | Learning and adaptive systems in artificial intelligence |
| 65K10 | Numerical optimization and variational techniques |
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