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Flexible sparsity regularization means stably approximating sparse solutionsof operator equations by using coefficient-dependent penalizations. We proposeand analyse a general nonconvex approach in this respect, from both theoreticaland numerical perspectives. Namely, we show convergence of the regularizationmethod and establish convergence properties of a couple of majorizationapproaches for the associated nonconvex problems. We also test a monotonealgorithm for an academic example where the operator is an $M$ matrix, and on atime-dependent optimal control problem, pointing out the advantages ofemploying variable penalties over a fixed penalty.

Nonconvex flexible sparsity regularization: theory and monotone numerical schemes