Contents

Motivated by variational models in continuum mechanics, we introduce a novel algorithm for performing nonsmooth and nonconvex minimizations with linear con-straints. We show how this algorithm is actually a natural generalization of well-known non-stationary augmented Lagrangian methods for convex optimization. The relevant features of this approach are its applicability to a large variety of nonsmooth and non-convex objective functions, its guaranteed global convergence to critical points of the objective energy, and its simplicity of implementation. In fact, the algorithm results in a nested double loop iteration, where in the inner loop an augmented Lagrangian algorithm performs an adaptive finite number of iterations on a fixed quadratic and strictly convex perturbation of the objective energy, while the external loop performs an adaptation of the quadratic perturbation. To show the versatility of this new algo-rithm, we exemplify how it can be easily used for computing critical points in inverse free-discontinuity variational models, such as the Mumford-Shah functional, and, by doing so, we also derive and analyze new iterative thresholding algorithms