Constrained and Sparse Switching Times Optimization via Augmented Lagrangian Proximal Methods

In this paper we reformulate a switching times optimization problem with non-uniform switching costs and dwell-time constraints via direct multiple shooting, sparsity-inducing regularization and semi-continuous variables. The transformed problem has composite smooth/nonsmooth objective function and smooth constraints. Necessary optimality conditions for such problems are derived, resembling results from smooth optimization. A safeguarded, primal-dual, augmented Lagrangian proximal method is proposed for its numerical solution, and the global convergence toward points satisfying the necessary conditions is detailed. Finally, numerical results demonstrate the efficacy and limitations of the method.