joint work with Thorsten Raasch
We consider the numerical minimization of Tikhonov-type functionals with discrete total generalized variation (TGV) penalties. The minimization of the Tikhonov functional can be interpreted as a bilevel optimization problem. The upper level problem is unconstrained, and the lower level problem is a mixed l1-TV separation problem. By exploiting the structure of the lower level problem, we can reformulate the first-order optimality conditions as an equivalent piecewise smooth system of equations which can then be treated by semismooth Newton methods.
joint work with Maria do Rosário de Pinho, Mostafa Shamsi
A common approach to determine efficient solutions of a Multi-Objective Optimization Problems is to reformulate it as a parameter dependent Single-Objective Optimization: this is the scalarization method. Here, we consider the well-known Pascoletti-Serafini Scalarization approach to solve the nonconvex Multi-Objective Optimal Control Problems (MOOCPs). We show that by restricting the parameter sets of the Pascoletti-Serafini Scalarization method, we overcome some of the difficulties. We illustrate the efficiency of the method for two MOOCPs comprising both two and three different costs.
joint work with Matthias Gerdts
We discuss the linear-quadratic optimal control of continuous-time dynamical systems via continuous and discrete-valued control variables, also considering state-independent switching costs. This is interpreted as a bilevel problem, involving switching times optimization, for a given mode sequence, and continuous optimal control. Switching costs are formulated in terms of cardinality, via the L0 ''norm''. An efficient routine for the simplex-constrained L0 proximal operator is devised and adopted to solve some numerical examples through an accelerated proximal gradient method.