During the last decade many efforts have been made to translate the powerful proximal algorithms for convex optimization problems to the solving of nonconvex optimization problems. In this talk we will review such iterative schemes which posses provable convergence properties and are designed to solve nonsmooth nonconvex optimization problems with structures which range from simple to complex. We will discuss the pillars of the convergence analysis, like the boundedness of the generated iterates, techniques to prove that cluster points of this sequence are stationary points of the optimization problem under investigation, and elements of nonconvex subdifferential calculus. We will also emphasize the role played by the Kurdyka-Lojasiewicz property when proving global convergence, and when deriving convergence rates in terms of the Lojasiewicz exponent.
In this talk, we will give an overview over optimization problems under uncertainty as they appear in energy management, together with global solution approaches. One way of protecting against uncertainties that occur in real-world applications is to apply and to develop methodologies from robust optimization. The latter takes these uncertainties into account already in the mathematical model. The task then is to determine solutions that are feasible for all considered realizations of the uncertain parameters, and among them one with best guaranteed solution value. We will introduce a number of electricity and gas network optimization problems for which robust protection is appropriate. Already for simplified cases, the design of algorithmically tractable robust counterparts and global solution algorithms is challenging. As an example, the stationary case in gas network operations is complex due to non-convex dependencies in the physical state variables pressures and flows. In the robust version, two-stage robust problems with a non-convex second stage are to be solved, and new solution methodologies need to be developed. We will highlight robust optimization approaches and conclude by pointing out some challenges and generalizations in the field.