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Titel der Dissertation: „Lagrange Multiplier Methods for Constrained Optimization and Variational Problems in Banach Spaces“

Abstract: One considers a class of variational systems the constraints of which depend both on a parameter as

well as on the decision variable itself. In this way one can model, e.g., quasi-variational inequalities

or implicit complementarity problems. On the basis of the recently developed directional limiting calculus

a new sufficient condition for the Aubin/Lipschitz like property of the respective solution map will be

derived. At the same time this approach leads to a workable formula for the graphical derivative of this

solution map which may be used in various sensitivity issues. The results can be applied in post-optimal

analysis or in some problems with equilibrium constraints and will be illustrated by an academic example.

Abstract: Complementarity constraints on control functions can be used to model switching requirements or result from transferring a bilevel optimal control problem with lower level control constraints into a single-level optimization problem using lower level optimality conditions.

In this talk, several different features of optimal control problems with complementarity constraints are addressed. First, it will be demonstrated that the existence of optimal solutions is closely related to the underlying function space setting. Due to the complementarity structure, constraint qualifications of reasonable strength fail to hold at all feasible points of the optimal control problem. However, it will be shown that necessary optimality conditions of strong stationarity-type are valid at locally optimal solutions. Finally, a penalty method based on smoothed NCP-functions is presented which can be used to compute globally optimal solutions of optimal control problems with control complementarity constraints. Some convergence results are discussed as well.

This talk is based on joint ongoing work with Christian Clason.

Abstract: Multicriteria optimization problems occur in many real life applications, for example in cancer radiotherapy treatment and in particular in intensity modulated radiation therapy (IMRT). In this talk we focus on optimization problems with multiple objectives that are ranked according to their importance. We solve these problems numerically by combining lexicographic optimization with our recently proposed level set scheme, which yields a sequence of auxiliary convex feasibility problems; solved here via projection methods. The projection enables us to combine the newly introduced superiorization methodology with multicriteria optimization methods to speed up computation while guaranteeing convergence of the optimization.

Nonsmooth optimization is a highly active field of research in the subject of applied and numerical mathematics. It requires sound knowledge of convex and nonsmooth analysis for the derivation and convergence analysis of modern methods to solve difficult and often nondifferentiable optimization problems. With Amir Beck (Israel), Christian Clason, Anton Schiela, Alexandra Schwartz (Germany) and Tuomo Valkonen (England) we were able to acquire five internationally renowned researchers as speakers. The lectures cover the range from theoretical foundations to the derivation and convergence analysis of optimization methods as well as their numerical realization and application. First, these topics will be explored for finite-dimensional optimization problems. Afterwards, the appropriate ideas and techniques will be transferred to infinite-dimensional problems.

Im Birkhäuser-Verlag ist das neue Buch von Christian Kanzow und Alexandra Schwartz erschienen: Spieltheorie. Theorie und Verfahren zur Lösung von Nash- und verallgemeinerten Nash-Gleichgewichtsproblemen