Coordination Funds

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Distributed parameter systems with non-smooth structures are among the most challenging problems in theory as well as in industrial, medical or economics applications. In particular, in elasto-plastic contact, thermoplasticity or phase separation problems, or in optimal system design in robotics and biomechanics, the transition from regularization or smoothing-based analytical and numerical treatments to genuinely non-smooth formulations and the shift from model-based numerical simulation to model-based multilevel optimization for problems in infinite dimensions is of utmost importance. All of these applications involve partial differential equations or (quasi)variational inequalities ((Q)VIs) leading to distributed parameter systems. Furthermore, in some cases primal-dual formulations result in complementarity systems, which are non-smooth. Yet other applications require to solve an optimization problem with another optimization as a constraint. Such hierarchical optimization problems are highly complex, and typically non-smooth and non-convex, and they are special instances of the more comprehensive class of mathematical programs with equilibrium constraints (MPECs). The latter also includes general optimal control problems for VIs or optimization problems with complementarity (problem) constraints (MPCCs).The non-smoothness of the objects mentioned above refers to typically continuous, but not necessarily differentiable system components, which may arise (i) directly, through variational formulations involving non-smooth functions or operators, (ii) through inequality constraints, nonlinear complementarity or switching systems, or(iii) through competition and hierarchy.The aim of this SPP is thus to combine non-smooth (numerical) analysis of non-linear complementarity or quasi-variational inequality problems as well as of hierarchical optimization with the development of robust solution algorithms. Specifically, the targeted topics of the envisaged SPP are the analysis, numerical solution, and applications of large-scale and infinite-dimensional problems where non-smoothness and/or switching occurs in(a) systems governing an optimization problem,(b) lower level problems of bi- or multilevel equilibrium problems,(c) coupled systems of equilibrium problems (in particular (generalized) Nash games),(d) systems that require robust solutions,(e) quasi-variational inequalities.The research of the SPP is governed by prototypical applications so that the most recent activities in the field will be merged and further explored, new analytic and algorithmic paradigms will be developed, implemented and validated in the context of real-world applications.