Coordination Funds

具有非光滑结构的分布参数系统在理论上以及在工业、医学或经济应用中都是最具挑战性的问题之一。特别是，在弹塑性接触、热塑性或相分离问题中，或者在机器人和生物力学中的优化系统设计中，从正则化或基于平滑的分析和数值处理过渡到真正的非光滑公式，以及从基于模型的数值模拟转向基于模型的多层优化对于无限维问题是至关重要的。所有这些应用都涉及到导致分布参数系统的偏微分方程或(准)变分不等式((Q)VIS)。此外，在某些情况下，原始-对偶公式导致互补系统，这是非光滑的。还有一些应用需要用另一个优化作为约束来解决优化问题。这种递阶优化问题非常复杂，而且通常是非光滑和非凸的，它们是更全面的具有平衡约束的数学规划(MPEC)的特殊实例。后者还包括VIS或具有互补(问题)约束的最优化问题(MPC)的一般最优控制问题。上述对象的非光滑性通常是指连续的，但不一定是可微的系统部件，它可能(I)通过涉及非光滑函数或算子的变分公式直接产生，(Ii)通过不等式约束、非线性互补或切换系统，或(Iii)通过竞争和分层。因此，该SPP的目的是将对非线性互补或准变分不等式问题的非光滑(数值)分析以及递阶优化与稳健求解算法的发展相结合。具体地说，设想的SPP的目标主题是大规模和无限维问题的分析、数值解和应用，其中在(A)控制优化问题的系统中发生非光滑和/或切换，(B)双层或多水平平衡问题的较低级别问题，(C)平衡问题的耦合系统(特别是(广义)纳什博弈)，(D)需要稳健解的系统，(E)拟变分不等式。SPP的研究由典型应用支配，从而该领域的最新活动将被合并和进一步探索，将在现实世界应用的背景下开发、实施和验证新的分析和算法范例。

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.