Coordination Funds

协调基金

• 批准号: 314302824
• 负责人: Professor Dr. Michael Hintermüller
• 金额: --
• 依托单位: Forschungsgruppe Nichtglatte Variationsprobleme und Operatorgleichungen
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2016
• 资助国家: 德国
• 起止时间: 2015-12-31 至 2023-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/314302824?language=en
• 关键词: Coordination; Funds

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.

具有非光滑结构的分布参数系统是理论以及工业、医学或经济应用中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，或在机器人和生物力学的优化系统设计中，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到基于模型的多层次优化的转变是至关重要的。所有这些应用涉及偏微分方程或（拟）变分不等式（(Q)VIs）导致分布参数系统。此外，在某些情况下，原对偶公式会产生非光滑的互补系统。然而，其他应用程序需要用另一种优化作为约束来解决优化问题。这种层次优化问题非常复杂，通常是非光滑和非凸的，它们是具有平衡约束（mpec）的更全面的数学规划类的特殊实例。后者还包括VIs的一般最优控制问题或具有互补（问题）约束的优化问题（mpcc）。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组件，它可能(i)直接产生，通过涉及非光滑函数或算子的变分公式，（ii）通过不等式约束，非线性互补或切换系统，或（iii）通过竞争和层次。因此，本SPP的目的是将非线性互补或拟变分不等式问题的非光滑（数值）分析以及分层优化与鲁棒解算法的发展结合起来。具体来说，设想的SPP的目标主题是分析，数值解决和大规模和无限维问题的应用，其中非光滑和/或切换发生在(a)控制优化问题的系统，(b)双或多层平衡问题的较低水平问题，(c)平衡问题的耦合系统（特别是（广义）纳什博弈），(d)需要鲁棒解的系统，(e)拟变分不等式。SPP的研究是由原型应用程序控制的，因此该领域的最新活动将被合并和进一步探索，新的分析和算法范例将在现实世界的应用环境中开发、实现和验证。