Optimization methods for mathematical programs with equilibrium constraints in function spaces based on adaptive error control and reduced order or low rank tensor approximations

基于自适应误差控制和降阶或低秩张量近似的函数空间中具有平衡约束的数学程序的优化方法

• 批准号: 314151277
• 负责人: Professor Dr. Michael Ulbrich
• 金额: --
• 依托单位: Arbeitsgruppe Nichtlineare Optimierung
• 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 2019-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/314151277?language=en
• 关键词: Optimization; methods; mathematical; programs; equilibrium

This project investigates optimization methods for mathematical programs with equilibrium constraints (MPECs) in function space that adaptively control the accuracy of the underlying discretization and of inexact subproblem solves in such a way that convergence is ensured. This enables the use of adaptive discretizations, reduced order models, and low rank tensor methods, thus making the solution of MPECs with high dimensional equilibrium constraints tractable and efficient. Two prototype classes of MPECs in function space are considered in the project: One with a family of parametric variational inequalities as constraints and the other constrained by a parabolic variational inequality. Based on a rigorous analytical foundation in function space, the project will develop and analyze inexact bundle methods combined with an implicit programming approach. In addition, inexact all-at-once methods will be considered. In both cases, the evaluation of cost function, constraints, and derivatives is carried out on discretizations which are adaptively refined during optimization and can further be approximated by reduced order models or low rank tensor methods. We will develop implementable control mechanisms for the inexactness, which are tailored to the needs of the optimization methods and can be based on a posteriori error estimators. The algorithms will be implemented and tested for the considered prototype classes of MPECs.

本项目研究了函数空间中具有平衡约束（mpec）的数学程序的优化方法，该方法自适应地控制底层离散化和不精确子问题解的精度，从而确保收敛。这使得自适应离散化、降阶模型和低秩张量方法的使用成为可能，从而使具有高维平衡约束的mpec的求解变得易于处理和高效。本文研究了函数空间中两类mpec的原型：一类以参数变分不等式为约束，另一类以抛物线型变分不等式为约束。基于功能空间的严格分析基础，该项目将开发和分析与隐式编程方法相结合的不精确束方法。此外，还将考虑不精确的一次性方法。在这两种情况下，成本函数、约束和导数的评估都是在离散化的基础上进行的，这些离散化在优化过程中自适应地改进，并可以通过降阶模型或低秩张量方法进一步逼近。我们将开发可实现的不精确性控制机制，根据优化方法的需要量身定制，并可以基于后验误差估计器。这些算法将在mpec的原型类中实现和测试。