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

本项目研究了函数空间中具有平衡约束（MPECs）的数学规划的优化方法，该方法自适应地控制底层离散化和不精确子问题解决的精度，以确保收敛。这使得使用自适应离散化，降阶模型，和低秩张量方法，从而使解决方案的MPEC与高维平衡约束的易处理和有效的。在该项目中考虑了函数空间中的两类原型MPEC：一类是以一族参数变分不等式作为约束，另一类是以抛物变分不等式作为约束。基于函数空间中严格的分析基础，该项目将开发和分析结合隐式编程方法的不精确束方法。此外，将考虑不精确的一次性方法。在这两种情况下，成本函数，约束和导数的评估是在优化过程中自适应细化的离散化，可以进一步近似降阶模型或低秩张量方法。我们将开发可实施的控制机制的不准确性，这是量身定制的优化方法的需要，可以基于后验误差估计。该算法将被实施和测试所考虑的原型类的MPEC。

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.