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

这个项目研究函数空间中带平衡约束的数学规划的优化方法，这种优化方法自适应地控制底层离散化的精度和以确保收敛的方式求解不确定子问题。这使得可以使用自适应离散化、降阶模型和低阶张量方法，从而使得求解具有高维平衡约束的MPEC变得容易和有效。该项目考虑了函数空间中的两类MPEC：一类是以参数变分不等式为约束的，另一类是以抛物型变分不等式为约束的。基于函数空间的严格分析基础，该项目将结合隐式编程方法开发和分析不精确束方法。此外，还将考虑不精确的一次性方法。在这两种情况下，代价函数、约束和导数的计算都是在离散化的基础上进行的，这些离散化在优化过程中被自适应地细化，并且可以进一步用降阶模型或低阶张量方法来逼近。我们将开发可实现的不精确度控制机制，这些机制是根据优化方法的需要量身定做的，可以基于后验误差估计器。这些算法将针对所考虑的MPEC原型类进行实施和测试。

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.