标题： 通过投影到全局最小值器来评估逆最优控制的基础函数集的质量 显示英文标题

标题： Assessing the Quality of a Set of Basis Functions for Inverse Optimal Control via Projection onto Global Minimizers

摘要： 逆优化（逆最优控制）是推断一个成本函数的任务，使得给定的测试点（轨迹）相对于发现的成本是（几乎）最优的。 逆优化中的先验方法假设真实成本是凸基函数集合的凸组合，并且该基与测试点是一致的。 然而，在许多应用中，数据生成的原则并不明确，因此一致性假设并不总是合理的。 这项工作提出使用测试点与由凸基函数的凸组合生成的全局最优集之间的距离，作为衡量基函数相对于测试点表达质量的指标。 较大的最小距离会否定基函数集。 引入了全局最优集的概念，并在无约束和有约束的情况下探索了其性质。 在凸二次情况下，通过双层梯度下降和增强的线性矩阵不等式分别实现了最小距离的上下界。 该框架的扩展包括最大可表示基函数、非凸基函数（局部极小值）以及应用多项式优化技术。 显示英文摘要

摘要： Inverse optimization (Inverse optimal control) is the task of imputing a cost function such that given test points (trajectories) are (nearly) optimal with respect to the discovered cost. Prior methods in inverse optimization assume that the true cost is a convex combination of a set of convex basis functions and that this basis is consistent with the test points. However, the consistency assumption is not always justified, as in many applications the principles by which the data is generated are not well understood. This work proposes using the distance between a test point and the set of global optima generated by the convex combinations of the convex basis functions as a measurement for the expressive quality of the basis with respect to the test point. A large minimal distance invalidates the set of basis functions. The concept of a set of global optima is introduced and its properties are explored in unconstrained and constrained settings. Upper and lower bounds for the minimum distance in the convex quadratic setting are implemented by bi-level gradient descent and an enriched linear matrix inequality respectively. Extensions to this framework include max-representable basis functions, nonconvex basis functions (local minima), and applying polynomial optimization techniques.