Parametric Methodology in Optimization

当问题所依赖的参数被摄动时，优化问题的解会发生什么的问题不仅在评估数学模型的稳定性中是重要的，而且在计算程序的设计和证明中也是重要的。无论是在最优控制中还是在随机规划中，参数相关性问题也是有效利用动态最优化中“待用成本”函数的关键。这个项目将推进处理这种相关性的方法，并探索其对计算的影响。变分分析中的最新工具是必不可少的，因为参数相关性具有内在的非光滑性。最优性条件的替代形式，特别是扰动稳健性，将在有限和无限维的广泛框架中进行研究。本文将从新的角度研究凸动力优化中的成本-去函数，包括确定性连续时间模型和随机离散时间模型，前者与哈密顿-雅可比理论有关，后者以数值势为主。

Questions about what happens to a solution to a problem of optimization when the parameters on which the problem depends are perturbed are important not only in assessing the stability of mathematical models, but also in the design and justification of computational procedures. Issues of parametric dependence are also the key to effective utilization of `cost-to-go' functions in dynamical optimization, whether in optimal control or stochastic programming. This project would advance the methodology for dealing with such dependence and explore its consequences for computation.The latest tools in variational analysis, which are essential becauseof inherent nonsmoothness in parametric dependence, would be applied.Alternative forms of optimality conditions, focused especially on perturbational robustness, would be investigated in a broad framework,both finite- and infinite-dimensional. Cost-to-go functions in convexdynamical optimization would be studied from the new perspective both in deterministic continuous-time models, with their connections to Hamilton-Jacobi theory, and in stochastic discrete-time models, wherethe numerical potential will dominate.