论文标题
非线性编程和时变优化的定量灵敏度界限
Quantitative Sensitivity Bounds for Nonlinear Programming and Time-varying Optimization
论文作者
论文摘要
受非线性优化的经典灵敏度结果的启发，我们得出并讨论了新的定量界限，以表征解决方案映射和参数化非线性程序的双重变量。特别是，我们分别为非凸或凸优化问题的解决方案图的局部和全局Lipschitz常数提供了明确的表达式。我们的结果旨在研究随时间变化的优化问题，这些问题在在线优化的各种应用中都很普遍，包括电力系统，机器人技术，信号处理等。在这种情况下，我们的结果可用于约束优化器的变化速率。为了说明我们的灵敏度界限的使用，我们概括了现有参数，以量化连续时间，单调运行算法的跟踪性能。此外，我们引入了一种新的连续运行算法，以进行时变的约束优化，我们将其建模为所谓的扰动扫描过程。对于这种不连续的方案，我们建立了一个明确的绑定在一类凸问题的渐近解决方案跟踪上。
Inspired by classical sensitivity results for nonlinear optimization, we derive and discuss new quantitative bounds to characterize the solution map and dual variables of a parametrized nonlinear program. In particular, we derive explicit expressions for the local and global Lipschitz constants of the solution map of non-convex or convex optimization problems, respectively. Our results are geared towards the study of time-varying optimization problems which are commonplace in various applications of online optimization, including power systems, robotics, signal processing and more. In this context, our results can be used to bound the rate of change of the optimizer. To illustrate the use of our sensitivity bounds we generalize existing arguments to quantify the tracking performance of continuous-time, monotone running algorithms. Further, we introduce a new continuous-time running algorithm for time-varying constrained optimization which we model as a so-called perturbed sweeping process. For this discontinuous scheme, we establish an explicit bound on the asymptotic solution tracking for a class of convex problems.
