This paper develops an online algorithm to solve a time-varying optimization problem with an objective that comprises a known time-varying cost and an unknown function. This problem structure arises in a number of engineering systems and cyber–physical systems where the known function captures time-varying engineering costs, and the unknown function models user’s satisfaction; in this context, the objective is to strike a balance between given performance metrics and user’s satisfaction. Key challenges related to the problem at hand are related to (1) the time variability of the problem, and (2) the fact that learning of the user’s utility function is performed concurrently with the execution of the online algorithm. This paper leverages Gaussian processes (GP) to learn the unknown cost function from noisy functional evaluation and build pertinent upper confidence bounds. Using the GP formalism, the paper then advocates time-varying optimization tools to design an online algorithm that exhibits tracking of the oracle-based optimal trajectory within an error ball, while learning the user’s satisfaction function with no-regret. The algorithmic steps are inexact, to account for possible limited computational budgets or real-time implementation considerations. Numerical examples are illustrated based on a problem related to vehicle control.

本文开发了一种在线算法来解决时变优化问题，其目标包括已知的时变成本和未知函数。这种问题结构出现在许多工程系统和网络物理系统中，其中已知功能捕获随时间变化的工程成本，而未知功能模拟用户的满意度；在这种情况下，目标是在给定的性能指标和用户满意度之间取得平衡。与手头问题相关的关键挑战与 (1) 问题的时间可变性有关，以及 (2) 用户效用函数的学习与在线算法的执行同时进行。本文利用高斯过程 (GP) 从嘈杂的功能评估中学习未知的成本函数，并建立相关的置信上限。然后，使用 GP 形式主义，论文提倡时变优化工具来设计一种在线算法，该算法展示了在错误球内跟踪基于预言机的最佳轨迹，同时无悔地学习用户的满意度函数。算法步骤不准确，以考虑可能有限的计算预算或实时实现考虑。基于与车辆控制相关的问题来说明数值示例。同时无悔地学习用户的满意度函数。算法步骤不准确，以考虑可能有限的计算预算或实时实现考虑。基于与车辆控制相关的问题来说明数值示例。同时无悔地学习用户的满意度函数。算法步骤不准确，以考虑可能有限的计算预算或实时实现考虑。基于与车辆控制相关的问题来说明数值示例。