Trading-off safety, exploration, and exploitation in learning-based optimization: a Set Membership approach

We propose a technique for global optimization considering black-box cost function and constraints, which have to be learned from data during the optimization process, arising for example in plant-control co-design of complex systems or controller tuning based on experiments. Assuming Lipschitz continuity of the cost function and constraints, we build a surrogate model and derive tight bounds on such functions based on a Set Membership framework. An exploitation step is designed to improve on the current best feasible candidate solution, searching in regions where all constraints are estimated as fulfilled, thus preserving safety. On the other hand, an exploration routine aims to discover the shape of the cost and constraint functions by picking points with large uncertainty, prioritizing regions where more constraints are predictably satisfied. The proposed algorithm can intuitively trade-off safety, exploration, and exploitation. The performance is evaluated on the problem of model predictive control tuning for a servomechanism with plant uncertainties and task-level constraints.