Cautious optimization via data informativity

This paper deals with the problem of accurately determining guaranteed suboptimal values of an unknown cost function on the basis of noisy measurements. We consider a set-valued variant to regression where, instead of finding a best estimate of the cost function, we reason over all functions compatible with the measurements and apply robust methods explicitly in terms of the data. Our treatment provides data-based conditions under which closed-forms expressions of upper bounds of the unknown function can be obtained, and regularity properties like convexity and Lipschitzness can be established. These results allow us to provide tests for point- and set-wise verification of suboptimality, and tackle the cautious optimization of the unknown function in both one-shot and online scenarios. We showcase the versatility of the proposed methods in two control-relevant problems: data-driven contraction analysis of unknown nonlinear systems and suboptimal regulation with unknown dynamics and cost. Simulations illustrate our results.