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-form 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 perform 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.