Gray-Box Nonlinear Feedback Optimization

Feedback optimization enables autonomous optimality seeking of a dynamical system through its closed-loop interconnection with iterative optimization algorithms. Among various iteration structures, model-based approaches require the input-output sensitivity matrix of the system to construct gradients, whereas model-free approaches eliminate this need by estimating gradients from real-time objective evaluations. These approaches offer complementary benefits in sample efficiency and accuracy against model mismatch, i.e., sensitivity errors. To achieve balanced closed-loop performance, we propose a gray-box feedback optimization controller, featuring systematic incorporation of approximate sensitivities into model-free updates via a tunable convex combination. We provide unified performance characterizations covering different approaches. We elucidate how cumulative sensitivity errors (model-based) and variances due to stochastic exploration (model-free) shape the closed-loop behavior and induce a trade-off between iteration and dimensional dependence. The proposed controller retains sample efficiency and provable (local) optimality for nonconvex problems despite inaccurate sensitivities. We further develop and characterize a running gray-box controller that handles constrained time-varying problems with changing objectives and steady-state input-output maps.