Gray-Box Nonlinear Feedback Optimization

TL;DR

This paper introduces gray-box feedback optimization controllers that combine model-based and model-free methods, improving sample efficiency and robustness in solving nonconvex, constrained, and time-varying optimization problems.

Contribution

It proposes a systematic approach to incorporate approximate sensitivities into model-free updates, with theoretical guarantees and adaptability for dynamic problems.

Findings

The gray-box approach balances sensitivity accuracy and sample efficiency.

Performance depends on iteration count, problem size, and sensitivity errors.

The method extends to constrained and time-varying optimization scenarios.

Abstract

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 of the system to construct gradients, whereas model-free approaches bypass this need by estimating gradients from real-time evaluations of the objective. These approaches own complementary benefits in sample efficiency and accuracy against model mismatch, i.e., errors of sensitivities. To achieve the best of both worlds, we propose gray-box feedback optimization controllers, featuring systematic incorporation of approximate sensitivities into model-free updates via adaptive convex combination. We quantify conditions on the accuracy of the sensitivities that render the gray-box approach preferable. We elucidate how the closed-loop performance is determined by the number of iterations, the problem dimension, and the cumulative effect of inaccurate sensitivities. The proposed controller contributes to a balanced closed-loop behavior, which retains provable sample efficiency and optimality guarantees for nonconvex problems. We further develop a running gray-box controller to handle constrained time-varying problems with changing objectives and steady-state maps.