Continuous-Time Zeroth-Order Dynamics with Projection Maps: Model-Free Feedback Optimization with Safety Guarantees

This paper introduces a class of model-free feedback methods for solving generic constrained optimization problems where the mathematical forms of the cost and constraint functions are not available. The proposed methods, termed Projected Zeroth-Order (P-ZO) dynamics, incorporate projection maps into a class of continuous-time zeroth-order dynamics that use direct measurements of the cost function and periodic dithering for the purpose of gradient learning. In particular, the proposed P-ZO algorithms can be interpreted as new extremum-seeking algorithms that autonomously drive an unknown system toward a neighborhood of the set of solutions of an optimization problem using only output feedback, while simultaneously guaranteeing that the input trajectories remain in a feasible set for all times. In this way, the P-ZO algorithms can properly handle hard and asymptotic constraints in model-free optimization problems without using penalty terms or barrier functions. Moreover, the proposed dynamics have suitable robustness properties with respect to small bounded additive disturbances on the states and dynamics, a property that is fundamental for practical real-world implementations. Additional tracking results for time-varying and switching cost functions are also derived under stronger convexity and smoothness assumptions and using tools from hybrid dynamical systems. Numerical examples are presented throughout the paper to illustrate the above results.