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

TL;DR

The paper addresses model-free constrained optimization where the objective and constraint functions are unknown and only zeroth-order evaluations are available. It introduces continuous-time Projected Zeroth-Order (P-ZO) dynamics that leverage projection maps (Lipschitz or discontinuous) and high-frequency dithering to learn gradients and steer a plant toward the solution set while guaranteeing feasibility (hard constraints) and safety (via shrinking/projection). Through averaging and singular perturbation analysis, the authors prove practical convergence to neighborhoods of the optimum, robustness to small disturbances, and tracking under time-varying and switching objectives, extending to primal-dual and Krasovskii-regularized non-smooth projections. The framework provides a principled approach for safe, model-free optimization in safety-critical domains such as power systems and robotics, with complete proofs and numerical examples illustrating the guarantees. By combining ES concepts with projection-based safety and zeroth-order estimation, the work delivers a versatile toolkit for real-time, model-free optimization under hard and soft constraints.

Abstract

This paper introduces a class of model-free feedback methods for solving generic constrained optimization problems where the specific mathematical forms of the objective 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 model-free dynamics that make use of 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 systematically 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 asymptotical 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.

Figures (6)

• Figure 1: Block diagram of P-GZO dynamics.
• Figure 2: Trajectory $\hat{\mathbf{x}}$ of the P-GZO algorithm on a regionally convex landscape. The safe region $\mathcal{X}$ is delimited by the red dashed line. All trajectories remain in $\mathcal{X}$ and converge to a neighborhood of $\mathbf{x}^*$.
• Figure 3: Trajectories of P-GZO dynamics using a shrunk feasible set $\mathcal{X}_{\varepsilon_a}$, satisfying $\mathcal{X}_{\varepsilon_a}+\varepsilon_a\mathbb{B}\subset\mathcal{X}$. Left: The trajectories generated by the algorithm track the minimizer of $f$ inside the feasible set $\mathcal{X}$. Right: Evolution in time of the trajectories $x$. The optimal trajectories are shown with dotted lines.
• Figure 4: Scheme of P-GZO dynamics with switching objectives.
• Figure 5: Block diagram of the DP-GZO algorithm.

Theorems & Definitions (25)

• Remark 1
• Remark 2
• Lemma 1
• Theorem 1
• Remark 3
• Theorem 2
• Example 1
• Remark 4
• Theorem 3
• Lemma 2