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Zeroth-Order Constrained Optimization from a Control Perspective via Feedback Linearization

Runyu Zhang, Gioele Zardini, Asuman Ozdaglar, Jeff Shamma, Na Li

TL;DR

This work tackles safe derivative-free constrained optimization with unknown constraints by introducing zeroth-order feedback-linearization (ZOFL), which leverages two-point zeroth-order estimators to form Jacobian–vector products and enforces feasibility via a linearized constraint dynamics. By modeling the optimization as a control system and applying a feedback-linearization design, ZOFL achieves high-probability constraint satisfaction with exponential contraction plus controllable residuals that scale with the zeroth-order radii and step size. The authors provide both equality- and inequality-constrained extensions, a midpoint discretization variant to reduce discretization bias, and extensive numerical validations showing improved feasibility relative to baselines while maintaining competitive objective values. The results offer a principled, theory-backed route for safe, derivative-free constrained optimization in nonconvex settings, with practical implications for safe learning and control under black-box constraints.

Abstract

Safe derivative-free optimization under unknown constraints is a fundamental challenge in modern learning and control. Existing zeroth-order (ZO) methods typically still assume access to a first-order oracle of the constraint functions or restrict attention to convex settings, leaving nonconvex optimization with black-box constraints largely unexplored. We propose the zeroth-order feedback-linearization (ZOFL) algorithm for ZO constrained optimization that enforces feasibility without access to the first-order oracle of the constraint functions and applies to both equality and inequality constraints. The proposed approach relies only on noisy, sample-based gradient estimates obtained via two-point estimators, yet provably guarantees constraint satisfaction under mild regularity conditions. It adopts a control-theoretic perspective on ZO constrained optimization and leverages feedback linearization, a nonlinear control technique, to enforce feasibility. Finite-time bounds on constraint violation and asymptotic global convergence guarantees are established for the ZOFL algorithm. A midpoint discretization variant is further developed to improve feasibility without sacrificing optimality. Empirical results demonstrate that ZOFL consistently outperforms standard ZO baselines, achieving competitive objective values while maintaining feasibility.

Zeroth-Order Constrained Optimization from a Control Perspective via Feedback Linearization

TL;DR

This work tackles safe derivative-free constrained optimization with unknown constraints by introducing zeroth-order feedback-linearization (ZOFL), which leverages two-point zeroth-order estimators to form Jacobian–vector products and enforces feasibility via a linearized constraint dynamics. By modeling the optimization as a control system and applying a feedback-linearization design, ZOFL achieves high-probability constraint satisfaction with exponential contraction plus controllable residuals that scale with the zeroth-order radii and step size. The authors provide both equality- and inequality-constrained extensions, a midpoint discretization variant to reduce discretization bias, and extensive numerical validations showing improved feasibility relative to baselines while maintaining competitive objective values. The results offer a principled, theory-backed route for safe, derivative-free constrained optimization in nonconvex settings, with practical implications for safe learning and control under black-box constraints.

Abstract

Safe derivative-free optimization under unknown constraints is a fundamental challenge in modern learning and control. Existing zeroth-order (ZO) methods typically still assume access to a first-order oracle of the constraint functions or restrict attention to convex settings, leaving nonconvex optimization with black-box constraints largely unexplored. We propose the zeroth-order feedback-linearization (ZOFL) algorithm for ZO constrained optimization that enforces feasibility without access to the first-order oracle of the constraint functions and applies to both equality and inequality constraints. The proposed approach relies only on noisy, sample-based gradient estimates obtained via two-point estimators, yet provably guarantees constraint satisfaction under mild regularity conditions. It adopts a control-theoretic perspective on ZO constrained optimization and leverages feedback linearization, a nonlinear control technique, to enforce feasibility. Finite-time bounds on constraint violation and asymptotic global convergence guarantees are established for the ZOFL algorithm. A midpoint discretization variant is further developed to improve feasibility without sacrificing optimality. Empirical results demonstrate that ZOFL consistently outperforms standard ZO baselines, achieving competitive objective values while maintaining feasibility.

Paper Structure

This paper contains 20 sections, 14 theorems, 108 equations, 3 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. First-order Constrained Optimization: A Control Perspective
  4. Zeroth-order Constrained Optimization: Baseline and Challenges
  5. Feedback-Linearization-Inspired Zeroth-order Algorithm
  6. Algorithm
  7. Theoretical Guarantees on Constraint Satisfaction
  8. Global Convergence
  9. Extension to Inequality-constrained Setting
  10. Exploring midpoint methods for zeroth-order optimization
  11. Numerical Validations
  12. Conclusions
  13. Other Related Works on Derivative Free Constrained Optimization
  14. Proof of Theorem \ref{['thm:zeroth-order-eq']}
  15. Bounding $\|x_{t+1} - x_t\|$
  16. ...and 5 more sections

Key Result

Theorem 1

Suppose Assumptions assump:boundedness--assump:h-norms hold and $K\succ 0$. Run alg:equality-zeroth-order with $u_i$'s n eq:two-point-estimator drawn i.i.d. from the unit sphere. Fix $\delta\in(0,1)$ and horizon $T_G\in\mathbb N$. If the batch size $T_B$ and probe radii $r_1,r_2$ satisfy (cf. Append and the stepsizes obey the stability condition $0<\eta_t\,\lambda_{\min}(K)<1$ for all $t$, then wi

Figures (3)

  • Figure 1: Control Perspective for constrained optimization.
  • Figure 2: Control Perspective for Zeroth-Order Constrained Optimization
  • Figure :

Theorems & Definitions (29)

  • Definition 1: Second and Third-order directional derivative norm
  • Theorem 1
  • Theorem 2
  • Theorem 1: Feasibility with Inequality Constraints
  • Proof 1: Proof of Theorem \ref{['thm:zeroth-order-eq']}
  • Lemma 1
  • Proof 2
  • Lemma 2
  • Proof 3
  • Proof 4: Proof of Theorem \ref{['thm:global-convergence']}
  • ...and 19 more