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Generalized Multi-Constraint Extremum Seeking

Alan Williams, Jorge Cortés, Alexander Scheinker

TL;DR

The paper addresses constrained optimization where both the objective and the gradients of multiple unknown inequality and equality constraints are unavailable. It introduces Generalized Multi-Constraint Extremum Seeking (GMC-ES), which fuses ES-based gradient estimation with a real-time QP safety filter grounded in a penalty Lyapunov function $V_\epsilon$ and the set-valued Lie derivative to enforce feasibility. By leveraging MFCQ–QP, a unique KKT point $\theta^*$, averaging theory, and singular perturbation arguments, the authors prove semiglobal practical asymptotic stability (SPA) and practical constraint maintenance for a broad class of nonconvex problems, and they demonstrate the approach on a 2D example. The resulting method yields a real-time, constraint-enforcing optimizer capable of handling multiple unknown constraints, with significant potential impact for safety-critical autonomous and decision-making systems, where gradients are not readily available.

Abstract

We generalize the Safe Extremum Seeking algorithm to address the minimization of an unknown objective function subject to multiple unknown inequality and equality constraints, relying on recent results of gradient flow systems. These constraints may represent safety or other critical conditions. The proposed ES algorithm functions as a general nonlinear programming tool, offering practical maintenance of all constraints and semiglobal practical asymptotic stability, utilizing a Lyapunov argument on the penalty function and the set-valued Lie derivative. The efficacy of the algorithm is demonstrated on a 2D problem.

Generalized Multi-Constraint Extremum Seeking

TL;DR

The paper addresses constrained optimization where both the objective and the gradients of multiple unknown inequality and equality constraints are unavailable. It introduces Generalized Multi-Constraint Extremum Seeking (GMC-ES), which fuses ES-based gradient estimation with a real-time QP safety filter grounded in a penalty Lyapunov function and the set-valued Lie derivative to enforce feasibility. By leveraging MFCQ–QP, a unique KKT point , averaging theory, and singular perturbation arguments, the authors prove semiglobal practical asymptotic stability (SPA) and practical constraint maintenance for a broad class of nonconvex problems, and they demonstrate the approach on a 2D example. The resulting method yields a real-time, constraint-enforcing optimizer capable of handling multiple unknown constraints, with significant potential impact for safety-critical autonomous and decision-making systems, where gradients are not readily available.

Abstract

We generalize the Safe Extremum Seeking algorithm to address the minimization of an unknown objective function subject to multiple unknown inequality and equality constraints, relying on recent results of gradient flow systems. These constraints may represent safety or other critical conditions. The proposed ES algorithm functions as a general nonlinear programming tool, offering practical maintenance of all constraints and semiglobal practical asymptotic stability, utilizing a Lyapunov argument on the penalty function and the set-valued Lie derivative. The efficacy of the algorithm is demonstrated on a 2D problem.

Paper Structure

This paper contains 10 sections, 10 theorems, 39 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries on Nonsmooth Analysis
  3. Stability of the safe gradient flow
  4. Lipschitzness, Equilibrium, and Invariance
  5. The Set-Valued Lie Derivative and Stability
  6. Extremum Seeking
  7. Design
  8. Analysis
  9. 2D Example
  10. Conclusion

Key Result

Proposition 1

Let $g_k: \mathbb{R}^n \rightarrow \mathbb{R}$ be differentiable functions for $k = 1,2, \ldots,m$, and let $V: \mathbb{R}^n \rightarrow \mathbb{R}$ be defined as the maximum of the collection of functions Let $I_V(x)$ denote the set of indices $k$ for which $V(x) = g_k(x)$ --- the "active" functions. Then:

Figures (1)

  • Figure 1: Trajectories $\hat{\theta}(t)$ plotted from various initial conditions along with the zero level sets of $g_1, g_2$ and level sets of $V_\epsilon$ with $\epsilon = 0.1$. The inequality set $g(\theta) \leq 0$ is shaded in green, and the point $\theta^*$ is the blue star.

Theorems & Definitions (17)

  • Proposition 1
  • Definition 1
  • Theorem 1: Invariance Principle
  • Theorem 2: Nagumo's Theorem
  • Proposition 2
  • proof
  • Proposition 3: MFCQ implies bounded multipliers
  • proof
  • Proposition 4: Lipschitzness of safe gradient flow
  • proof
  • ...and 7 more