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Safe Zeroth-Order Optimization Using Quadratic Local Approximations

Baiwei Guo, Yuning Jiang, Giancarlo Ferrari-Trecate, Maryam Kamgarpour

TL;DR

This work tackles safe optimization for black-box constrained problems by introducing Safe Zeroth-Order Sequential QCQP (SZO-QQ), a method that builds local quadratic proxies of constraint functions around strictly feasible points and optimizes over the resulting local feasible sets via QCQP subproblems. The approach guarantees sample feasibility and, under mild assumptions, that accumulation points are KKT pairs; it further provides explicit termination criteria that yield an $\eta$-KKT pair in $O\left(\frac{d}{\eta^{2}}\right)$ iterations with $O\left(\frac{d^{2}}{\eta^{2}}\right)$ samples. Empirical results in unknown nonconvex QCQPs, open-loop control with unmodeled disturbances, and OPTF demonstrate competitive convergence, strong safety guarantees, and performance close to model-based baselines. The method’s safety, modular subproblem structure, and dependence on minimal gradient information make it attractive for real-world, hardware-in-the-loop optimization in domains like power systems and autonomous control.

Abstract

This paper addresses black-box smooth optimization problems, where the objective and constraint functions are not explicitly known but can be queried. The main goal of this work is to generate a sequence of feasible points converging towards a KKT primal-dual pair. Assuming to have prior knowledge on the smoothness of the unknown objective and constraints, we propose a novel zeroth-order method that iteratively computes quadratic approximations of the constraint functions, constructs local feasible sets and optimizes over them. Under some mild assumptions, we prove that this method returns an $η$-KKT pair (a property reflecting how close a primal-dual pair is to the exact KKT condition) within $O({1}/{η^{2}})$ iterations. Moreover, we numerically show that our method can achieve faster convergence compared with some state-of-the-art zeroth-order approaches. The effectiveness of the proposed approach is also illustrated by applying it to nonconvex optimization problems in optimal control and power system operation.

Safe Zeroth-Order Optimization Using Quadratic Local Approximations

TL;DR

This work tackles safe optimization for black-box constrained problems by introducing Safe Zeroth-Order Sequential QCQP (SZO-QQ), a method that builds local quadratic proxies of constraint functions around strictly feasible points and optimizes over the resulting local feasible sets via QCQP subproblems. The approach guarantees sample feasibility and, under mild assumptions, that accumulation points are KKT pairs; it further provides explicit termination criteria that yield an -KKT pair in iterations with samples. Empirical results in unknown nonconvex QCQPs, open-loop control with unmodeled disturbances, and OPTF demonstrate competitive convergence, strong safety guarantees, and performance close to model-based baselines. The method’s safety, modular subproblem structure, and dependence on minimal gradient information make it attractive for real-world, hardware-in-the-loop optimization in domains like power systems and autonomous control.

Abstract

This paper addresses black-box smooth optimization problems, where the objective and constraint functions are not explicitly known but can be queried. The main goal of this work is to generate a sequence of feasible points converging towards a KKT primal-dual pair. Assuming to have prior knowledge on the smoothness of the unknown objective and constraints, we propose a novel zeroth-order method that iteratively computes quadratic approximations of the constraint functions, constructs local feasible sets and optimizes over them. Under some mild assumptions, we prove that this method returns an -KKT pair (a property reflecting how close a primal-dual pair is to the exact KKT condition) within iterations. Moreover, we numerically show that our method can achieve faster convergence compared with some state-of-the-art zeroth-order approaches. The effectiveness of the proposed approach is also illustrated by applying it to nonconvex optimization problems in optimal control and power system operation.
Paper Structure (29 sections, 13 theorems, 58 equations, 6 figures, 1 algorithm)

This paper contains 29 sections, 13 theorems, 58 equations, 6 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Safe Zeroth-Order Algorithm
  4. Local feasible set construction
  5. The proposed algorithm
  6. Properties of SZO-QQ's iterates and output
  7. On the accumulation points of $\{x_k\}_{k\geq 1}$
  8. The output of Algorithm \ref{['alg:sslo']} is an $\eta$-KKT pair
  9. Complexity analysis
  10. Numerical Results
  11. Solving an unknown non-convex QCQP
  12. Selection of $\xi$ for deriving a $10^{-2}$-KKT pair
  13. Performance comparison with other methods
  14. Impact of the parameters $L_i$ and $M_i$
  15. Open-loop optimal control with unmodelled disturbance
  16. ...and 14 more sections

Key Result

Lemma 1

Under Assumption ass: smoothness, we have

Figures (6)

  • Figure 1: Objective value as a function of computation time.
  • Figure 2: Objective value as a function of the number of samples, where the curve for LB-SGD overlaps with that for Extremum Seeking.
  • Figure 3: Objective value as a function of computation time:$\text{ }$$L_0=5$ fixed and $M_0$ varied
  • Figure 4: Objective value as a function of computation time: $\text{ }$$M_0=3$ fixed and $L_0$ varied
  • Figure 5: Objective value as a function of the number of samples
  • ...and 1 more figures

Theorems & Definitions (21)

  • Definition 1
  • Remark 1
  • Remark 2
  • Lemma 1: berahas2022theoretical, Theorem 3.2
  • Theorem 1
  • Remark 3
  • Remark 4
  • Proposition 1
  • Proposition 2
  • Theorem 2
  • ...and 11 more