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An Interior-Point Algorithm for Continuous Nonlinearly Constrained Optimization with Noisy Function and Derivative Evaluations

Frank E. Curtis, Shima Dezfulian, Andreas Waechter

TL;DR

This work develops an interior-point algorithm for continuous nonlinear inequality optimization when objective and constraint evaluations, including derivatives, are contaminated by deterministic noise. By solving a fixed-barrier subproblem with slack variables and a merit-function framework, the method balances objective progress and constraint satisfaction using a trust-region-based normal step and a tangential step under noisy derivatives. The authors provide convergence analysis showing that, under realistic bounded-noise and well-posedness assumptions, the noiseless barrier-subproblem stationarity measure can be driven to a threshold that scales with the noise, and they demonstrate practical effectiveness on a large suite of nonlinear inequality problems. The results indicate robustness to noise and suggest that adaptively reducing the barrier parameter may be beneficial in some cases, though excessive barrier shrinking offers limited gains when evaluations are noisy. Overall, the approach offers a principled, implementable pathway for reliable optimization in simulations where noise is inherent.

Abstract

An algorithm based on the interior-point methodology for solving continuous nonlinearly constrained optimization problems is proposed, analyzed, and tested. The distinguishing feature of the algorithm is that it presumes that only noisy values of the objective and constraint functions and their first-order derivatives are available. The algorithm is based on a combination of a previously proposed interior-point algorithm that allows inexact subproblem solutions and recently proposed algorithms for solving bound- and equality-constrained optimization problems with only noisy function and derivative values. It is shown that the new interior-point algorithm drives a stationarity measure below a threshold that depends on bounds on the noise in the function and derivative values. The results of numerical experiments show that the algorithm is effective across a wide range of problems.

An Interior-Point Algorithm for Continuous Nonlinearly Constrained Optimization with Noisy Function and Derivative Evaluations

TL;DR

This work develops an interior-point algorithm for continuous nonlinear inequality optimization when objective and constraint evaluations, including derivatives, are contaminated by deterministic noise. By solving a fixed-barrier subproblem with slack variables and a merit-function framework, the method balances objective progress and constraint satisfaction using a trust-region-based normal step and a tangential step under noisy derivatives. The authors provide convergence analysis showing that, under realistic bounded-noise and well-posedness assumptions, the noiseless barrier-subproblem stationarity measure can be driven to a threshold that scales with the noise, and they demonstrate practical effectiveness on a large suite of nonlinear inequality problems. The results indicate robustness to noise and suggest that adaptively reducing the barrier parameter may be beneficial in some cases, though excessive barrier shrinking offers limited gains when evaluations are noisy. Overall, the approach offers a principled, implementable pathway for reliable optimization in simulations where noise is inherent.

Abstract

An algorithm based on the interior-point methodology for solving continuous nonlinearly constrained optimization problems is proposed, analyzed, and tested. The distinguishing feature of the algorithm is that it presumes that only noisy values of the objective and constraint functions and their first-order derivatives are available. The algorithm is based on a combination of a previously proposed interior-point algorithm that allows inexact subproblem solutions and recently proposed algorithms for solving bound- and equality-constrained optimization problems with only noisy function and derivative values. It is shown that the new interior-point algorithm drives a stationarity measure below a threshold that depends on bounds on the noise in the function and derivative values. The results of numerical experiments show that the algorithm is effective across a wide range of problems.

Paper Structure

This paper contains 13 sections, 26 theorems, 128 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contributions
  3. Notation
  4. Outline
  5. Problem Description
  6. Algorithm Description, Fixed Barrier Parameter
  7. Analysis of Algorithm \ref{['alg.exact_alg']}
  8. Discussion of Aims of Convergence Analysis
  9. Assumptions and Well-Posedness
  10. Convergence Analysis
  11. Numerical Experiments
  12. Conclusion and Future Work
  13. Convergence of $\|{\bar{J}}_k^T{\bar{c}}_k\|^2$

Key Result

Lemma 4.1

Suppose that Assumptions assumption.err0 and assumption.bounded hold. Then, if Algorithm alg.exact_alg reaches iteration $k \in \mathbb{N}$ such that $s_k \geq 0$ and $c(z_k) \geq 0$, then it either terminates finitely or proceeds to iteration $k+1$ such that $s_{k+1} \geq 0$ and $c(z_{k+1}) \geq 0$

Figures (4)

  • Figure 1: $\mu=10^{-1}$, measured with noisy function and derivative values
  • Figure 2: $\mu=10^{-1}$, measured with noiseless function and derivative values
  • Figure 3: $\mu=10^{-4}$, measured with noisy function and derivative values
  • Figure 4: $\mu=10^{-4}$, measured with noiseless function and derivative values

Theorems & Definitions (52)

  • Remark 3.1
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • Lemma 4.4
  • proof
  • Lemma 4.5
  • proof
  • ...and 42 more