Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Feasible Method for Constrained Derivative-Free Optimization

Melody Qiming Xuan, Jorge Nocedal

TL;DR

A method for solving constrained optimization problems when the derivatives of the objective function are unavailable, while the derivatives of the constraints are known, which allows the objective and constraint function to be nonconvex.

Abstract

This paper explores a method for solving constrained optimization problems when the derivatives of the objective function are unavailable, while the derivatives of the constraints are known. We allow the objective and constraint function to be nonconvex. The method constructs a quadratic model of the objective function via interpolation and computes a step by minimizing this model subject to the original constraints in the problem and a trust region constraint. The step computation requires the solution of a general nonlinear program, which is economically feasible when the constraints and their derivatives are very inexpensive to compute compared to the objective function. The paper includes a summary of numerical results that highlight the method's promising potential.

A Feasible Method for Constrained Derivative-Free Optimization

TL;DR

A method for solving constrained optimization problems when the derivatives of the objective function are unavailable, while the derivatives of the constraints are known, which allows the objective and constraint function to be nonconvex.

Abstract

This paper explores a method for solving constrained optimization problems when the derivatives of the objective function are unavailable, while the derivatives of the constraints are known. We allow the objective and constraint function to be nonconvex. The method constructs a quadratic model of the objective function via interpolation and computes a step by minimizing this model subject to the original constraints in the problem and a trust region constraint. The step computation requires the solution of a general nonlinear program, which is economically feasible when the constraints and their derivatives are very inexpensive to compute compared to the objective function. The paper includes a summary of numerical results that highlight the method's promising potential.
Paper Structure (10 sections, 8 equations, 6 figures, 10 tables, 1 algorithm)

This paper contains 10 sections, 8 equations, 6 figures, 10 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. The FIBO Algorithm
  3. Specification of the Algorithm
  4. Feasibility
  5. Per-Iteration Cost
  6. Related Work
  7. Numerical Experiments
  8. Final Remarks
  9. Numerical Results for Feasible Initial Points
  10. Numerical Results for Infeasible Initial Point

Figures (6)

  • Figure 3.1: Log-ratio plot \ref{['acc']} comparing the final accuracy achieved by FIBO and FD.
  • Figure 3.2: Log-ratio plot comparing FIBO and FD in terms of the number of objective function evaluations required to satisfy \ref{['efficiency']} for $\tau = 10^{-1}\text{ (upper left)},$$10^{-3}\text{ (upper right)},\\$$10^{-5}$$\text{ (bottom left)}, 10^{-7}\text{ (bottom right)}$.
  • Figure 3.3: Log-ratio plot comparing FIBO and FD in terms of the number of constraint evaluations to satisfy \ref{['efficiency']} for $\tau = 10^{-1} \text{ (upper left)}$, $10^{-3}\text{ (upper right)},$$10^{-5}\text{ (bottom left)}$, $10^{-7}\text{ (bottom right)}$.
  • Figure B.1: Infeasible $x_0$. Log-ratio Plot for Comparing the Final Accuracy \ref{['acc']} of FIBO and FD.
  • Figure B.2: Infeasible $x_0$. Log-ratio plot comparing FIBO and FD in terms of the number of function evaluations to satisfy \ref{['efficiency']} for $\tau = 10^{-1} \text{ (upper left)}, 10^{-3}\text{ (upper right)}, 10^{-5}\text{ (bottom left)}, 10^{-7}\text{ (bottom right)}$.
  • ...and 1 more figures