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An augmented Lagrangian trust-region method with inexact gradient evaluations to accelerate constrained optimization problems using model hyperreduction

Tianshu Wen, Matthew J. Zahr

TL;DR

This work develops an augmented Lagrangian trust-region method integrated with on-the-fly EQP-based hyperreduction to accelerate constrained PDE optimization without offline training. By solving AL subproblems with a bound-constrained trust-region that accommodates inexact gradients, and by building reduced models on-the-fly, the approach guarantees convergence to a critical point of the original problem while dramatically reducing HDM evaluations. The authors prove global convergence for the inexact-trust-region framework and demonstrate speedups up to ~12.7x on aerodynamic inverse-design problems with one or two side constraints, highlighting practical efficiency and robustness. The methodology opens paths to extensions to unsteady problems, higher-order quadrature hyperreduction, and broader PDE-constrained optimization applications.

Abstract

We present an augmented Lagrangian trust-region method to efficiently solve constrained optimization problems governed by large-scale nonlinear systems with application to partial differential equation-constrained optimization. At each major augmented Lagrangian iteration, the expensive optimization subproblem involving the full nonlinear system is replaced by an empirical quadrature-based hyperreduced model constructed on-the-fly. To ensure convergence of these inexact augmented Lagrangian subproblems, we develop a bound-constrained trust-region method that allows for inexact gradient evaluations, and specialize it to our specific setting that leverages hyperreduced models. This approach circumvents a traditional training phase because the models are built on-the-fly in accordance with the requirements of the trust-region convergence theory. Two numerical experiments (constrained aerodynamic shape design) demonstrate the convergence and efficiency of the proposed work. A speedup of 12.7x (for all computational costs, even costs traditionally considered "offline" such as snapshot collection and data compression) relative to a standard optimization approach that does not leverage model reduction is shown.

An augmented Lagrangian trust-region method with inexact gradient evaluations to accelerate constrained optimization problems using model hyperreduction

TL;DR

This work develops an augmented Lagrangian trust-region method integrated with on-the-fly EQP-based hyperreduction to accelerate constrained PDE optimization without offline training. By solving AL subproblems with a bound-constrained trust-region that accommodates inexact gradients, and by building reduced models on-the-fly, the approach guarantees convergence to a critical point of the original problem while dramatically reducing HDM evaluations. The authors prove global convergence for the inexact-trust-region framework and demonstrate speedups up to ~12.7x on aerodynamic inverse-design problems with one or two side constraints, highlighting practical efficiency and robustness. The methodology opens paths to extensions to unsteady problems, higher-order quadrature hyperreduction, and broader PDE-constrained optimization applications.

Abstract

We present an augmented Lagrangian trust-region method to efficiently solve constrained optimization problems governed by large-scale nonlinear systems with application to partial differential equation-constrained optimization. At each major augmented Lagrangian iteration, the expensive optimization subproblem involving the full nonlinear system is replaced by an empirical quadrature-based hyperreduced model constructed on-the-fly. To ensure convergence of these inexact augmented Lagrangian subproblems, we develop a bound-constrained trust-region method that allows for inexact gradient evaluations, and specialize it to our specific setting that leverages hyperreduced models. This approach circumvents a traditional training phase because the models are built on-the-fly in accordance with the requirements of the trust-region convergence theory. Two numerical experiments (constrained aerodynamic shape design) demonstrate the convergence and efficiency of the proposed work. A speedup of 12.7x (for all computational costs, even costs traditionally considered "offline" such as snapshot collection and data compression) relative to a standard optimization approach that does not leverage model reduction is shown.
Paper Structure (24 sections, 8 theorems, 122 equations, 6 figures, 8 tables, 3 algorithms)

This paper contains 24 sections, 8 theorems, 122 equations, 6 figures, 8 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Governing equations
  4. Optimization formulation
  5. Hyperreduction
  6. Projection-based model reduction
  7. An optimization-aware empirical quadrature procedure
  8. Formulation
  9. Training
  10. Error estimation
  11. Augmented Lagrangian trust-region method based on hyperreduced models
  12. An inexact trust-region method to solve bound-constrained problems
  13. Augmented Lagrangian framework
  14. Accelerated bound-constrained optimization using trust regions and on-the-fly model hyperreduction
  15. Numerical experiments
  16. ...and 9 more sections

Key Result

Theorem 1

Under Assumptions assum:hdm-assum:eqp, there exist constants $c_1,c_2>0$ such that for any ${\boldsymbol{\mu}}\in\mathcal{D}$, ${\boldsymbol{\rho}}\in\mathcal{R}$, ${\boldsymbol{\theta}}\in\mathbb{R} ^{N_{\bm{c}}}$, and $\tau > 0$ Furthermore, there exist constants $c_1',c_2',c_3',c_4'>0$ such that

Figures (6)

  • Figure 1: Convergence history (only major iterations shown) of the objective value (left) and constraint violation (right) of the EQP/TR method applied the inverse design problem (\ref{['eqn:contin_inv']}) for combinations of $(\tau_0, a)$. Legend: $(\tau_0=10,a=10)$ (\ref{['line:fp10f10']}), $(\tau_0=10,a=50)$ (\ref{['line:fp10f50']}), $(\tau_0=10,a=100)$ (\ref{['line:fp10f100']}), $(\tau_0=25,a=10)$ (\ref{['line:fp25f10']}), $(\tau_0=25,a=50)$ (\ref{['line:fp25f50']}), $(\tau_0=25,a=100)$ (\ref{['line:fp25f100']}), $(\tau_0=50,a=10)$ (\ref{['line:fp50f10']}), $(\tau_0=50,a=50)$ (\ref{['line:fp50f50']}), $(\tau_0=50,a=100)$ (\ref{['line:fp50f100']}), and HDM solution (\ref{['line:fhdm']}).
  • Figure 2: Convergence history (only major iterations shown) of the objective value (left) and constraint violation (right) of the EQP/TR method applied the inverse design problem (\ref{['eqn:contin_inv']}) when inheriting a different number of snapshots from the previous AL iteration. Legend: 0 snapshots (\ref{['line:fsz0']}), 5 snapshots (\ref{['line:fsz5']}), 15 snapshots (\ref{['line:fsz15']}), 20 snapshots (\ref{['line:fsz20']}), and HDM solution (\ref{['line:fhdm']}).
  • Figure 3: Convergence history (only major iterations shown) of the objective value (left) and constraint violation (right) of the HDM-based interior point solver (\ref{['line:fhdm']}), ROM/TR method (\ref{['line:from']}), and EQP/TR method (\ref{['line:feqp']}) applied to the inverse design problem in (\ref{['eqn:contin_inv']}).
  • Figure 4: The domain shape and density for the inverse design problem at the starting configuration (top-left), HDM optimal solution (top-right), ROM optimal solution (bottom-left), and EQP optimal solution (bottom-right).
  • Figure 5: Convergence history (only major iterations shown) of the objective value (left) and constraint violation (right) of the HDM-based interior point solver (\ref{['line:fhdm']}), ROM/TR method (\ref{['line:from']}), and EQP/TR method (\ref{['line:feqp']}) applied to the inverse design problem in (\ref{['airfoil:eqn:prob']}).
  • ...and 1 more figures

Theorems & Definitions (34)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Remark 7
  • Theorem 1
  • proof
  • Corollary 1.1
  • ...and 24 more