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An Inexact Weighted Proximal Trust-Region Method

Leandro Farias Maia, Robert Baraldi, Drew P. Kouri

TL;DR

The paper tackles nonconvex nonsmooth optimization of $F(x)=f(x)+\phi(x)$ by extending a proximal trust-region method to accommodate inexact proximity operators via the $\delta$-Fréchet subdifferential and weighted inner products. It establishes a theoretical link between $\delta$-proximal objects and $\delta$-subgradients, adapts the trust-region framework to use inexact proximals through $\tilde{h}_k$ and $\tilde{Q}_k$, and develops a weighted proximal-gradient approach to compute $\mathrm{Prox}_{r\phi}^a$ with provable accuracy guarantees. The method is demonstrated on a Burgers' equation PDE-constrained control problem with an $L^1$-norm penalty, showing robustness to proximity inexactness and to inexact PDE solves, including a notable reduction in linear-system solves. These results broaden the applicability of proximal-trust-region methods to problems with non-diagonal inner products and inexact proximal computations, with practical impact for large-scale PDE-constrained optimization.

Abstract

In [R. J. Baraldi and D. P. Kouri, Math. Program., 201:1 (2023), pp. 559-598], the authors introduced a trust-region method for minimizing the sum of a smooth nonconvex and a nonsmooth convex function, the latter of which has an analytical proximity operator. While many functions satisfy this criterion, e.g., the $\ell_1$-norm defined on $\ell_2$, many others are precluded by either the topology or the nature of the nonsmooth term. Using the $δ$-Fréchet subdifferential, we extend the definition of the inexact proximity operator and enable its use within the aforementioned trust-region algorithm. Moreover, we augment the analysis for the standard trust-region convergence theory to handle proximity operator inexactness with weighted inner products. We first introduce an algorithm to generate a point in the inexact proximity operator and then apply the algorithm within the trust-region method to solve an optimal control problem constrained by Burgers' equation.

An Inexact Weighted Proximal Trust-Region Method

TL;DR

The paper tackles nonconvex nonsmooth optimization of by extending a proximal trust-region method to accommodate inexact proximity operators via the -Fréchet subdifferential and weighted inner products. It establishes a theoretical link between -proximal objects and -subgradients, adapts the trust-region framework to use inexact proximals through and , and develops a weighted proximal-gradient approach to compute with provable accuracy guarantees. The method is demonstrated on a Burgers' equation PDE-constrained control problem with an -norm penalty, showing robustness to proximity inexactness and to inexact PDE solves, including a notable reduction in linear-system solves. These results broaden the applicability of proximal-trust-region methods to problems with non-diagonal inner products and inexact proximal computations, with practical impact for large-scale PDE-constrained optimization.

Abstract

In [R. J. Baraldi and D. P. Kouri, Math. Program., 201:1 (2023), pp. 559-598], the authors introduced a trust-region method for minimizing the sum of a smooth nonconvex and a nonsmooth convex function, the latter of which has an analytical proximity operator. While many functions satisfy this criterion, e.g., the -norm defined on , many others are precluded by either the topology or the nature of the nonsmooth term. Using the -Fréchet subdifferential, we extend the definition of the inexact proximity operator and enable its use within the aforementioned trust-region algorithm. Moreover, we augment the analysis for the standard trust-region convergence theory to handle proximity operator inexactness with weighted inner products. We first introduce an algorithm to generate a point in the inexact proximity operator and then apply the algorithm within the trust-region method to solve an optimal control problem constrained by Burgers' equation.
Paper Structure (7 sections, 6 theorems, 63 equations, 1 table, 2 algorithms)

This paper contains 7 sections, 6 theorems, 63 equations, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Notation and Problem Assumptions
  3. A Proximal Trust-Region Method
  4. Inexact Proximity Operator
  5. Weighted Proximal Operators
  6. Numerics
  7. Conclusion and Future Work

Key Result

Lemma 3.1

\newlabellem:qk The function $r \mapsto Q_k (r )$ is continuous and nonincreasing. In fact, if $r >t > 0$ then $Q_k (t) > Q_k (r )$ whenever $x_k (t) \neq x_k(r)$. Moreover, if $h_k > 0$ then

Theorems & Definitions (13)

  • Lemma 3.1
  • Definition 4.1: $\delta$-Proximity Operator
  • Definition 4.2: $\delta$-Subgradient
  • Lemma 4.3
  • proof
  • Theorem 4.4
  • proof
  • Lemma 5.1
  • proof
  • Lemma 5.2
  • ...and 3 more