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On Convergence of Binary Trust-Region Steepest Descent

Paul Manns, Mirko Hahn, Christian Kirches, Sven Leyffer, Sebastian Sager

TL;DR

It is concluded that BTR also constitutes a descent algorithm on the continuous relaxation and its iterates converge weakly-$^*$ to stationary points of the latter.

Abstract

Binary trust-region steepest descent (BTR) and combinatorial integral approximation (CIA) are two recently investigated approaches for the solution of optimization problems with distributed binary-/discrete-valued variables (control functions). We show improved convergence results for BTR by imposing a compactness assumption that is similar to the convergence theory of CIA. As a corollary we conclude that BTR also constitutes a descent algorithm on the continuous relaxation and its iterates converge weakly-$^*$ to stationary points of the latter. We provide computational results that validate our findings. In addition, we observe a regularizing effect of BTR, which we explore by means of a hybridization of CIA and BTR.

On Convergence of Binary Trust-Region Steepest Descent

TL;DR

It is concluded that BTR also constitutes a descent algorithm on the continuous relaxation and its iterates converge weakly- to stationary points of the latter.

Abstract

Binary trust-region steepest descent (BTR) and combinatorial integral approximation (CIA) are two recently investigated approaches for the solution of optimization problems with distributed binary-/discrete-valued variables (control functions). We show improved convergence results for BTR by imposing a compactness assumption that is similar to the convergence theory of CIA. As a corollary we conclude that BTR also constitutes a descent algorithm on the continuous relaxation and its iterates converge weakly- to stationary points of the latter. We provide computational results that validate our findings. In addition, we observe a regularizing effect of BTR, which we explore by means of a hybridization of CIA and BTR.
Paper Structure (37 sections, 12 theorems, 46 equations, 2 figures, 3 tables, 5 algorithms)

This paper contains 37 sections, 12 theorems, 46 equations, 2 figures, 3 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Combinatorial integral approximation hante2013relaxationkirches2020compactnesssager2012integersager2011combinatorial
  3. Binary trust-region steepest descent hahn2022binarysharma2020inversionvogt2020mixed
  4. Standing assumptions
  5. Contributions
  6. Structure of the paper
  7. Notation
  8. Combinatorial Integral Approximation
  9. Inputs of \ref{['alg:cia']}
  10. The subroutine Round and order-conserving domain dissections
  11. Asymptotics of \ref{['alg:binary_tr_steepest_descent']} under \ref{['ass:tr_convergence']}
  12. Binary Trust-Region Steepest Descent
  13. Inputs of \ref{['alg:binary_tr_steepest_descent']}
  14. Trust-region subproblem and subroutine FindStep
  15. Description of the steps of \ref{['alg:binary_tr_steepest_descent']}
  16. ...and 22 more sections

Key Result

Proposition 2.1

Let Assumptions ass:tr_convergence, itm:Jdifferentiable, and itm:Jprime_cc hold. Then $J : L^2(\Omega) \to \mathbb{R}$ is weakly continuous.

Figures (2)

  • Figure 1: Visualization of the resulting control functions for Rel. (= continuous relaxation of \ref{['eq:poisson-bvp']}), CIA (SUR) / (COR) (= continuous relaxation and SUR / COR), CIA (SHG) (= continuous relaxation and SHG), CIA (SUR) / (COR) + BTR, Rel. + BTR (= continuous relaxation and BTR started from a cellwise rounding), and BTR (= BTR started from zero), where the value one is colored black and the value zero is colored white. For Rel., the intermediate values in $[0,1]$ are depicted in grayscale.
  • Figure 2: Visualization of the resulting control functions for SUR on uniform $8\times 8$ and $128\times 128$ square grids (top row) and BTR initialized with them (bottom row).

Theorems & Definitions (28)

  • Remark 1.1
  • Proposition 2.1
  • proof
  • Remark 2.2
  • Proposition 4.1
  • proof
  • Definition 4.2
  • Proposition 4.3
  • proof
  • Theorem 4.4
  • ...and 18 more