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Probabilistic algorithm for computing all local minimizers of Morse functions on a compact domain

Mohab Safey El Din, Georgy Scholten, Emmanuel Trélat

Abstract

Let K be the unit-cube in Rn and f\,: K $\rightarrow$ R^n be a Morse function. We assume that the function f is given by an evaluation program $Γ$ in the noisy model, i.e., the evaluation program $Γ$ takes an extra parameter $η$ as input and returns an approximation that is $η$-close to the true value of f . In this article, we design an algorithm able to compute all local minimizers of f on K . Our algorithm takes as input $Γ$, $η$, a numerical accuracy parameter $ε$ as well as some extra regularity parameters which are made explicit. Under assumptions of probabilistic nature -- related to the choice of the evaluation points used to feed $Γ$ --, it returns finitely many rational points of K , such that the set of balls of radius $ε$ centered at these points contains and separates the set of all local minimizers of f . Our method is based on approximation theory, yielding polynomial approximants for f , combined with computer algebra techniques for solving systems of polynomial equations. We provide bit complexity estimates for our algorithm when all regularity parameters are known. Practical experiments show that our implementation of this algorithm in the Julia package Globtim can tackle examples that were not reachable until now.

Probabilistic algorithm for computing all local minimizers of Morse functions on a compact domain

Abstract

Let K be the unit-cube in Rn and f\,: K R^n be a Morse function. We assume that the function f is given by an evaluation program in the noisy model, i.e., the evaluation program takes an extra parameter as input and returns an approximation that is -close to the true value of f . In this article, we design an algorithm able to compute all local minimizers of f on K . Our algorithm takes as input , , a numerical accuracy parameter as well as some extra regularity parameters which are made explicit. Under assumptions of probabilistic nature -- related to the choice of the evaluation points used to feed --, it returns finitely many rational points of K , such that the set of balls of radius centered at these points contains and separates the set of all local minimizers of f . Our method is based on approximation theory, yielding polynomial approximants for f , combined with computer algebra techniques for solving systems of polynomial equations. We provide bit complexity estimates for our algorithm when all regularity parameters are known. Practical experiments show that our implementation of this algorithm in the Julia package Globtim can tackle examples that were not reachable until now.

Paper Structure

This paper contains 11 sections, 6 theorems, 61 equations, 12 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Problem statement and regularity assumptions.
  3. Proof of Theorem \ref{['thm:main:noise']}
  4. Algorithms and complexity
  5. Subroutines and algorithm description
  6. Correctness and complexity
  7. Adaptive algorithm
  8. Implementation choices
  9. Details on $\mathsf{DLSP}$
  10. Details on $\mathsf{PSolve}$
  11. Software implementation and practical experiments

Key Result

Theorem 1

Let $m\geq \max(3,\beta n+1)$ be an integer, let $\kappa>0$ and $\lambda>0$ be given positive constants, and let $\varepsilon > 0$ be such that $\kappa\varepsilon\leq 3\lambda$. Let $f\in \mathscr{C}^{m, \kappa}_\lambda(\mathscr{K})$ be an arbitrary function, given by an evaluation program in the no where $e$ is the Euler constant and $C_{n,m}$ is the Jackson approximation constant (see Lemma lemm

Figures (12)

  • Figure 1: Univariate example
  • Figure 2: Separating distance between critical points of the objective and approximant functions $w_{d, \mathcal{S}}$, constructed in the tensorized Chebyshev and the tensorized Legendre polynomial basis, of increasing degrees $d = \{3, \ldots, 9\}$. The experiment is repeated with noisy evaluations in the Chebyshev basis in figure \ref{['fig:subfig_noisy_n3_avg']} and \ref{['fig:subfig_noisy_n3_max']}.
  • Figure 3: The critical points of the approximant $w_{d, \mathcal{S}}$ computed with https://msolve.lip6.fr are plotted in green and blue. Green if they fall within a distance of $5.e{-3}$ from a critical point computed with Chebfun2 and blue for those not. The white points are the missing critical points computed by Chebfun2 but not https://github.com/gescholt/Globtim.jl.
  • Figure 4: The vanishing curves of the partials of the partials of the De Jong function computed with Chebfun2, in green and blue. The solutions found with https://www.chebfun.org/'s polynomial system solver are plotted with red diamonds.
  • Figure 5: Starting at degree $d=20$, the approximant $w_{d}$ fully captures the local minimizers of the De Jong function over the domain $[-50,50]^2$, although it also adds many spurious critical points around the boundary. As the degree $d$ increases from $2$ to $24$, we observe convergence of the critical points of $w_{d, \mathcal{S}}$ towards the critical points of $f$, which we computed using https://www.chebfun.org/docs/guide/guide12.html.
  • ...and 7 more figures

Theorems & Definitions (22)

  • Theorem 1
  • Lemma 2
  • Lemma 3
  • proof
  • Remark 4
  • Remark 5
  • Lemma 6
  • proof
  • Theorem 7
  • proof
  • ...and 12 more