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

2025-07-17

AI Summary

We address global optimization of a Morse function $ f: K o mathbb{R} $, where $ K subset mathbb{R}^n $ is the unit cube and function evaluations are corrupted by noise. Given numerical precision $ varepsilon $ and a regularization parameter, our probabilistic algorithm reliably locates *all* local minima. The method integrates high-accuracy polynomial approximation of $ f $ with symbolic computation: critical points are computed by solving the gradient system via a computer algebra system, followed by interval-based verification and probabilistic separation to guarantee—under noise—with high probability that all minima are enclosed and none omitted. Our key contribution is the first synergistic use of approximation theory and symbolic methods for enumerating minima of noisy Morse functions, circumventing classical assumptions of exact evaluation and infinite differentiability. The algorithm features analyzable bit-complexity and is implemented in Julia as the package *Globtim*, successfully solving previously intractable high-dimensional nonconvex problems.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

Let K be the unit-cube in Rn and f,: K $ ightarrow$ 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.

Problem

• Computes all local minimizers of Morse functions on a compact domain
• Handles noisy function evaluations with probabilistic assumptions
• Uses polynomial approximation and computer algebra for solutions

Innovation

• Probabilistic algorithm for Morse function minimizers
• Combines polynomial approximation and computer algebra
• Handles noisy evaluations with accuracy parameters