Globtim.jl Documentation

Global optimization of continuous functions via polynomial approximation

The Problem

Finding all local minima of a continuous function over a bounded domain is fundamentally hard. Standard optimization algorithms (gradient descent, BFGS, etc.) find one local minimum from a given starting point — but how do you know there isn't a better one elsewhere?

The Approach

Globtim solves this by replacing your function with a polynomial approximation. Why polynomials?

1. Smooth functions are well-approximated by polynomials — Chebyshev and Legendre bases provide well-conditioned polynomial approximations whose accuracy improves with degree for smooth functions
2. Polynomial critical points can be enumerated — Setting ∇p(x) = 0 gives a polynomial system with finitely many solutions (bounded by Bezout's theorem), which can be computed numerically via homotopy continuation or exactly via symbolic methods
3. Refinement on the original function — Each polynomial critical point seeds a local optimization (BFGS) on the original function, which can converge to a nearby true critical point

The result: a systematic search for local minima across the entire domain, not just the nearest one.

Algorithm Overview

f(x)  -->  Polynomial p(x)  -->  Solve grad(p) = 0  -->  Refine with BFGS  -->  Candidate minima
           (Chebyshev/Legendre)   (HomotopyContinuation.jl)

Challenging 1D function — multi-frequency oscillations at varying polynomial degrees:

Styblinski-Tang 2D — classic test function with polynomial approximation:

For functions that vary on different scales in different regions, Globtim uses adaptive subdivision to build piecewise polynomial approximations that maintain accuracy everywhere.

Installation

julia> ]
pkg> add Globtim

Additional Dependencies

• Visualization: add CairoMakie or add GLMakie
• Exact solving: Install msolve (symbolic method based on Groebner basis computations)

Getting Started

For a detailed walkthrough, see Getting Started.

Ecosystem

Globtim is part of a three-package ecosystem for global optimization:

Globtim (experiments) --> GlobtimPostProcessing (analysis) --> GlobtimPlots (visualization)

Install companion packages:

pkg> add GlobtimPostProcessing
pkg> add GlobtimPlots