Function Minimization in Rust, Simplified

ganesh, (/ɡəˈneɪʃ/) named after the Hindu god of wisdom, provides several common minimization algorithms as well as a straightforward, trait-based interface to create your own extensions. This crate is intended to be as simple as possible. The user needs to implement the Function trait on some struct which will take a vector of parameters and return a single-valued Result ($f(\mathbb{R}^n) \to \mathbb{R}$). Users can optionally provide a gradient function to speed up some algorithms, but a default central finite-difference implementation is provided so that all algorithms will work out of the box.

Caution

This crate is still in an early development phase, and the API is not stable. It can (and likely will) be subject to breaking changes before the 1.0.0 version release (and hopefully not many after that).

Table of Contents

• Key Features
• Quick Start
• Bounds
• Future Plans

Key Features

• Simple but powerful trait-oriented library which tries to follow the Unix philosophy of "do one thing and do it well".
• Generics to allow for different numeric types to be used in the provided algorithms.
• Algorithms that are simple to use with sensible defaults.
• Traits which make developing future algorithms simple and consistent.

Quick Start

This crate provides some common test functions in the test_functions module. Consider the following implementation of the Rosenbrock function:

use std::convert::Infallible;
use ganesh::prelude::*;

pub struct Rosenbrock {
    pub n: usize,
}
impl Function<f64, (), Infallible> for Rosenbrock {
    fn evaluate(&self, x: &[f64], _user_data: &mut ()) -> Result<f64, Infallible> {
        Ok((0..(self.n - 1))
            .map(|i| 100.0 * (x[i + 1] - x[i].powi(2)).powi(2) + (1.0 - x[i]).powi(2))
            .sum())
    }
}

To minimize this function, we could consider using the Nelder-Mead algorithm:

use ganesh::prelude::*;
use ganesh::algorithms::NelderMead;

fn main() -> Result<(), Infallible> {
    let problem = Rosenbrock { n: 2 };
    let nm = NelderMead::default();
    let mut m = Minimizer::new(nm, 2);
    let x0 = &[2.0, 2.0];
    m.minimize(&problem, x0, &mut ())?;
    println!("{}", m.status);
    Ok(())
}

This should output

MSG:       term_f = STDDEV
X:         +1.000 ± 0.707
           +1.000 ± 1.416
F(X):      +0.000
N_F_EVALS: 159
N_G_EVALS: 0
CONVERGED: true
COV:       
  ┌             ┐
  │ 0.500 1.000 │
  │ 1.000 2.005 │
  └             ┘

Bounds

All minimizers in ganesh have access to a feature which allows algorithms which usually function in unbounded parameter spaces to only return results inside a bounding box. This is done via a parameter transformation, the same one used by LMFIT and MINUIT. This transform is not enacted on algorithms which already have bounded implementations, like L-BFGS-B. While the user inputs parameters within the bounds, unbounded algorithms can (and in practice will) convert those values to a set of unbounded "internal" parameters. When functions are called, however, these internal parameters are converted back into bounded "external" parameters, via the following transformations:

Upper and lower bounds:

$$x_\text{int} = \arcsin\left(2\frac{x_\text{ext} - x_\text{min}}{x_\text{max} - x_\text{min}} - 1\right)$$ $$x_\text{ext} = x_\text{min} + \left(\sin(x_\text{int}) + 1\right)\frac{x_\text{max} - x_\text{min}}{2}$$

Upper bound only:

$$x_\text{int} = \sqrt{(x_\text{max} - x_\text{ext} + 1)^2 - 1}$$ $$x_\text{ext} = x_\text{max} + 1 - \sqrt{x_\text{int}^2 + 1}$$

Lower bound only:

$$x_\text{int} = \sqrt{(x_\text{ext} - x_\text{min} + 1)^2 - 1}$$ $$x_\text{ext} = x_\text{min} - 1 + \sqrt{x_\text{int}^2 + 1}$$

As noted in the documentation for both LMFIT and MINUIT, these bounds should be used with caution. They turn linear problems into nonlinear ones, which can mess with error propagation and even fit convergence, not to mention increase function complexity. Methods which output covariance matrices need to be adjusted if bounded, and MINUIT recommends fitting a second time near a minimum without bounds to ensure proper error propagation.

Future Plans

• Eventually, I would like to implement MCMC algorithms and some more modern gradient-free optimization techniques.
• There are probably many optimizations and algorithm extensions that I'm missing right now because I just wanted to get it working first.
• A test suite