MGH: Moré, Garbow, and Hillstrom Test Functions

A pure Rust implementation of the classic Moré, Garbow, and Hillstrom (MGH) collection of test functions for unconstrained optimization algorithms.

This crate provides a standard suite of benchmark problems, essential for developers and researchers who need to test, validate, and compare the performance of optimization algorithms. The implementations are based on the original Fortran package and the definitions in the seminal paper:

Moré, J. J., Garbow, B. S., & Hillstrom, K. E. (1981). Testing Unconstrained Optimization Software. ACM Transactions on Mathematical Software (TOMS), 7(1), 17–41. https://dl.acm.org/doi/pdf/10.1145/355934.355936

Core Concepts

The library is organized into three main structs:

1. MGH: Evaluates the objective functions. Most functions are formulated as a sum of squares, $F(x) = \sum_{i=1}^{m} f_i(x)^2$. All methods in MGH return a scalar f64.
2. MGHInit: Provides the standard initial starting points ($x_0$) for each test function, as specified in the MGH paper.
3. MGHMin: Provides known solution vectors ($x^*$) for functions where the global minimum is documented.

Installation

Add the crate to your Cargo.toml file:

[dependencies]
mgh = "0.1.0"

Usage and Examples

Example 1: Fixed-Dimension Function (Gaussian)

Functions with fixed $n$ and $m$ use static methods.

use mgh::{MGH, MGHInit, MGHMin};

// 1. Get the standard starting point (n=3)
let x0 = MGHInit::gaussian();

// 2. Evaluate the objective function
let value = MGH::gaussian(&x0);
println!("Value at x0: {}", value);

// 3. (Optional) Verify at the known minimum
let x_min = MGHMin::gaussian();
let min_value = MGH::gaussian(&x_min);
assert!(min_value < 1e-10);

Example 2: Variable-Dimension Function (Extended Rosenbrock)

Functions where $n$ can be specified. $m$ is determined by $n$.

use mgh::{MGH, MGHInit, MGHMin};

let n = 10; // Must be even for Extended Rosenbrock

let x0 = MGHInit::extended_rosenbrock(n);
let value = MGH::extended_rosenbrock(&x0);

let x_min = MGHMin::extended_rosenbrock(n);
let min_value = MGH::extended_rosenbrock(&x_min);
assert_eq!(min_value, 0.0);

Example 3: Specifying Residuals (Biggs EXP6)

Functions requiring an explicit $m$ parameter use MGH::aux(m).

use mgh::{MGH, MGHInit};

let m = 20; // Number of residuals
let x0 = MGHInit::biggs_exp6();

// Use .aux(m) to specify the number of residuals
let value = MGH::aux(m).biggs_exp6(&x0);

---

API Reference

The tables below list all available test functions and their properties.

1. Fixed-Dimension Functions

These use simple static methods like MGH::gaussian(&x).

Function	$n$	$m$	Initial Vector	Solution Vector
bard	3	15	✅	✅
beale	2	3	✅	✅
brown_badly_scaled	2	3	✅	✅
freudenstein_and_roth	2	2	✅	✅
gaussian	3	15	✅	✅
hellical_valley	3	3	✅	✅
kowalik_and_osborne	4	11	✅	❌
meyer	3	16	✅	❌
osborne_1	5	33	✅	❌
osborne_2	11	65	✅	❌
powell_badly_scaled	2	2	✅	✅
powell_singular	4	4	✅	✅
rosenbrock	2	2	✅	✅
wood	4	6	✅	✅

2. Variable-Dimension Functions

These use static methods but MGHInit and MGHMin require an n argument.

Function	$m$	Initial Vector	Solution Vector
brown_almost_linear	$n$	✅	✅
broyden_banded	$n$	✅	❌
broyden_tridiagonal	$n$	✅	❌
discrete_boundary_value	$n$	✅	❌
discrete_integral_equation	$n$	✅	❌
extended_powell_singular	$n$ (mult of 4)	✅	✅
extended_rosenbrock	$n$ (even)	✅	✅
penalty1	$n + 1$	✅	❌
penalty2	$2n$	✅	❌
trigonometric	$n$	✅	❌
variably_dimensioned	$n + 2$	✅	✅
watson	31 ($2 \le n \le 31$)	✅	❌

3. Functions with Auxiliary $m$ Parameter

These require MGH::aux(m) to specify the number of residuals.

Function	Dimension	Initial Vector	Solution Vector
biggs_exp6	$n = 6$	✅	✅
box_3d	$n = 3$	✅	✅
brown_and_dennis	$n = 4$	✅	✅
gulf_research_and_development	$n = 3$	✅	✅
jennrich_and_sampson	$n = 2$	✅	✅
chebyquad	Variable $n$	✅	❌
linear_full_rank	Variable $n$	✅	✅
linear_rank_1	Variable $n$	✅	❌
linear_rank_1_zero	Variable $n$	✅	❌

License

• MIT license (LICENSE-MIT)

Contributing

Contributions are welcome! Feel free to open an issue or submit a pull request.