argmin_testfunctions/
booth.rs

1// Copyright 2018-2024 argmin developers
2//
3// Licensed under the Apache License, Version 2.0 <LICENSE-APACHE or
4// http://apache.org/licenses/LICENSE-2.0> or the MIT license <LICENSE-MIT or
5// http://opensource.org/licenses/MIT>, at your option. This file may not be
6// copied, modified, or distributed except according to those terms.
7
8//! # Booth test function
9//!
10//! Defined as
11//!
12//! $$
13//! f(x_1, x_2) = (x_1 + 2x_2 - 7)^2 + (2x_1 + x_2 - 5)^2
14//! $$
15//!
16//! where $x_i \in [-10,\\,10]$.
17//!
18//! The global minimum is at $f(x_1, x_2) = f(1, 3) = 0$.
19
20use num::{Float, FromPrimitive};
21
22/// Booth test function
23///
24/// Defined as
25///
26/// $$
27/// f(x_1, x_2) = (x_1 + 2x_2 - 7)^2 + (2x_1 + x_2 - 5)^2
28/// $$
29///
30/// where $x_i \in [-10,\\,10]$.
31///
32/// The global minimum is at $f(x_1, x_2) = f(1, 3) = 0$.
33pub fn booth<T>(param: &[T; 2]) -> T
34where
35    T: Float + FromPrimitive,
36{
37    let n2 = T::from_f64(2.0).unwrap();
38    let n5 = T::from_f64(5.0).unwrap();
39    let n7 = T::from_f64(7.0).unwrap();
40
41    let [x1, x2] = *param;
42    (x1 + n2 * x2 - n7).powi(2) + (n2 * x1 + x2 - n5).powi(2)
43}
44
45/// Derivative of Booth test function
46pub fn booth_derivative<T>(param: &[T; 2]) -> [T; 2]
47where
48    T: Float + FromPrimitive,
49{
50    let n8 = T::from_f64(8.0).unwrap();
51    let n10 = T::from_f64(10.0).unwrap();
52    let n34 = T::from_f64(34.0).unwrap();
53    let n38 = T::from_f64(38.0).unwrap();
54
55    let [x1, x2] = *param;
56
57    [n10 * x1 + n8 * x2 - n34, n8 * x1 + n10 * x2 - n38]
58}
59
60/// Hessian of Booth test function
61///
62/// Returns $\left(\begin{matrix}10 & 8\\\\8 & 10\end{matrix}\right)$ for every input.
63pub fn booth_hessian<T>(_param: &[T; 2]) -> [[T; 2]; 2]
64where
65    T: Float + FromPrimitive,
66{
67    let n8 = T::from_f64(8.0).unwrap();
68    let n10 = T::from_f64(10.0).unwrap();
69
70    [[n10, n8], [n8, n10]]
71}
72
73#[cfg(test)]
74mod tests {
75    use super::*;
76    use approx::assert_relative_eq;
77    use finitediff::FiniteDiff;
78    use proptest::prelude::*;
79    use std::{f32, f64};
80
81    #[test]
82    fn test_booth_optimum() {
83        assert_relative_eq!(booth(&[1_f32, 3_f32]), 0.0, epsilon = f32::EPSILON);
84        assert_relative_eq!(booth(&[1_f64, 3_f64]), 0.0, epsilon = f64::EPSILON);
85
86        let deriv = booth_derivative(&[1.0, 3.0]);
87        for i in 0..2 {
88            assert_relative_eq!(deriv[i], 0.0, epsilon = f64::EPSILON);
89        }
90    }
91
92    proptest! {
93        #[test]
94        fn test_booth_derivative_finitediff(a in -10.0..10.0, b in -10.0..10.0) {
95            let param = [a, b];
96            let derivative = booth_derivative(&param);
97            let derivative_fd = Vec::from(param).central_diff(&|x| booth(&[x[0], x[1]]));
98            for i in 0..derivative.len() {
99                assert_relative_eq!(
100                    derivative[i],
101                    derivative_fd[i],
102                    epsilon = 1e-4,
103                    max_relative = 1e-2
104                );
105            }
106        }
107    }
108
109    proptest! {
110        #[test]
111        fn test_booth_hessian(a in -10.0..10.0, b in -10.0..10.0) {
112            let param = [a, b];
113            let hessian = booth_hessian(&param);
114            let hessian_fd = [[10.0, 8.0], [8.0, 10.0]];
115            let n = hessian.len();
116            for i in 0..n {
117                assert_eq!(hessian[i].len(), n);
118                for j in 0..n {
119                    assert_relative_eq!(
120                        hessian[i][j],
121                        hessian_fd[i][j],
122                        epsilon = 1e-5,
123                        max_relative = 1e-2
124                    );
125                }
126            }
127        }
128    }
129}
argmin_testfunctions/booth.rs

argmin_testfunctions/
booth.rs
