Added derivative and Hessian for Bukin No.6 test function

crates/argmin-testfunctions/src/bukin.rs

@@ -37,15 +37,125 @@ where
     n100 * (x2 - n001 * x1.powi(2)).abs().sqrt() + n001 * (x1 + n10).abs()
 }
 
+/// Derivative of Bukin test function No. 6
+pub fn bukin_n6_derivative<T>(param: &[T; 2]) -> [T; 2]
+where
+    T: Float + FromPrimitive,
+{
+    let [x1, x2] = *param;
+
+    let n0 = T::from_f64(0.0).unwrap();
+    let n0_01 = T::from_f64(0.01).unwrap();
+    let n10 = T::from_f64(10.0).unwrap();
+    let n50 = T::from_f64(50.0).unwrap();
+    let eps = T::from_f64(std::f64::EPSILON).unwrap();
+
+    let denom = (x2 - n0_01 * x1.powi(2)).abs().powi(3).sqrt();
+    let tmp = x2 - n0_01 * x1.powi(2);
+
+    if denom.abs() <= eps {
+        // Deriviative is actually not defined at optimum. Therefore, as soon as we get close,
+        // we'll set the derivative to 0
+        [n0, n0]
+    } else {
+        [
+            n0_01 * (x1 + n10).signum() - x1 * tmp / denom,
+            n50 * tmp / denom,
+        ]
+    }
+}
+
+/// Hessian of Bukin test function No. 6
+pub fn bukin_n6_hessian<T>(param: &[T; 2]) -> [[T; 2]; 2]
+where
+    T: Float + FromPrimitive,
+{
+    let [x1, x2] = *param;
+
+    let n0 = T::from_f64(0.0).unwrap();
+    let n0_01 = T::from_f64(0.01).unwrap();
+    let n0_02 = T::from_f64(0.02).unwrap();
+    let n0_0001 = T::from_f64(0.0001).unwrap();
+    let n0_5 = T::from_f64(0.5).unwrap();
+    let n25 = T::from_f64(25.0).unwrap();
+    let eps = T::from_f64(std::f64::EPSILON * 1e-4).unwrap();
+
+    let tmp = x2 - n0_01 * x1.powi(2);
+    let denom = tmp.abs().powi(7).sqrt();
+
+    if denom.abs() <= eps {
+        // Hessian is actually not defined at optimum. Therefore, as soon as we get close,
+        // we'll set the Hessian to 0
+        [[n0, n0], [n0, n0]]
+    } else {
+        let offdiag = n0_5 * x1 * tmp.powi(2) / denom;
+        [
+            [
+                x2 * (-n0_0001 * x1.powi(4) + n0_02 * x2 * x1.powi(2) - x2.powi(2)) / denom,
+                offdiag,
+            ],
+            [offdiag, -n25 * tmp.powi(2) / denom],
+        ]
+    }
+}
+
 #[cfg(test)]
 mod tests {
     use super::*;
     use approx::assert_relative_eq;
+    use finitediff::FiniteDiff;
+    use proptest::prelude::*;
     use std::{f32, f64};
 
     #[test]
     fn test_bukin_n6_optimum() {
         assert_relative_eq!(bukin_n6(&[-10_f32, 1_f32]), 0.0, epsilon = f32::EPSILON);
         assert_relative_eq!(bukin_n6(&[-10_f64, 1_f64]), 0.0, epsilon = f64::EPSILON);
+
+        let deriv = bukin_n6_derivative(&[-10_f64, 1_f64]);
+        for i in 0..2 {
+            assert_relative_eq!(deriv[i], 0.0, epsilon = f64::EPSILON);
+        }
+
+        let hessian = bukin_n6_hessian(&[-10_f64, 1_f64]);
+        for i in 0..2 {
+            for j in 0..2 {
+                assert_relative_eq!(hessian[i][j], 0.0, epsilon = f64::EPSILON);
+            }
+        }
+    }
+
+    proptest! {
+        #[test]
+        fn test_bukin_n6_derivative_finitediff(a in -15.0..-5.0, b in -3.0..3.0) {
+            let param = [a, b];
+            let derivative = bukin_n6_derivative(&param);
+            let derivative_fd = Vec::from(param).central_diff(&|x| bukin_n6(&[x[0], x[1]]));
+            for i in 0..derivative.len() {
+                // finite differences aren't really that useful here, therefore the extremely
+                // high epsilon. We're happy if we're somewhat close.
+                assert_relative_eq!(derivative[i], derivative_fd[i], epsilon = 1.0);
+            }
+        }
+    }
+
+    proptest! {
+        #[test]
+        fn test_bukin_n6_hessian_finitediff(a in -15.0..-5.0, b in -3.0..3.0) {
+            let param = [a, b];
+            let hessian = bukin_n6_hessian(&param);
+            let hessian_fd = Vec::from(param).central_hessian(&|x| bukin_n6_derivative(&[x[0], x[1]]).to_vec());
+            let n = hessian.len();
+            println!("1: {a}/{b} {hessian:?}");
+            println!("2: {a}/{b} {hessian_fd:?}");
+            for i in 0..n {
+                assert_eq!(hessian[i].len(), n);
+                for j in 0..n {
+                    // finite differences aren't really that useful here, therefore the extremely
+                    // high epsilon. We're happy if we're somewhat close.
+                    assert_relative_eq!(hessian[i][j], hessian_fd[i][j], epsilon = 100.0);
+                }
+            }
+        }
     }
 }