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//! Hock-Schittkowski problem 71 (constrained NLP).
//!
//! min x1*x4*(x1+x2+x3) + x3
//! s.t. x1*x2*x3*x4 >= 25 (g1)
//! x1^2+x2^2+x3^2+x4^2 = 40 (g2)
//! 1 <= xi <= 5, i=1..4
//!
//! Starting point: (1.0, 5.0, 5.0, 1.0)
//! Known solution: x* = (1.0, 4.743, 3.821, 1.379), f* ~ 17.014
use ripopt::{NlpProblem, SolveStatus, SolverOptions};
struct Hs071;
impl NlpProblem for Hs071 {
fn num_variables(&self) -> usize {
4
}
fn num_constraints(&self) -> usize {
2
}
fn bounds(&self, x_l: &mut [f64], x_u: &mut [f64]) {
for i in 0..4 {
x_l[i] = 1.0;
x_u[i] = 5.0;
}
}
fn constraint_bounds(&self, g_l: &mut [f64], g_u: &mut [f64]) {
// g1: x1*x2*x3*x4 >= 25 => 25 <= g1 <= +inf
g_l[0] = 25.0;
g_u[0] = f64::INFINITY;
// g2: x1^2+x2^2+x3^2+x4^2 = 40 => 40 <= g2 <= 40
g_l[1] = 40.0;
g_u[1] = 40.0;
}
fn initial_point(&self, x0: &mut [f64]) {
x0[0] = 1.0;
x0[1] = 5.0;
x0[2] = 5.0;
x0[3] = 1.0;
}
fn objective(&self, x: &[f64], _new_x: bool, obj: &mut f64) -> bool {
*obj = x[0] * x[3] * (x[0] + x[1] + x[2]) + x[2];
true
}
fn gradient(&self, x: &[f64], _new_x: bool, grad: &mut [f64]) -> bool {
// df/dx1 = x4*(x1+x2+x3) + x1*x4 = x4*(2*x1+x2+x3)
grad[0] = x[3] * (2.0 * x[0] + x[1] + x[2]);
// df/dx2 = x1*x4
grad[1] = x[0] * x[3];
// df/dx3 = x1*x4 + 1
grad[2] = x[0] * x[3] + 1.0;
// df/dx4 = x1*(x1+x2+x3)
grad[3] = x[0] * (x[0] + x[1] + x[2]);
true
}
fn constraints(&self, x: &[f64], _new_x: bool, g: &mut [f64]) -> bool {
g[0] = x[0] * x[1] * x[2] * x[3];
g[1] = x[0] * x[0] + x[1] * x[1] + x[2] * x[2] + x[3] * x[3];
true
}
fn jacobian_structure(&self) -> (Vec<usize>, Vec<usize>) {
// g1 depends on all 4 vars, g2 depends on all 4 vars => 8 entries
(
vec![0, 0, 0, 0, 1, 1, 1, 1],
vec![0, 1, 2, 3, 0, 1, 2, 3],
)
}
fn jacobian_values(&self, x: &[f64], _new_x: bool, vals: &mut [f64]) -> bool {
// dg1/dx1 = x2*x3*x4
vals[0] = x[1] * x[2] * x[3];
// dg1/dx2 = x1*x3*x4
vals[1] = x[0] * x[2] * x[3];
// dg1/dx3 = x1*x2*x4
vals[2] = x[0] * x[1] * x[3];
// dg1/dx4 = x1*x2*x3
vals[3] = x[0] * x[1] * x[2];
// dg2/dx1 = 2*x1
vals[4] = 2.0 * x[0];
// dg2/dx2 = 2*x2
vals[5] = 2.0 * x[1];
// dg2/dx3 = 2*x3
vals[6] = 2.0 * x[2];
// dg2/dx4 = 2*x4
vals[7] = 2.0 * x[3];
true
}
fn hessian_structure(&self) -> (Vec<usize>, Vec<usize>) {
// Lower triangle entries of the 4x4 Hessian of the Lagrangian.
// We enumerate all lower-triangle positions that can be nonzero:
// (0,0), (1,0), (1,1), (2,0), (2,1), (2,2), (3,0), (3,1), (3,2), (3,3)
(
vec![0, 1, 1, 2, 2, 2, 3, 3, 3, 3],
vec![0, 0, 1, 0, 1, 2, 0, 1, 2, 3],
)
}
fn hessian_values(&self, x: &[f64], _new_x: bool, obj_factor: f64, lambda: &[f64], vals: &mut [f64]) -> bool {
// Hessian of objective:
// d2f/dx1dx1 = 2*x4
// d2f/dx1dx2 = x4 (sym)
// d2f/dx1dx3 = x4 (sym)
// d2f/dx1dx4 = 2*x1+x2+x3 (sym)
// d2f/dx2dx4 = x1 (sym)
// d2f/dx3dx4 = x1 (sym)
// all others = 0
// Hessian of g1 = x1*x2*x3*x4:
// d2g1/dx1dx2 = x3*x4
// d2g1/dx1dx3 = x2*x4
// d2g1/dx1dx4 = x2*x3
// d2g1/dx2dx3 = x1*x4
// d2g1/dx2dx4 = x1*x3
// d2g1/dx3dx4 = x1*x2
// diagonal = 0
// Hessian of g2 = x1^2+x2^2+x3^2+x4^2:
// d2g2/dxi dxi = 2, off-diagonal = 0
// Index 0: (0,0) = obj_factor*2*x4 + lambda[1]*2
vals[0] = obj_factor * 2.0 * x[3] + lambda[1] * 2.0;
// Index 1: (1,0) = obj_factor*x4 + lambda[0]*x3*x4
vals[1] = obj_factor * x[3] + lambda[0] * x[2] * x[3];
// Index 2: (1,1) = lambda[1]*2
vals[2] = lambda[1] * 2.0;
// Index 3: (2,0) = obj_factor*x4 + lambda[0]*x2*x4
vals[3] = obj_factor * x[3] + lambda[0] * x[1] * x[3];
// Index 4: (2,1) = lambda[0]*x1*x4
vals[4] = lambda[0] * x[0] * x[3];
// Index 5: (2,2) = lambda[1]*2
vals[5] = lambda[1] * 2.0;
// Index 6: (3,0) = obj_factor*(2*x1+x2+x3) + lambda[0]*x2*x3
vals[6] = obj_factor * (2.0 * x[0] + x[1] + x[2]) + lambda[0] * x[1] * x[2];
// Index 7: (3,1) = obj_factor*x1 + lambda[0]*x1*x3
vals[7] = obj_factor * x[0] + lambda[0] * x[0] * x[2];
// Index 8: (3,2) = obj_factor*x1 + lambda[0]*x1*x2
vals[8] = obj_factor * x[0] + lambda[0] * x[0] * x[1];
// Index 9: (3,3) = lambda[1]*2
vals[9] = lambda[1] * 2.0;
true
}
}
fn main() {
env_logger::init();
let problem = Hs071;
let options = SolverOptions {
max_iter: 500,
tol: 1e-8,
..SolverOptions::default()
};
println!("Solving HS071...");
println!(" Start: (1.0, 5.0, 5.0, 1.0)");
println!();
let result = ripopt::solve(&problem, &options);
println!("Status: {:?}", result.status);
println!("Iterations: {}", result.iterations);
println!("Objective: {:.10}", result.objective);
println!(
"Solution: x = ({:.6}, {:.6}, {:.6}, {:.6})",
result.x[0], result.x[1], result.x[2], result.x[3]
);
println!(
"Constraint multipliers: ({:.6}, {:.6})",
result.constraint_multipliers[0], result.constraint_multipliers[1]
);
println!(
"Constraint values: g1 = {:.6}, g2 = {:.6}",
result.constraint_values[0], result.constraint_values[1]
);
assert_eq!(result.status, SolveStatus::Optimal);
}