Module constrained 

Constrained test problems with known minima on their constraints’ edges, for Run::tell_constrained. Each gives an Evaluation with constraints written g(x) ≤ 0.

Problem	Constraint	Minimum
constrained::sphere_above	x₀ ≥ 1	1 at x = (1, 0, …, 0)
constrained::tangent	Σ xᵢ ≥ n	n at x = 1
constrained::g06	two circles	(x₀* − 10)³ + (x₁* − 20)³ at their crossing

Constants

G06_X0

g06’s minimum’s first coordinate, x₀* = 14.095.

Functions

g06

Problem g06 of the CEC 2006 constrained benchmark (J. J. Liang et al., “Problem definitions and evaluation criteria for the CEC 2006 special session on constrained real-parameter optimization”, Nanyang Technological University (2006)), two variables: f = (x₀ − 10)³ + (x₁ − 20)³, with g₁ = 100 − (x₀ − 5)² − (x₁ − 5)² ≤ 0 (outside one circle) and g₂ = (x₀ − 6)² + (x₁ − 5)² − 82.81 ≤ 0 (inside another), and bounds 13 ≤ x₀ ≤ 100, 0 ≤ x₁ ≤ 100 for the caller to set. The minimum is where the circles cross: subtracting the two edges gives 2 x₀ − 11 = 17.19, so G06_X0, and g06_x1 below the centers.

g06_x1

g06’s minimum’s second coordinate, x₁* = 5 − √(100 − (x₀* − 5)²).

sphere_above

The sphere Σ xᵢ² with x₀ ≥ 1, as g = 1 − x₀ ≤ 0. Its minimum is 1, at x = (1, 0, …, 0): the constraint binds, as the sphere’s own minimum breaks it.

tangent

The tangent problem: the sphere with Σ xᵢ ≥ n, as g = n − Σ xᵢ ≤ 0, a constraint along no variable’s axis. By symmetry and Lagrange’s condition 2 xᵢ = λ, the minimum is n, at x = 1.