Lis–Eiben (LE)
This family is represented by the LE1 constructor. It implements Test 1 from J. Lis and A. E. Eiben's “A multi-sexual genetic algorithm for multiobjective optimization” [17].
Overview
LE1 has nvar = 2 and nobj = 2. The componentwise bounds are shown below.
| Problem | nvar | nobj | Lower bound | Upper bound |
|---|---|---|---|---|
LE1 | 2 | 2 | -5.0 | 10.0 |
An analytical Jacobian is registered. Hessians are not registered. The catalog metadata classifies both objectives as not strictly convex (:not_strictly_convex).
Both objective values are defined throughout $[-5,10]^2$. The first objective is not differentiable at $(0,0)$, and the second is not differentiable at $(0.5,0.5)$. Consequently, Jacobian row 1 throws a DomainError at $(0,0)$, whereas row 2 throws a DomainError at $(0.5,0.5)$. The other row remains available at each point through eval_jacobian_row.
A full eval_jacobian(prob, x) call throws at either singular point. No positive tolerance is imposed: all other points are evaluated by the analytical formulas, subject to the range and precision of the input floating-point type.
Mathematical formulation
The formulas below describe the objective functions implemented by the constructor. Let $F:[-5,10]^2\to\mathbb{R}^2$ be defined by $F(x)=(f_1(x),f_2(x))$. The objectives are
\[\begin{aligned} f_1(x) &= \left(x_1^2+x_2^2\right)^{1/8},\\ f_2(x) &= \left((x_1-\tfrac{1}{2})^2 +(x_2-\tfrac{1}{2})^2\right)^{1/4}. \end{aligned}\]
Usage
using MOProblems
prob = LE1()
x = [0.25, 0.25]
values = eval_f(prob, x)
J = eval_jacobian(prob, x)Constructor reference
MOProblems.LE1 — Function
LE1()Construct the fixed two-variable, two-objective LE1 problem.
The variables are bounded in [-5, 10]^2. An analytical Jacobian is registered; objective Hessians are not registered. Both objective values are defined throughout the box, but the first Jacobian row is undefined at x == [0, 0] and the second at x == [0.5, 0.5]. Evaluating an undefined row throws a DomainError.