docs(le1): document source formulation and Jacobian domain

Author: Souza-DR

docs/make.jl

@@ -43,6 +43,7 @@ makedocs(
             "IM" => "problems/im.md",
             "JOS" => "problems/jos.md",
             "KW" => "problems/kw.md",
+            "LE" => "problems/le.md",
         ],
         "API Reference" => "api.md",
         "References" => "references.md",

docs/src/problems/index.md

@@ -48,3 +48,5 @@ when available, and registered analytical derivatives when implemented.
 - [Kim–de Weck (KW)](@ref): one fixed-dimension problem from the second
   numerical example of I. Y. Kim and O. L. de Weck [KW2005](@cite), expressed
   in the package's minimization convention.
+- [Lis–Eiben (LE)](@ref): one fixed-dimension problem from Test 1 of J. Lis and
+  A. E. Eiben [LE1997](@cite).

docs/src/problems/le.md

@@ -0,0 +1,62 @@
+# 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” [LE1997](@cite).
+
+## 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`).
+
+!!! warning "Jacobian domain"
+    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
+
+```math
+\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
+
+```julia
+using MOProblems
+
+prob = LE1()
+x = [0.25, 0.25]
+
+values = eval_f(prob, x)
+J = eval_jacobian(prob, x)
+```
+
+## Constructor reference
+
+```@docs
+MOProblems.LE1
+```

docs/src/refs.bib

@@ -186,6 +186,16 @@ @article{KW2005
   url = {https://doi.org/10.1007/s00158-004-0465-1}
 }
 
+@inproceedings{LE1997,
+  author = {Lis, J. and Eiben, A. E.},
+  booktitle = {Proceedings of 1997 IEEE International Conference on Evolutionary Computation (ICEC '97)},
+  title = {A multi-sexual genetic algorithm for multiobjective optimization},
+  year = {1997},
+  pages = {59--64},
+  keywords = {Genetic algorithms;Optimization methods;Biomedical engineering;Computer science;Decision making;Sorting;Constraint optimization;Design optimization;Decision feedback equalizers},
+  doi = {10.1109/ICEC.1997.592269}
+}
+
 @article{ZDT2000,
   author = {Zitzler, Eckart and Deb, Kalyanmoy and Thiele, Lothar},
   journal = {Evolutionary Computation},

src/Meta/LE1.jl

@@ -1,7 +1,7 @@
 LE1_meta = ProblemMeta(
     dimension = FixedDimension(2, 2),
-    name = "LE1",                # Official problem name
-    has_bounds = true,           # Box constraints present
-    has_jacobian = true,         # Analytical Jacobian available
+    name = "LE1",
+    has_bounds = true,
+    has_jacobian = true,
     strict_convexity = [:not_strictly_convex, :not_strictly_convex],
 )