Laumanns–Thiele–Deb–Zitzler (LTDZ)
This family is represented by the canonical LTDZ1 constructor. The test problem originates in Equation (8) of “Combining Convergence and Diversity in Evolutionary Multiobjective Optimization” by Laumanns, Thiele, Deb, and Zitzler [20].
The public LTDZ() constructor is an alias for LTDZ1(). Fliege, Drummond, and Svaiter refer to the benchmark as LTDZ [10], whereas Huband et al. assign it the catalog name LTDZ1 [9]. MOProblems.jl uses LTDZ1 as the canonical catalog identity while supporting both constructor names.
Equation (8) of Laumanns et al. is formulated for maximization and prints $f_3$ with $\cos(\pi x_1/2)\sin(\pi x_1/2)$. Huband et al. classify LTDZ1 as apparently containing a typographical error and list $f_3$ with only $\sin(\pi x_1/2)$ in Table XVI. LTDZ1 adopts the latter expression and negates the objectives for minimization.
Overview
LTDZ1 has nvar = 3 and nobj = 3. The alias LTDZ() constructs the same canonical problem and therefore also reports prob.name == "LTDZ1". Its componentwise bounds are shown below.
| Problem | nvar | nobj | Lower bound | Upper bound |
|---|---|---|---|---|
LTDZ1 | 3 | 3 | 0.0 | 1.0 |
An analytical Jacobian is registered. Hessians are not registered. The catalog metadata classifies all three objectives as not strictly convex (:not_strictly_convex).
Mathematical formulation
The formulas below describe the minimization objectives implemented by both constructors. Let $F:\mathbb{R}^3\to\mathbb{R}^3$ be defined by $F(x)=(f_1(x),f_2(x),f_3(x))$, where $x=(x_1,x_2,x_3)\in[0,1]^3$. The objectives are
\[\begin{aligned} f_1(x) &= -3+(1+x_3)\cos\left(\frac{\pi x_1}{2}\right) \cos\left(\frac{\pi x_2}{2}\right),\\ f_2(x) &= -3+(1+x_3)\cos\left(\frac{\pi x_1}{2}\right) \sin\left(\frac{\pi x_2}{2}\right),\\ f_3(x) &= -3+(1+x_3)\sin\left(\frac{\pi x_1}{2}\right). \end{aligned}\]
Usage
using MOProblems
prob = LTDZ1()
x = [0.5, 0.5, 0.0]
values = eval_f(prob, x)
huband_maximization_values = -values
J = eval_jacobian(prob, x)
same_prob = LTDZ()Constructor reference
MOProblems.LTDZ1 — Function
LTDZ1()Construct the fixed-dimension LTDZ1 benchmark with three variables and three objectives. Each variable is bounded by [0, 1]. An analytical Jacobian is registered; objective Hessians are not registered.
The constructor uses the third objective given for LTDZ1 in Table XVI of Huband et al. (2006), rather than a literal transcription of Equation (8) in Laumanns et al. (2002). It negates all three maximization objectives to follow the package's minimization convention.
MOProblems.LTDZ — Function
LTDZ()Alias for LTDZ1(). Fliege, Drummond, and Svaiter (2009) use the name LTDZ for this benchmark, while Huband et al. (2006) catalog it as LTDZ1.