Zitzler–Deb–Thiele (ZDT)
This family comprises ZDT1, ZDT2, ZDT3, ZDT4, and ZDT6. These are the real-valued test functions $\mathcal{T}_1$–$\mathcal{T}_4$ and $\mathcal{T}_6$ of Definition 4, Equations (7)–(10) and (12), of “Comparison of Multiobjective Evolutionary Algorithms: Empirical Results” [19]. The source's remaining test function, $\mathcal{T}_5$ (Equation (11)), is defined over binary strings rather than real variables, and the package provides no constructor for it.
The formulations of ZDT1, ZDT2, and ZDT3 also appear as $F_2$, $F_3$, and $F_5$ of Jin, Olhofer, and Sendhoff [15], which attributes them to this source. See the Jin–Olhofer–Sendhoff (JOS) page.
Overview
Every constructor takes the number of variables nvar as its only parameter and requires nvar >= 2, giving nobj = 2. The default dimensions and componentwise bounds are shown below.
| Problem | Source function | Default nvar | nobj | Lower bounds | Upper bounds |
|---|---|---|---|---|---|
ZDT1 | $\mathcal{T}_1$ | 30 | 2 | $[0,\ldots,0]$ | $[1,\ldots,1]$ |
ZDT2 | $\mathcal{T}_2$ | 30 | 2 | $[0,\ldots,0]$ | $[1,\ldots,1]$ |
ZDT3 | $\mathcal{T}_3$ | 30 | 2 | $[0,\ldots,0]$ | $[1,\ldots,1]$ |
ZDT4 | $\mathcal{T}_4$ | 10 | 2 | $[0,-5,\ldots,-5]$ | $[1,5,\ldots,5]$ |
ZDT6 | $\mathcal{T}_6$ | 10 | 2 | $[0,\ldots,0]$ | $[1,\ldots,1]$ |
The default dimensions are the ones used by the source for each function.
Analytical Jacobians are registered for all five constructors. Hessians are not registered. The catalog metadata classifies every objective in ZDT1 through ZDT6 as not strictly convex (:not_strictly_convex).
Within the registered boxes, the full Jacobian is undefined on the following boundary sets, although both objective values remain defined:
| Problems | Boundary set where the full Jacobian is undefined |
|---|---|
ZDT1, ZDT3, ZDT4 | The entire face $x_1=0$ |
ZDT6 | The segment $x_2=\cdots=x_n=0$, with $x_1\in[0,1]$ |
For ZDT1 and ZDT4, $\partial f_2/\partial x_1$ equals $-\tfrac{1}{2}\sqrt{g(x)/x_1}$ for $x_1>0$ and tends to $-\infty$ as $x_1\downarrow0$, since $g(x)\geq1$. The additional trigonometric terms in ZDT3 do not cancel this divergence. On the face $x_1=0$, the partial derivatives of $f_2$ with respect to $x_2,\ldots,x_n$ remain defined and equal the corresponding partial derivatives of $g$.
For ZDT6, let $\mu=\frac{1}{n-1}\sum_{i=2}^{n}x_i$. For $\mu>0$, the partial derivatives of $f_2$ with respect to $x_i$, $i\geq2$, equal $\left(1+(f_1/g)^2\right)\frac{9}{4(n-1)}\mu^{-3/4}$ and tend to $+\infty$ as $\mu\downarrow0$. The positive factor $1+(f_1/g)^2$ prevents cancellation. Within the box, $\mu=0$ is equivalent to $x_2=\cdots=x_n=0$; the partial derivative with respect to $x_1$ remains defined there and equals $-2f_1f_1'$.
Thus, $f_2$ has no finite full gradient on these sets, even when considering one-sided variations within the box. Evaluating Jacobian row 2 there throws a DomainError, whereas row 1 remains available through eval_jacobian_row(prob, x, 1). A full eval_jacobian(prob, x) call also throws. ZDT2 has a finite Jacobian throughout its registered box.
No positive tolerance is imposed: at all other points within the boxes, the Jacobians are evaluated by the analytical formulas, subject to the range and precision of the input floating-point type.
Mathematical formulations
All five problems share the structure
\[F(x) = (f_1(x), f_2(x)), \qquad f_2(x) = g(x)\,h(f_1(x), g(x)),\]
where $F:\Omega\to\mathbb{R}^2$ with $n\geq2$ and $\Omega\subset\mathbb{R}^n$ is the registered box listed above. The function $f_1$ depends only on $x_1$, and $g$ depends only on $x_2,\ldots,x_n$. The constructors differ in $f_1$, $h$, and $g$.
ZDT1
The objectives are
\[\begin{aligned} f_1(x) &= x_1,\\ f_2(x) &= g(x)\left(1-\sqrt{\frac{f_1(x)}{g(x)}}\right), \end{aligned}\]
where
\[g(x) = 1 + \frac{9}{n-1}\sum_{i=2}^{n}x_i.\]
ZDT2
The objectives are
\[\begin{aligned} f_1(x) &= x_1,\\ f_2(x) &= g(x)\left(1-\left(\frac{f_1(x)}{g(x)}\right)^{2}\right), \end{aligned}\]
where
\[g(x) = 1 + \frac{9}{n-1}\sum_{i=2}^{n}x_i.\]
ZDT3
The objectives are
\[\begin{aligned} f_1(x) &= x_1,\\ f_2(x) &= g(x)\left(1-\sqrt{\frac{f_1(x)}{g(x)}} -\frac{f_1(x)}{g(x)}\sin\left(10\pi f_1(x)\right)\right), \end{aligned}\]
where
\[g(x) = 1 + \frac{9}{n-1}\sum_{i=2}^{n}x_i.\]
ZDT4
The objectives are
\[\begin{aligned} f_1(x) &= x_1,\\ f_2(x) &= g(x)\left(1-\sqrt{\frac{f_1(x)}{g(x)}}\right), \end{aligned}\]
where
\[g(x) = 1 + 10(n-1) + \sum_{i=2}^{n} \left(x_i^{2}-10\cos\left(4\pi x_i\right)\right).\]
ZDT6
The objectives are
\[\begin{aligned} f_1(x) &= 1-\exp(-4x_1)\sin^{6}\left(6\pi x_1\right),\\ f_2(x) &= g(x)\left(1-\left(\frac{f_1(x)}{g(x)}\right)^{2}\right), \end{aligned}\]
where
\[g(x) = 1 + 9\left(\frac{1}{n-1}\sum_{i=2}^{n}x_i\right)^{0.25}.\]
Usage
julia> using MOProblems
julia> using Random
julia> prob = ZDT1(nvar = 30);
julia> lower, upper = recommended_bounds(prob);
julia> rng = MersenneTwister(1234);
julia> α = rand(rng, prob.nvar);
julia> x = lower .+ α .* (upper .- lower);
julia> values = eval_f(prob, x);
julia> J = eval_jacobian(prob, x);
julia> (length(values), size(J))
(2, (2, 30))Constructor reference
MOProblems.ZDT1 — Function
ZDT1(; nvar::Int = 30)Construct the two-objective ZDT1 problem.
Requires nvar >= 2. Each variable is bounded in [0, 1]. An analytical Jacobian is registered; objective Hessians are not registered.
At x[1] == 0, the second Jacobian row is undefined and its evaluation throws a DomainError. Objective values and the first row remain defined.
MOProblems.ZDT2 — Function
ZDT2(; nvar::Int = 30)Construct the two-objective ZDT2 problem.
Requires nvar >= 2. Each variable is bounded in [0, 1]. An analytical Jacobian is registered; objective Hessians are not registered.
MOProblems.ZDT3 — Function
ZDT3(; nvar::Int = 30)Construct the two-objective ZDT3 problem.
Requires nvar >= 2. Each variable is bounded in [0, 1]. An analytical Jacobian is registered; objective Hessians are not registered.
At x[1] == 0, the second Jacobian row is undefined and its evaluation throws a DomainError. Objective values and the first row remain defined.
MOProblems.ZDT4 — Function
ZDT4(; nvar::Int = 10)Construct the two-objective ZDT4 problem.
Requires nvar >= 2. The first variable is bounded in [0, 1]; the remaining variables are bounded in [-5, 5]. An analytical Jacobian is registered; objective Hessians are not registered.
At x[1] == 0, the second Jacobian row is undefined and its evaluation throws a DomainError. Objective values and the first row remain defined.
MOProblems.ZDT6 — Function
ZDT6(; nvar::Int = 10)Construct the two-objective ZDT6 problem.
Requires nvar >= 2. Each variable is bounded in [0, 1]. An analytical Jacobian is registered; objective Hessians are not registered.
When all variables except the first are zero, the second Jacobian row is undefined and its evaluation throws a DomainError. Objective values and the first row remain defined.