Van Veldhuizen (VV)

This family is represented by the MOP2, MOP3, MOP5, MOP6, and MOP7 constructors. They come from Table 5.3 of the doctoral thesis of Van Veldhuizen [36], a suite that collects multiobjective problems already published elsewhere and renumbers them. The names are positions in that table rather than names given by the authors who created the problems; this package implements five of its seven entries, which is why the numbering has gaps. The formulations follow Table VIII of Huband et al. [9], which reproduces the suite and identifies the source of each problem.

MOP6 is the one entry that was not published as a test problem in its own right: Van Veldhuizen derived it from the problem construction method of Deb [41]. Huband et al. describe that method and the resulting problem in detail.

Overview

MOP2 takes the number of variables n as its only parameter and requires n >= 1; the other four have fixed dimensions. MOP6 takes a shape parameter q, described with its formulation below, which leaves its dimensions unchanged. The registered bounds are the ones given by Van Veldhuizen and reproduced by Huband et al.

The last column reports the catalog metadata; every objective not listed there is classified as :not_strictly_convex. An analytical Jacobian is registered for all five problems, and objective Hessians are not registered.

Mathematical formulations

MOP2

Let $F:\mathbb{R}^n \to \mathbb{R}^2$ be defined by $F(x)=(f_1(x),f_2(x))$, where $n$ is the number of variables. The objectives are

\[\begin{aligned} f_1(x) &= 1-\exp\!\left(-\sum_{i=1}^{n}\left(x_i-\tfrac{1}{\sqrt{n}}\right)^2\right),\\ f_2(x) &= 1-\exp\!\left(-\sum_{i=1}^{n}\left(x_i+\tfrac{1}{\sqrt{n}}\right)^2\right). \end{aligned}\]

MOP3

Let $F:\mathbb{R}^2 \to \mathbb{R}^2$ be defined by $F(x)=(f_1(x),f_2(x))$. The objectives are

\[\begin{aligned} f_1(x) &= 1+\left(A_1-B_1(x)\right)^2+\left(A_2-B_2(x)\right)^2,\\ f_2(x) &= (x_1+3)^2+(x_2+1)^2, \end{aligned}\]

where the constants $A_1$, $A_2$ and the functions $B_1$, $B_2$ are

\[\begin{aligned} A_1 &= 0.5\sin 1-2\cos 1+\sin 2-1.5\cos 2,\\ A_2 &= 1.5\sin 1-\cos 1+2\sin 2-0.5\cos 2,\\ B_1(x) &= 0.5\sin x_1-2\cos x_1+\sin x_2-1.5\cos x_2,\\ B_2(x) &= 1.5\sin x_1-\cos x_1+2\sin x_2-0.5\cos x_2. \end{aligned}\]

MOP5

Let $F:\mathbb{R}^2 \to \mathbb{R}^3$ be defined by $F(x)=(f_1(x),f_2(x),f_3(x))$. The objectives are

\[\begin{aligned} f_1(x) &= 0.5\left(x_1^2+x_2^2\right)+\sin\!\left(x_1^2+x_2^2\right),\\ f_2(x) &= \frac{(3x_1-2x_2+4)^2}{8}+\frac{(x_1-x_2+1)^2}{27}+15,\\ f_3(x) &= \frac{1}{x_1^2+x_2^2+1}-1.1\exp\!\left(-x_1^2-x_2^2\right). \end{aligned}\]

MOP6

Let $F:\mathbb{R}^2 \to \mathbb{R}^2$ be defined by $F(x)=(f_1(x),f_2(x))$. The objectives are

\[\begin{aligned} f_1(x) &= x_1,\\ f_2(x) &= g(x)\left(1-\left(\frac{x_1}{g(x)}\right)^{2}-\frac{x_1}{g(x)}\sin\!\left(2\pi q x_1\right)\right), \end{aligned}\]

where

\[g(x) = 1+10x_2.\]

The frequency $q$ is the constructor parameter, an integer of at least 1 whose default is the value $q = 4$ used by Van Veldhuizen. It sets how many disconnected pieces the Pareto front has, which is the respect in which the source describes this problem as scalable.

MOP7

Let $F:\mathbb{R}^2 \to \mathbb{R}^3$ be defined by $F(x)=(f_1(x),f_2(x),f_3(x))$. The objectives are

\[\begin{aligned} f_1(x) &= \frac{(x_1-2)^2}{2}+\frac{(x_2+1)^2}{13}+3,\\ f_2(x) &= \frac{(x_1+x_2-3)^2}{36}+\frac{(-x_1+x_2+2)^2}{8}-17,\\ f_3(x) &= \frac{(x_1+2x_2-1)^2}{175}+\frac{(-x_1+2x_2)^2}{17}-13. \end{aligned}\]

Usage

julia> using MOProblems

julia> using Random

julia> prob = MOP5();

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))
(3, (3, 2))

Constructor reference

MOP2(; nvar::Int = 3)

MOP3()

MOP5()

MOP6(; q::Int = 4)

MOP7()