Preuss–Naujoks–Rudolph (PNR)

This family is represented by the PNR constructor. It implements Case 1 of the TWO-ON-ONE test problem introduced in “Pareto Set and EMOA Behavior for Simple Multimodal Multiobjective Functions” [26].

Overview

PNR has fixed dimensions and is unconstrained.

The recommended box provides a practical finite search region when an algorithm requires one. An analytical Jacobian is registered. Hessians are not registered. The catalog metadata classifies the first objective as not strictly convex (:not_strictly_convex) and the second as strictly convex (:strictly_convex).

Mathematical formulation

The source parameterization is

\[\begin{aligned} f_1(x) &= x_1^4+x_2^4-x_1^2+x_2^2-cx_1x_2+dx_1+20,\\ f_2(x) &= (x_1-k)^2+(x_2-l)^2. \end{aligned}\]

Case 1 sets $c=10$ and $d=k=l=0$. Therefore, the constructor implements $F:\mathbb{R}^2\to\mathbb{R}^2$, with $F(x)=(f_1(x),f_2(x))$ and

\[\begin{aligned} f_1(x) &= x_1^4+x_2^4-x_1^2+x_2^2-10x_1x_2+20,\\ f_2(x) &= x_1^2+x_2^2. \end{aligned}\]

Usage

julia> using MOProblems

julia> using Random

julia> prob = PNR();

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, 2))

Constructor reference

PNR()