Valenzuela-Rendón–Uresti-Charre (VU)

This family is represented by the VU1 and VU2 constructors. The problems are the two test problems of “A nongenerational genetic algorithm for multiobjective optimization” [35], in the explicit form cataloged under the same names in Table XVI of Huband et al. [9], because the original text does not state them in a directly verifiable form.

Overview

VU1 and VU2 have nvar = 2 and nobj = 2. Their componentwise variable bounds are shown below.

An analytical Jacobian is registered for both problems. Hessians are not registered. In VU1, the catalog metadata classifies $f_1$ as not strictly convex (:not_strictly_convex) and $f_2$ as strictly convex (:strictly_convex). In VU2, it classifies both objectives as not strictly convex.

Mathematical formulations

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

VU1

The objectives are

\[\begin{aligned} f_1(x) &= \frac{1}{x_1^2+x_2^2+1},\\ f_2(x) &= x_1^2+3x_2^2+1. \end{aligned}\]

VU2

The objectives are

\[\begin{aligned} f_1(x) &= x_1+x_2+1,\\ f_2(x) &= x_1^2+2x_2-1. \end{aligned}\]

Usage

julia> using MOProblems

julia> using Random

julia> prob = VU1();

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

VU1()

VU2()