Sefrioui–Perlaux (SP)

This family is represented by the SP1 constructor. It implements the simple mathematical example introduced in Section 2.4 of “Nash genetic algorithms: examples and applications” [31].

Overview

SP1 has fixed dimensions and is unconstrained.

The source presents the objective functions without specifying a domain or a finite search box. Accordingly, SP1 does not register variable bounds. The box $[-100,100]^2$ is recommended only as a practical finite search region when an algorithm requires one; it is not part of the problem definition.

An analytical Jacobian is registered. Hessians are not registered. The catalog metadata classifies both objectives as strictly convex (:strictly_convex).

Mathematical formulation

Let $F:\mathbb{R}^2 \to \mathbb{R}^2$ be defined by $F(x)=(f_1(x),f_2(x))$, where $x=(x_1,x_2)$. The constructor implements

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

Usage

julia> using MOProblems

julia> using Random

julia> prob = SP1();

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

SP1()