Socha–Kisiel-Dorohinicki (SK)

This family is represented by the SK1 and SK2 constructors. Socha and Kisiel-Dorohinicki used both problems as test cases in "Agent-based evolutionary multiobjective optimisation" [29]. Huband et al. later cataloged them under the SK1 and SK2 names in Table XVI [9].

Overview

Both problems have fixed dimensions and are unconstrained.

An analytical Jacobian is registered for both problems. Hessians are not registered. The catalog metadata classifies both objectives of SK1 as not strictly convex (:not_strictly_convex); for SK2 it classifies $f_1$ as strictly convex (:strictly_convex) and $f_2$ as not strictly convex (:not_strictly_convex).

Mathematical formulations

The formulas below describe the minimization objectives implemented by the constructors, that is, the negatives of the corrected source objectives.

SK1

Let $F:\mathbb{R}\to\mathbb{R}^2$ be the implemented minimization vector $F(x)=(f_1(x),f_2(x))$, where $x = (x_1)$. Its components are the quartic polynomials

\[\begin{aligned} f_1(x) &= x_1^4 + 3x_1^3 - 10x_1^2 - 10x_1 - 10,\\ f_2(x) &= 0.5\,x_1^4 - 2x_1^3 - 10x_1^2 + 10x_1 - 5. \end{aligned}\]

SK2

Let $F:\mathbb{R}^4\to\mathbb{R}^2$ be the implemented minimization vector $F(x)=(f_1(x),f_2(x))$, where $x = (x_1,x_2,x_3,x_4)$. Its components are

\[\begin{aligned} f_1(x) &= (x_1-2)^2 + (x_2+3)^2 + (x_3-5)^2 + (x_4-4)^2 - 5,\\ f_2(x) &= -\frac{\sin(x_1)+\sin(x_2)+\sin(x_3)+\sin(x_4)} {1 + \dfrac{x_1^2+x_2^2+x_3^2+x_4^2}{100}}. \end{aligned}\]

Usage

julia> using MOProblems

julia> using Random

julia> prob = SK1();

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> source_values = -values;  # values in the source maximization convention

julia> J = eval_jacobian(prob, x);

julia> prob2 = SK2();

julia> lower2, upper2 = recommended_bounds(prob2);

julia> α2 = rand(rng, prob2.nvar);

julia> y = lower2 .+ α2 .* (upper2 .- lower2);

julia> values2 = eval_f(prob2, y);

julia> J2 = eval_jacobian(prob2, y);

julia> (length(values), size(J), length(source_values), length(values2), size(J2))
(2, (2, 1), 2, 2, (2, 4))

Constructor reference

SK1()

SK2()