Ishibuchi–Murata (IM)

This family is represented by the IM1 constructor. The problem is drawn from “A multi-objective genetic local search algorithm and its application to flowshop scheduling” [14].

Overview

IM1 has nvar = 2 and nobj = 2. Its componentwise bounds are shown below.

An analytical Jacobian is registered. Hessians are not registered. The catalog metadata classifies both objectives as not strictly convex (:not_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 $x=(x_1,x_2) \in \mathbb{R}^2$. The objectives are

\[\begin{aligned} f_1(x) &= 2\sqrt{x_1},\\ f_2(x) &= x_1(1-x_2)+5. \end{aligned}\]

Usage

julia> using MOProblems

julia> using Random

julia> prob = IM1();

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

IM1()