Mao–Hirasawa–Hu–Murata (MHHM)

The MHHM1 and MHHM2 constructors implement the optimization problems used in Simulations 1 and 2, respectively, by Jiangming Mao, K. Hirasawa, Jinlu Hu, and J. Murata in “Genetic symbiosis algorithm for multiobjective optimization problem” [23].

Overview

Both constructors have three objectives and componentwise variable bounds of [0, 1]. Their fixed dimensions are shown below.

Analytical Jacobians are registered for both constructors. Hessians are not registered. The catalog metadata classifies every objective in MHHM1 and MHHM2 as strictly convex (:strictly_convex).

Mathematical formulations

MHHM1

Let $F:[0,1]\to\mathbb{R}^3$ be defined by $F(x)=(f_1(x),f_2(x),f_3(x))$. The objectives are

\[\begin{aligned} f_1(x) &= (x_1-0.8)^2,\\ f_2(x) &= (x_1-0.85)^2,\\ f_3(x) &= (x_1-0.9)^2. \end{aligned}\]

MHHM2

Let $F:[0,1]^2\to\mathbb{R}^3$ be defined by $F(x)=(f_1(x),f_2(x),f_3(x))$. The objectives are

\[\begin{aligned} f_1(x) &= (x_1-0.8)^2+(x_2-0.6)^2,\\ f_2(x) &= (x_1-0.85)^2+(x_2-0.7)^2,\\ f_3(x) &= (x_1-0.9)^2+(x_2-0.6)^2. \end{aligned}\]

Usage

julia> using MOProblems

julia> using Random

julia> prob = MHHM2();

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))
(3, (3, 2))

Constructor reference

MHHM1()

MHHM2()