Miglierina–Molho–Recchioni (MMR)

This family comprises the MMR1 through MMR4 constructors. They implement, in order, Tests 1–4 from “Box-constrained multi-objective optimization: A gradient-like method without ‘a priori’ scalarization” [25].

Overview

All four constructors have two objectives and fixed dimensions. Their bounds are shown below.

Analytical Jacobians are registered for all four constructors. Hessians are not registered. The catalog metadata classifies every objective in MMR1 through MMR4 as not strictly convex (:not_strictly_convex).

Mathematical formulations

The formulas below describe the objective functions implemented by the constructors. For each problem, let $F:\mathbb{R}^n \to \mathbb{R}^2$ be defined by $F(x)=(f_1(x),f_2(x))$.

MMR1

For $x \in [0.1,1]\times[0,1]$, the objectives are

\[\begin{aligned} f_1(x) &= x_1,\\ f_2(x) &= \frac{\psi(x_2)}{x_1}, \end{aligned}\]

where

\[\psi(x_2) = 2 - 0.8\exp\left[-\left(\frac{x_2-0.6}{0.4}\right)^2\right] - \exp\left[-\left(\frac{x_2-0.2}{0.04}\right)^2\right].\]

MMR2

For $x \in [0,1]^2$, the objectives are

\[\begin{aligned} f_1(x) &= x_1,\\ f_2(x) &= \psi(x_2)r(x_1,x_2), \end{aligned}\]

where

\[\begin{aligned} r(x_1,x_2) &= 1-\left(\frac{x_1}{\psi(x_2)}\right)^\alpha -\frac{x_1}{\psi(x_2)}\sin(2\pi q x_1),\\ \psi(x_2) &= 1+10x_2,\\ \alpha &= 2,\\ q &= 4. \end{aligned}\]

MMR3

For $x \in [-1,1]^2$, the objectives are

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

MMR4

For $x \in [0,4]^3$, the objectives are

\[\begin{aligned} f_1(x) &= x_1-2x_2-x_3-\frac{36}{2x_1+x_2+2x_3+1},\\ f_2(x) &= -3x_1+x_2-x_3. \end{aligned}\]

Usage

julia> using MOProblems

julia> using Random

julia> prob = MMR1();

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

MMR1()

MMR2()

MMR3()

MMR4()