Lovison (Lov)

This family comprises Lov1 through Lov6, corresponding to Examples 1–6 in Alberto Lovison's “Singular Continuation: Generating Piecewise Linear Approximations to Pareto Sets via Global Analysis” [18]. Lov6 is Lovison's smooth regularization of the third ZDT problem introduced by Zitzler, Deb, and Thiele [19].

Overview

All constructors have two objectives, fixed dimensions, and analytical Jacobians. Objective Hessians are not registered.

For Lov1 through Lov5, the returned boxes are recommendations of the package developers, not constraints in the problem definitions. They do not imply containment of the Pareto set.

Mathematical formulations

The formulas below describe the minimization objectives implemented by the constructors.

Lov1

For $x=(x_1,x_2)\in\mathbb{R}^2$,

\[\begin{aligned} f_1(x) &= 1.05x_1^2+0.98x_2^2,\\ f_2(x) &= 0.99(x_1-3)^2+1.03(x_2-2.5)^2. \end{aligned}\]

Lov2

For $x=(x_1,x_2)\in\mathbb{R}^2$ with $x_1\ne-1$,

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

Lov3

For $x=(x_1,x_2)\in\mathbb{R}^2$,

\[\begin{aligned} f_1(x) &= x_1^2+x_2^2,\\ f_2(x) &= (x_1-6)^2-(x_2+0.3)^2. \end{aligned}\]

Lov4

For $x=(x_1,x_2)\in\mathbb{R}^2$,

\[\begin{aligned} f_1(x)={}&x_1^2+x_2^2\\ &+4\left[ \exp\left(-(x_1+2)^2-x_2^2\right) +\exp\left(-(x_1-2)^2-x_2^2\right) \right],\\ f_2(x)={}&(x_1-6)^2+(x_2+0.5)^2. \end{aligned}\]

Lov5

For $x=(x_1,x_2,x_3)\in\mathbb{R}^3$, let

\[p_0= \begin{pmatrix} 0\\ 0.15\\ 0 \end{pmatrix}, \qquad p_1= \begin{pmatrix} 0\\ -1.1\\ 0 \end{pmatrix}.\]

Define

\[M= \begin{pmatrix} -1.0 & -0.03 & 0.011\\ -0.03 & -1.0 & 0.07\\ 0.011 & 0.07 & -1.01 \end{pmatrix}.\]

The auxiliary function in Equation (4.5) is

\[g(x_1,x_2,x_3;M,p,\sigma) = \sqrt{\frac{2\pi}{\sigma}} \exp\left( \frac{ \left( \begin{pmatrix} x_1\\ x_2\\ x_3 \end{pmatrix} -p \right)^\top M \left( \begin{pmatrix} x_1\\ x_2\\ x_3 \end{pmatrix} -p \right) }{\sigma^2} \right).\]

Using h for the function denoted by $f$ in the article, define

\[h(x_1,x_2,x_3) = g(x_1,x_2,x_3;M,p_0,0.35) +g(x_1,x_2,0.5x_3;M,p_1,3.0).\]

Lovison writes the two objectives of Example 5 for maximization. The implementation negates them to follow the package's minimization convention:

\[\begin{aligned} f_1(x_1,x_2,x_3) &=-\frac{\sqrt{2}}{2}x_1 -\frac{\sqrt{2}}{2}h(x_1,x_2,x_3),\\ f_2(x_1,x_2,x_3) &=\frac{\sqrt{2}}{2}x_1 -\frac{\sqrt{2}}{2}h(x_1,x_2,x_3). \end{aligned}\]

Lov6

For $x_1\in[0.1,0.425]$ and $x_i\in[-0.16,0.16]$ for $i=2,\ldots,6$,

\[\begin{aligned} f_1(x)&=x_1,\\ f_2(x)&=1-\sqrt{x_1}-x_1\sin(10\pi x_1) +\sum_{i=2}^{6}x_i^2. \end{aligned}\]

Usage

julia> using MOProblems

julia> using Random

julia> prob = Lov5();

julia> lower, upper = recommended_bounds(prob);

julia> rng = MersenneTwister(1234);

julia> α = rand(rng, prob.nvar);

julia> x = lower .+ α .* (upper .- lower);

julia> @assert isnothing(prob.bounds)

julia> values = eval_f(prob, x);

julia> J = eval_jacobian(prob, x);

julia> (length(values), size(J))
(2, (2, 3))

Constructor reference

Lov1()

Lov2()

Lov3()

Lov4()

Lov5()

Lov6()