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 six constructors have fixed dimensions, two objectives, and registered analytical Jacobians. Lov1 through Lov5 have no registered variable bounds. Lov6 retains the bounds that are part of Equation (4.6). Hessians are not registered.

The catalog strict-convexity classifications are:

Suggested experimental boxes

Examples 1–5 do not include box constraints in the source formulation, and the implementation does not apply the intervals below. They are retained only as suggested finite regions for numerical experiments:

Each interval applies componentwise. These boxes are practical exploration windows, not part of the mathematical definitions and not claims about the location of complete Pareto sets. Lovison discusses multiple or local branches in some individual examples, but does not state a general repetition property for Pareto structures outside these boxes [18].

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

using MOProblems

prob = Lov5()
x = zeros(prob.nvar)

@assert isnothing(prob.bounds)
values = eval_f(prob, x)
J = eval_jacobian(prob, x)

Constructor reference

Lov1()

Lov2()

Lov3()

Lov4()

Lov5()

Lov6()