Farina (Far)
This family is represented by the Far1 constructor. The analytical test case originates in "A neural network based generalized response surface multiobjective evolutionary algorithm" [8]. The name Far1 and the corrected formulation implemented by MOProblems.jl follow the catalog of Huband et al. [9].
Farina presents the analytical test case in Equation (4) but does not name it Far1. Huband et al. assign that catalog name and explicitly identify apparent typographical errors in the equation as printed, providing a corrected formulation in Table XVI. MOProblems.jl implements the Huband et al. formulation, not a literal transcription of Equation (4).
In particular, the fourth exponential term of $f_1$ is implemented as a bounded exponential centered at $(0.6,-0.6)$, with the negative sum of the two squared offsets in its exponent. The fifth exponential term of $f_2$ has a positive coefficient and likewise uses the negative sum of the squared offsets from $(-0.4,-0.8)$. These are the two corrected terms reported by Huband et al. and used by the constructor.
Overview
The constructor has nvar = 2 and nobj = 2. It has no general constraints; the componentwise variable bounds are shown below.
| Problem | nvar | nobj | Lower bound | Upper bound |
|---|---|---|---|---|
Far1 | 2 | 2 | -1.0 | 1.0 |
An analytical Jacobian is registered. Hessians are not registered. The catalog metadata classifies both objectives as not strictly convex (:not_strictly_convex).
Mathematical formulations
The formulas below describe the objective functions implemented by the constructor. Let $F:\mathbb{R}^2 \to \mathbb{R}^2$ be defined by $F(x)=(f_1(x),f_2(x))$, where $x=(x_1,x_2)\in[-1,1]^2$.
Far1
The objectives are
\[\begin{aligned} f_1(x) ={}& -2\exp\left(15\left(-(x_1-0.1)^2-x_2^2\right)\right)\\ &-\exp\left(20\left(-(x_1-0.6)^2-(x_2-0.6)^2\right)\right)\\ &+\exp\left(20\left(-(x_1+0.6)^2-(x_2-0.6)^2\right)\right)\\ &+\exp\left(20\left(-(x_1-0.6)^2-(x_2+0.6)^2\right)\right)\\ &+\exp\left(20\left(-(x_1+0.6)^2-(x_2+0.6)^2\right)\right),\\[0.5em] f_2(x) ={}& 2\exp\left(20\left(-x_1^2-x_2^2\right)\right)\\ &+\exp\left(20\left(-(x_1-0.4)^2-(x_2-0.6)^2\right)\right)\\ &-\exp\left(20\left(-(x_1+0.5)^2-(x_2-0.7)^2\right)\right)\\ &-\exp\left(20\left(-(x_1-0.5)^2-(x_2+0.7)^2\right)\right)\\ &+\exp\left(20\left(-(x_1+0.4)^2-(x_2+0.8)^2\right)\right). \end{aligned}\]
Usage
using MOProblems
prob = Far1()
x = [0.0, 0.0]
values = eval_f(prob, x)
J = eval_jacobian(prob, x)Constructor reference
MOProblems.Far1 — Function
Far1()Construct the fixed-dimension Far1 benchmark with two variables and two objectives. Each variable is bounded by [-1, 1]. An analytical Jacobian is registered; Hessians are not registered.
The constructor follows the corrected Far1 transcription cataloged by Huband et al. (2006), rather than a literal transcription of the equation as printed by Farina (2002).