Fonseca–Fleming (FF)

This family is represented by the FF1 constructor. The test problem is presented in "An Overview of Evolutionary Algorithms in Multiobjective Optimization" [11].

Overview

FF1 has nvar = 2 and nobj = 2. It has no explicit variable bounds.

Fonseca and Fleming present the objective functions without explicit variable bounds. The FF1 constructor follows that formulation.

An analytical Jacobian is registered. Hessians are not registered. The catalog metadata classifies both objectives as not strictly convex (:not_strictly_convex).

Mathematical formulation

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 $x=(x_1,x_2)\in\mathbb{R}^2$. The objectives are

\[\begin{aligned} f_1(x) &= 1-\exp\left(-(x_1-1)^2-(x_2+1)^2\right),\\ f_2(x) &= 1-\exp\left(-(x_1+1)^2-(x_2-1)^2\right). \end{aligned}\]

Usage

using MOProblems

prob = FF1()
x = [0.0, 0.0]

values = eval_f(prob, x)
J = eval_jacobian(prob, x)

Constructor reference

FF1()