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, and the FF1 constructor follows that formulation. For experiments that need a bounded region, recommended_bounds returns $[-1,1]^2$, a box recommended by the package developers; prob.bounds remains nothing.

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

julia> using MOProblems

julia> using Random

julia> prob = FF1();

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

FF1()