Fliege–Drummond–Svaiter (FDS)

This family is represented by the FDS constructor. The scalable test problem was introduced in "Newton's Method for Multiobjective Optimization" [10].

Overview

FDS(; nvar) requires nvar >= 1 and has nobj = 3; the default is n = 5. Its componentwise bounds are shown below.

The objective functions in equations (8.2)–(8.4) of Fliege, Drummond, and Svaiter are presented without explicit variable bounds. In their numerical experiments, the authors sampled starting points from $[-2,2]^n$ and used the same interval as box constraints. The FDS constructor follows this experimental specification.

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

Mathematical formulation

The formulas below describe the objective functions implemented by the constructor. Let $F:\mathbb{R}^n \to \mathbb{R}^3$ be defined by $F(x)=(f_1(x),f_2(x),f_3(x))$, where $n \geq 1$ and $x=(x_1,\ldots,x_n)\in[-2,2]^n$. The objectives are

\[\begin{aligned} f_1(x) &= \frac{1}{n^2}\sum_{i=1}^{n} i(x_i-i)^4,\\ f_2(x) &= \exp\left(\frac{1}{n}\sum_{i=1}^{n}x_i\right) + \lVert x\rVert_2^2,\\ f_3(x) &= \frac{1}{n(n+1)}\sum_{i=1}^{n} i(n-i+1)\exp(-x_i). \end{aligned}\]

Usage

julia> using MOProblems

julia> using Random

julia> prob = FDS();

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))
(3, (3, 5))

Constructor reference

FDS(; nvar::Int = 5)