Hillermeier (Hil)

This family is represented by the Hil1 constructor. It implements the academic example presented as Example 4.1 in C. Hillermeier's “Generalized Homotopy Approach to Multiobjective Optimization” [12].

Overview

Hil1 has nvar = 2 and nobj = 2. It has no registered variable bounds.

The source problem is defined for $x \in \mathbb{R}^2$. Both objectives are 1-periodic in each variable, so the square $[0,1]^2$ covers one complete period in each variable and is sufficient to represent the full image set $F(\mathbb{R}^2)$. recommended_bounds returns this square; it is a sampling window, not a variable-bound specification.

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

Mathematical formulation

Let $F:\mathbb{R}^2 \to \mathbb{R}^2$ be defined by $F(x)=(f_1(x),f_2(x))$. The objectives are

\[\begin{aligned} f_1(x) &= \cos(a(x))b(x),\\ f_2(x) &= \sin(a(x))b(x), \end{aligned}\]

where

\[\begin{aligned} a(x) &= \frac{2\pi}{360}\left(45 +40\sin(2\pi x_1)+25\sin(2\pi x_2)\right),\\ b(x) &= 1+0.5\cos(2\pi x_1). \end{aligned}\]

Usage

julia> using MOProblems

julia> using Random

julia> prob = Hil1();

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

Hil1()