Binh–Korn (BK)

This family is represented by the BK1 constructor. The problem is drawn from “An evolution strategy for the multiobjective optimization” [3].

Overview

The constructor has nvar = 2 and nobj = 2. The componentwise bounds are shown below.

Analytical Jacobians are registered for the constructor. Hessians are not registered. The catalog metadata classifies both objectives in BK1 as strictly convex (: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 $x = (x_1,x_2) \in \mathbb{R}^2$.

BK1

The objectives are

\[\begin{aligned} f_1(x) &= x_1^2 + x_2^2,\\ f_2(x) &= (x_1 - 5)^2 + (x_2 - 5)^2. \end{aligned}\]

Usage

julia> using MOProblems

julia> using Random

julia> prob = BK1();

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

BK1()