Kim–de Weck (KW)

This family is represented by the KW2 constructor. It provides a minimization-equivalent implementation of the second numerical example in I. Y. Kim and O. L. de Weck's “Adaptive weighted-sum method for bi-objective optimization: Pareto front generation” [16].

Overview

KW2 has nvar = 2 and nobj = 2. Its componentwise variable bounds are shown below.

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 the implemented minimization vector, $F(x)=(f_1(x),f_2(x))=-J(x)$, where $x=(x_1,x_2)\in[-3,3]^2$. Its components are

\[\begin{aligned} f_1(x) ={}& -3(1-x_1)^2\exp\left(-x_1^2-(x_2+1)^2\right)\\ &+10\left(\frac{x_1}{5}-x_1^3-x_2^5\right) \exp\left(-x_1^2-x_2^2\right)\\ &+3\exp\left(-(x_1+2)^2-x_2^2\right) -\frac{1}{2}(2x_1+x_2),\\ f_2(x) ={}& -3(1+x_2)^2\exp\left(-x_2^2-(1-x_1)^2\right)\\ &+10\left(-\frac{x_2}{5}+x_2^3+x_1^5\right) \exp\left(-x_1^2-x_2^2\right)\\ &+3\exp\left(-(2-x_2)^2-x_1^2\right). \end{aligned}\]

Usage

using MOProblems

prob = KW2()
x = [0.0, 0.0]

values = eval_f(prob, x)
article_values = -values
jac = eval_jacobian(prob, x)

Constructor reference

KW2()