Dumitrescu-Grosan-Oltean (DGO)

This family comprises the DGO0, DGO1, and DGO2 constructors. The problems are drawn from "A New Evolutionary Approach for Multiobjective Optimization" [5].

The names DGO1 and DGO2 are common in later benchmark collections, but they correspond to Examples 2 and 3 in the original paper. DGO0 corresponds to Example 1.

Overview

All three constructors have nvar = 1 and nobj = 2. Their componentwise bounds are shown below.

Analytical Jacobians are registered for all three constructors. Hessians are not registered. The catalog metadata classifies both objectives in DGO0 and DGO2 as strictly convex (:strictly_convex) and both objectives in DGO1 as not strictly convex (:not_strictly_convex).

Mathematical formulations

The formulas below describe the objective functions implemented by the constructors. For each problem, let $F:\mathbb{R}^1 \to \mathbb{R}^2$ be defined by $F(x)=(f_1(x),f_2(x))$.

DGO0

The objectives are

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

DGO1

The objectives are

\[\begin{aligned} f_1(x) &= \sin(x_1),\\ f_2(x) &= \sin(x_1 + 0.7). \end{aligned}\]

DGO2

The objectives are

\[\begin{aligned} f_1(x) &= x_1^2,\\ f_2(x) &= 9 - \sqrt{81 - x_1^2}. \end{aligned}\]

Usage

using MOProblems

prob = DGO0()
x = [0.0]

values = eval_f(prob, x)
J = eval_jacobian(prob, x)

Constructor reference

DGO0()

DGO1()

DGO2()