Moré–Garbow–Hillstrom (MGH)

The MGH names retain the numbering of four problems in the nonlinear least-squares collection of Moré, Garbow, and Hillstrom [21]. In that source, the numbered functions are residuals, and the corresponding scalar unconstrained objective is their sum of squares. Mita, Fukuda, and Yamashita [22] report multiobjective formulations based on these problems in Appendix A of their numerical study. Let r_i denote a residual from the original collection. The formulations in [22] use f_i = r_i for MGH9 and MGH16, and f_i = r_i^2 for MGH26 and MGH33. The original problems are unconstrained [21], while the bounds adopted by the constructors are based on those reported in [22]. The objectives in MOProblems.jl follow these formulations, while the package makes documented changes to some dimensions. Thus, the MGH prefix records the historical origin and does not mean that every constructor literally reproduces either source.

Overview

The four constructors differ in whether the numbers of variables and objectives are fixed, coupled, or independently configurable.

MGH9 has fixed dimensions: the package exposes no dimension parameter and constructs a problem with nvar = 3 and nobj = 15. These dimensions agree with n = 3 and m = 15 in both [21] and [22].

MGH16 fixes nvar = 4 and accepts m >= 4, with nobj = m; its default is m = 5. The original problem fixes n = 4 and allows m >= n [21], while [22] uses n = 4 and m = 5.

MGH26 accepts n >= 1 and couples the dimensions as nvar = nobj = n; its default is n = 4. The original problem allows n to vary and defines m = n [21], while [22] uses n = m = 4.

MGH33 accepts n >= 2 and m >= 2 independently, with nvar = n and nobj = m; its defaults are n = m = 10. The original problem allows n to vary but requires m >= n [21], while [22] uses n = 10 and m = 4. Thus, the package supports the dimensions independently and includes (n, m) = (10, 4) as a nondefault configuration.

The following table summarizes the dimension behavior implemented by the constructors.

Analytical Jacobians are registered for all four constructors. Hessians are not registered. The catalog metadata classifies every objective in each default instance as not strictly convex (:not_strictly_convex).

Mathematical formulations

The formulas below reproduce the objectives implemented by the constructors.

MGH9 — Gaussian

Let $F:\mathbb{R}^3\to\mathbb{R}^{15}$ be defined by $F(x)=(f_1(x),\ldots,f_{15}(x))$ for $x\in[-2,2]^3$. The objectives are

\[f_i(x)=x_1\exp\left(-\frac{x_2(t_i-x_3)^2}{2}\right)-y_i, \qquad i=1,\ldots,15,\]

where $t_i=(8-i)/2$ and

\[\begin{aligned} y_1=y_{15}&=0.0009, & y_2=y_{14}&=0.0044,\\ y_3=y_{13}&=0.0175, & y_4=y_{12}&=0.0540,\\ y_5=y_{11}&=0.1295, & y_6=y_{10}&=0.2420,\\ y_7=y_9&=0.3521, & y_8&=0.3989. \end{aligned}\]

MGH16 — Brown–Dennis

For $m\geq4$, let $F:\mathbb{R}^4\to\mathbb{R}^m$ be defined by $F(x)=(f_1(x),\ldots,f_m(x))$. The objectives are

\[f_i(x)=\left(x_1+t_ix_2-\exp(t_i)\right)^2 +\left(x_3+x_4\sin(t_i)-\cos(t_i)\right)^2, \qquad i=1,\ldots,m,\]

where $t_i=i/5$ and $x\in[-25,25]\times[-5,5]\times[-5,5]\times[-1,1]$.

MGH26 — Trigonometric

For $n\geq1$, let $F:\mathbb{R}^n\to\mathbb{R}^n$ be defined by $F(x)=(f_1(x),\ldots,f_n(x))$ for $x\in[-1,1]^n$. The objectives are

\[f_i(x)=\left( n-\sum_{j=1}^{n}\cos(x_j) +i\left(1-\cos(x_i)\right)-\sin(x_i) \right)^2, \qquad i=1,\ldots,n.\]

MGH33 — Linear function, rank 1

For independent $n\geq2$ and $m\geq2$, let $F:\mathbb{R}^n\to\mathbb{R}^m$ be defined by $F(x)=(f_1(x),\ldots,f_m(x))$ for $x\in[-1,1]^n$. The objectives are

\[f_i(x)=\left(i\sum_{j=1}^{n}jx_j-1\right)^2, \qquad i=1,\ldots,m.\]

Usage

The following example uses the linear rank-1 dimensions reported in [22].

using MOProblems

prob = MGH33(n = 10, m = 4)
x = zeros(prob.nvar)

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

Constructor reference

MGH9()

MGH16(; m::Int = 5)

MGH26(; n::Int = 4)

MGH33(; n::Int = 10, m::Int = 10)