Stadler–Dauer (SD)

This family is represented by the SD constructor. The problem is the “Elastic Trusses” example of “Multicriteria Optimization in Engineering: A Tutorial and Survey” [28].

Overview

SD has nvar = 4 and nobj = 2. Its componentwise variable bounds are shown below.

The lower bounds keep the feasible set inside the positive orthant, away from the poles of $f_2$.

An analytical Jacobian is registered. Hessians are not registered. The catalog metadata classifies $f_1$ as not strictly convex (:not_strictly_convex) and $f_2$ as strictly convex (:strictly_convex).

Mathematical formulation

Let $F:\mathbb{R}^4 \to \mathbb{R}^2$ be defined by $F(x)=(f_1(x),f_2(x))$, where $x = (x_1,x_2,x_3,x_4)$. The objectives are

\[\begin{aligned} f_1(x) &= 2x_1 + \sqrt{2}\,(x_2 + x_3) + x_4,\\ f_2(x) &= \frac{2}{x_1} + \frac{2\sqrt{2}}{x_2} + \frac{2\sqrt{2}}{x_3} + \frac{2}{x_4}. \end{aligned}\]

Source formulation

The source [28] states the problem as the optimal design of the four-bar truss of its Fig. 9, loaded by forces $F$. The decision variables $x_1,\ldots,x_4$ are the cross-sectional areas of the four bars, and the two criteria are the volume of the truss and the deflection of its outermost joint,

\[\begin{aligned} g_1(x) &= L\left(2x_1 + \sqrt{2}\,x_2 + \sqrt{2}\,x_3 + x_4\right),\\ g_2(x) &= \frac{FL}{E}\left(\frac{2}{x_1} + \frac{2\sqrt{2}}{x_2} + \frac{2\sqrt{2}}{x_3} + \frac{2}{x_4}\right), \end{aligned}\]

over the feasible set

\[\begin{aligned} \frac{F}{\sigma} \le\;& x_1 \le \frac{3F}{\sigma}, & \sqrt{2}\,\frac{F}{\sigma} \le\;& x_2 \le \frac{3F}{\sigma},\\ \sqrt{2}\,\frac{F}{\sigma} \le\;& x_3 \le \frac{3F}{\sigma}, & \frac{F}{\sigma} \le\;& x_4 \le \frac{3F}{\sigma}, \end{aligned}\]

where $\sigma$ is a characteristic stress, $E$ is the elastic modulus, $L$ is the length of a truss section, and $F$ is the applied force.

SD implements the normalized instance $L = 1$, $F/\sigma = 1$, and $FL/E = 1$, so that $f_1 = g_1$ and $f_2 = g_2$ over the bounds tabulated above. The physical parameters are not exposed as constructor arguments. They would not produce a structurally different problem: writing $s = F/\sigma$ and substituting $x = s\,y$,

\[g_1(s\,y) = (L s)\, f_1(y), \qquad g_2(s\,y) = \left(\frac{L\sigma}{E}\right) f_2(y), \qquad y \in [1,3]\times[\sqrt{2},3]^2\times[1,3],\]

so every choice of $F$, $\sigma$, $E$, and $L$ is the implemented instance up to a positive scaling of the decision variables and a positive scaling of each objective. Pareto dominance is invariant under positive componentwise scaling of the objectives, so the Pareto-optimal set of the parametric problem is $s$ times the Pareto-optimal set of SD, and its Pareto front is the front of SD with the first objective scaled by $L s$ and the second by $L\sigma/E$.

Usage

julia> using MOProblems

julia> using Random

julia> prob = SD();

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, 4))

Constructor reference

SD()