A splitting algorithm for simulation-based optimization problems with categorical variables

Figures & data

Figure 1. Illustration of the quadratic underestimating function $\overline{F}$. The objective function F is evaluated at the set of sample points $\overline{V}$ (illustrated by the dots).

Figure 2. Illustration in ${\mathbb{R}}^2$ of the nearest edge projection splitting strategy. The split is identified across the edge $Z_i\left(3\right)$–$Z_i\left(5\right)$ of the minimum spanning tree corresponding to the designs $Z_i\left(X_i^s\right)=\left\{Z_i\right(1),\dots ,Z_i(6\left)\right\}$ by projecting (illustrated by the dashed line) the solution to (3(3a) $\underset{\mathbf{v},\mathit{\lambda }}{\mathrm{minimize}}\overline{F}({\mathbf{v}}^1,\dots ,{\mathbf{v}}^m),$(3a) ), in terms of ${\mathbf{v}}^i$, onto the tree.

Figure 3. Flowchart of the algorithm quadDS.

Figure 4. A stepped cantilever beam with m=5 segments.

Figure 5. Data profiles $d_a\left(\beta \right)$ and performance profiles ${\rho }_a\left(\alpha \right)$ for three variants of quadDS, the MATLAB GA and NOMAD applied to 120 instances of the sparse artificial problem with $\tau =0.1$.

Figure 6. Data profiles $d_a\left(\beta \right)$ and performance profiles ${\rho }_a\left(\alpha \right)$ for three variants of quadDS, the MATLAB GA and NOMAD applied to 120 instances of the fully artificial problem with $\tau =0.1$.

Figure 7. Data profiles $d_a\left(\beta \right)$ and performance profiles ${\rho }_a\left(\alpha \right)$ for three variants of quadDS, the MATLAB GA and NOMAD applied to 120 instances of the beam problem with $\tau =0.1$.

Figure 8. Data profiles $d_a\left(\beta \right)$ and performance profiles ${\rho }_a\left(\alpha \right)$ for three variants of quadDS, the MATLAB GA and NOMAD applied to 120 instances of the tyres selection problem with $\tau =0.001$. Note that $d_{\mathrm{NOMAD}}\left(\beta \right)=0$ for all values of β and ${\rho }_{\mathrm{NOMAD}}\left(\alpha \right)=0.5$ for all values of α, because NOMAD did not solve any of the instances of the tyres selection problem within the maximum allowed number of objective function evaluations.