# Data Reduction of Digital Twin Simulation Experiments Using Different Optimisation Methods
## 2 Literature Background
### Optimisation Methods
#### 2.2.2 Downhill Simplex  
This optimisation method uses the idea of the Nelder-Mead downhill simplex algorithm [34]. This heuristic method uses a set of \(n+1\) linearly independent points, i.e., possible solutions (individuals) in the search space--simplex,  
\[S=\left[\mathbf{X}_{1},\mathbf{X}_{2},\,\ldots,\,\mathbf{X}_{n+1}\right],\,S \subset\widetilde{X} \tag{5}\]  
where  
* \(S\) denotes the simplex;  
\begin{table}
\begin{tabular}{l c c c c} \hline \hline \multirow{2}{*}{**Optimisation Method**} & \multicolumn{4}{c}{**Total/\% of the Total**} \\ \cline{2-5}  & **Papers** & **Proceedings** & **Total** & **Cum. \%** \\ \hline _Heuristics_: & & & & \\ Local Search & 7/11.7\% & 3/5.0\% & 10/16.7\% & 18.2\% \\ Random Search & 5/8.3\% & 2/3.3\% & 7/11.7\% & 30.9\% \\ Hill Climbing & 3/5.0\% & 2/3.3\% & 5/8.3\% & 40.0\% \\ Others & 24/40.0\% & 14/23.3\% & 38/63.3\% & 100\% \\ _Metaheuristics_: & & & & \\ Evolutionary & 60/20.7\% & 65/22.4\% & 125/43.1\% & 43.1\% \\ Simulated Annealing & 11/3.8\% & 12/4.1\% & 23/7.9\% & 51.0\% \\ Tabu Search & 14/4.8\% & 6/2.1\% & 20/6.9\% & 57.9\% \\ VNS & 5/1.7\% & 1/0.3\% & 6/2.1\% & 60.0\% \\ Others & 87/30.0\% & 29/10.0\% & 116/40.0\% & 100\% \\ \hline \hline \end{tabular}
\end{table}
Table 1: DSBO methods used [32].  
\(\mathbf{X}_{1}\) denotes the first solution candidate;
* \(n\) denotes the dimension of the search space;
* \(\widetilde{X}\) denotes the search space--the domain of the input parameters of the discrete event simulation model--solution candidates.  
If we evaluate points with the objective function (objective function minimisation), we have  
\[\mathbf{X}_{H}=\text{argmax}F(\mathbf{X}),\ \mathbf{X}\in S\subset\widetilde{X} \tag{6}\]  
\[\mathbf{X}_{L}=\text{argmin}F(\mathbf{X}),\ \mathbf{X}\in S\subset\widetilde{X} \tag{7}\]  
\[\mathbf{X}_{\text{G}}=\frac{\sum_{i=1}^{n}(\mathbf{X}_{i}-\mathbf{X}_{H})}{n},\ \mathbf{X}_{i}\in S\subset\widetilde{X} \tag{8}\]  
where  
* \(\mathbf{X}_{H}\) denotes the worst solution candidate of the simplex;
* \(\mathbf{X}_{L}\) denotes the best solution candidate;
* \(F(\mathbf{X})\) denotes the objective function value of the solution candidate;
* \(\mathbf{X}_{\text{G}}\) denotes the simplex centroid--centre of gravity (calculated from the solution candidate apart from the best solution candidate).  
The method uses the following four basic phases of generating a solution candidate- see Figure 4 [35]:  
* Reflection--mirroring the worst solution candidate;
* Expansion--searching for a better solution in the reflection direction;
* Contraction--testing the solution candidate between reflection and simplex centroid
* Reduction--shrinkage towards the best solution candidate of the simplex.  
If we want to use this optimisation method for a discrete-event simulation optimisation, the problem is rounding of the point coordinates in the search space (the point coordinates are the vector of the values of the simulation model input parameters) to the nearest feasible coordinates in the search space. This leads to deviation from the original direction in our case if the step of the axis, the decision variable, in the search space is large. A detailed description of this implemented optimisation method is described in [36; 37].