# An Adaptive Linear Programming Algorithm with Parameter Learning
## 4 A Test Problem
### RMC Tuning  
We make rules for each procedure of RMC tuning based on heuristics. The heuristics are generalized from parameter learning and can be adjusted through the search process. C3: Evaluate the result of current RMC based on EIs and DEI. C3-1: Determine the next RMC value. Rule 1: Compare the performance of multiple EIs and define the comparison rules. Lines 22-30 in Algorithm A1 in Appendix A are an expansion of this rule. RMC A is better than RMC B because no less than \(\kappa\) of the EI(A) are better than EI(B), whereas other EI(A) do not exceed \(\gamma\) of the upper and lower bound of DEI. In this problem, we set \(\kappa=1/2\) and \(\gamma=3\ 0\%\). Rule 2: Determine when and how the RMC should be updated. Lines 6-13 in Appendix A explain this rule. We use a hill-climbing approach to update the RMC. If the updating in the previous RMC-tuning cycle improves the performance, then the previous updating is in the \"hill-climbing direction,\" and we continue updating the RMC in this \begin{table}
\begin{tabular}{c c c c c c c} \hline  & _W1_ & _W2_ & _W3_ & _W1_ & _W2_ & _W3_ \\ \hline
1 & 1 & 0 & 0 & 11 & 0 & 0.75 & 0.25 \\
2 & 0 & 1 & 0 & 12 & 0 & 0.25 & 0.75 \\
3 & 0 & 0 & 1 & 13 & 0.33 & 0.33 & 0.33 \\
4 & 0.5 & 0.5 & 0 & 14 & 0.2 & 0.2 & 0.6 \\
5 & 0.5 & 0 & 0.5 & 15 & 0.4 & 0.2 & 0.4 \\
6 & 0 & 0.5 & 0.5 & 16 & 0.2 & 0.4 & 0.4 \\
7 & 0.25 & 0.75 & 0 & 17 & 0.6 & 0.2 & 0.2 \\
8 & 0.25 & 0 & 0.75 & 18 & 0.4 & 0.4 & 0.2 \\
9 & 0.75 & 0 & 0.25 & 19 & 0.2 & 0.6 & 0.2 \\
10 & 0.75 & 0.25 & 0 & & & & \\ \hline \end{tabular}
\end{table} 
Table 7: Weight vectors used in [36] as different design scenarios. \begin{table}
\begin{tabular}{c c c c c c} \hline
**RMC** & **Statistics** & **Z** & **Nit** & **Nacc** & **Nab** & **Naoc** \\ \hline
0.1 & \(\mu\) & 0.1480 & 46.58 & 18.74 & 1.79 & 0.84 \\  & \(\sigma\) & 0.0679 & 5.95 & 0.87 & 0.63 & 0.50 \\ \hline
0.5 & \(\mu\) & 0.1467 & 20.42 & 19.47 & 2.05 & 0.79 \\  & \(\sigma\) & 0.0675 & 9.47 & 0.84 & 0.62 & 0.42 \\ \hline
0.8 & \(\mu\) & 0.1480 & 8.32 & 14.16 & 2.21 & 0.79 \\  & \(\sigma\) & 0.0675 & 5.56 & 6.94 & 0.63 & 0.71 \\ \hline \end{tabular}
\end{table} 
Table 8: Results of EIs using sample RMC values with nineteen design scenarios. \begin{table}
\begin{tabular}{c c c c c c} \hline
**DEI of \(\mu_{Z}\)** & **DEI of \(\sigma_{Z}\)** & **DEI of \(\mu_{Naoc}\)** & **DEI of \(\sigma_{Naoc}\)** & **DEI of \(\mu_{Nab}\)** & **DEI of \(\sigma_{Nab}\)** \\ \hline
[0, 0.1477] & [0, 0.0677] & [0, 0.82] & [0, 0.55] & [0, 1.95] & [0, 0.63] \\ \hline \end{tabular}
\end{table} 
Table 9: The initial DEIs. direction with a step size \(\alpha\); otherwise, we need the best RMC to \"pull us back\" to the right direction with a portion \(\beta\); hence, we update the RMC as a linear combination of the best RMC (elite) and the RMC two cycles ago (parent). In this problem, we set \(\alpha\) as a random value in [0, 1] and \(\beta\) as a random value in [0.5, 1]. In this way, we incorporate greediness, elitism, and randomness in evolution. C3-2: Evaluate whether DEI needs to be updated. Rule 3: Determine when and how the DEI should be updated. See Algorithm A1 in Appendix A, Lines 18-38. In an RMC-tuning cycle, if more than \(\kappa\) EIs are better than the previous cycle, and more than \(\iota\) EIs are in the DEI, whereas no more than \(\gamma\gamma\) EIs have minor violations, then the DEI is updated. Here, we set \(\kappa=\iota=2/3\). For a problem with more EIs, \(\kappa\) and \(\iota\) can be tuned using the performance improvement rate. This rule prevents an over restrictive DEI from blocking us from a better range while ensuring gradual and relatively conservative updating of the DEI. C3-3: Evaluate whether \"the best RMC\" needs to be updated. Rule 4: Determine when and how the best RMC is updated. See Algorithm A1 in Appendix A, Lines 34-36. We use a variable (\"best\") to store the best RMC. If more than \(\kappa\) EIs of the current RMC are better than the EIs of the best, the current RMC becomes the new best.