# An Adaptive Linear Programming Algorithm with Parameter Learning
## 3 Parameter Learning to Dynamically Change the RMC
### Step 3--Learning the DEI to Tune the RMC  
We identify the desired range of the EIs (DEI), learn the connections between the RMC and the EIs, and bring the EIs into the DEI by setting the RMC. To make the process \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Criteria** & **Information** & **EIs** & **Meaning** \\ \hline \multirow{3}{*}{Weight-sum deviation} & \multirow{3}{*}{\(Z=\sum_{k\in K}W_{k}\cdot(a_{k}^{-}+d_{k}^{+})\)} & \(\mu_{x}\) & The average weight-sum deviations \\ \cline{3-4}  & & \(\sigma_{x}\) & The stability of the weight-sum deviations when \\  & & & changing design scenarios \\ \hline \multirow{6}{*}{Robustness} & \multirow{3}{*}{The number of active bounds:} & \(\mu_{\text{Nab}}\) & The average sensitivity of the solutions under all \\  & & & design scenarios to variable bounds \\ \cline{3-4}  & & \(\sigma_{\text{Nab}}\) & The stability of the sensitivity of the solutions to \\  & & & variable bounds when changing design scenarios \\ \cline{3-4}  & & \(\mu_{\text{Nac}}\) & The average sensitivity of the solutions under all \\  & & & design scenarios to original constraints \\ \cline{3-4}  & & \(\sigma_{\text{Nac}}\) & The stability of the sensitivity of the solutions to \\  & & & original constraints when changing design scenarios \\ \hline \multirow{6}{*}{Approximation accuracy} & \multirow{3}{*}{The number of accumulated constraints:} & \(\mu_{\text{Nac}}\) & The average complexity of the approximated \\  & & & problem under all design scenarios \\ \cline{3-4}  & & \(\sigma_{\text{Nac}}\) & The stability of the complexity of the approximated \\  & & & problem when changing design scenarios \\ \cline{2-4}  & & \(\mu_{\text{Nat}}\) & The average convergence speed under all \\  & & & design scenarios \\ \cline{2-4}  & & \(\sigma_{\text{Nab}}\) & The stability of the convergence speed when \\  & & & changing design scenarios \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Develop evaluation indices (EIs)—mean (\(\mu\)) and standard deviation (SD) (\(\sigma\)). efficient, we combine off-line learning and the on-line learning. First, we use off-line learning using a sample of RMC values to initialize the DEI and parameters and then adopt on-line learning to update the DEI and tune the RMC. In Figure 10, we illustrate the two processes (Rectangles B and C) and show their relationship with the synthesis cycle A. During parameter initialization (B), we generate sample values of RMC (B1). By running the synthesis cycle (A) with each RMC value, we obtain the corresponding EIs (B2) and evaluate them (B3). We choose the RMC value associated with the best EIs as the starting RMC value for tuning (B3-1) and remove the EIs that are insensitive to RMC changes (B3-2). We initialize the DEI based on the sample results (B3-3) to allow a certain proportion (e.g., 75%) of RMC values to fall into the DEI. We choose the best RMC among the sample (B3-4) to start the on-line learning. These results (B3-1 to B3-4) are aggregated as the \"actions to be taken\" (B3-5) and the input of the RMC tuning cycle (C). For the RMC tuning cycle (C), with an RMC value (C1-1), we run the synthesis cycle (A) and obtain the results (C2). By evaluating the results using the DEI and comparing with previous iterations (C3), we determine the next RMC value (C3-1), update the DEI if necessary--either restrict or relax the DEI based on the tradeoffs among EIs (C3-2)--and update the best RMC if necessary (C3-3). These evaluation results are aggregated (C3-4) for judging whether to stop iterating (C4)--the program either goes to the next iteration of  
Figure 10: ALPPL including parameter initialization and RMC tuning. RMC tuning with the aggregated results (C3-4) as input or stops with the best RMC as the returned value. The stopping criteria include the number of total iterations and the number of iterations without updating the best RMC. This part is summarized as Algorithm 4.