# An Adaptive Linear Programming Algorithm with Parameter Learning
## Appendix A
We tune \(0\) maximizing the \(Lz\) - \(norm\) distance between EIs of two adjacent iterations and get \(10\%\)
21\(L<0\)// Initiate the number of EIs of \(RMC_{t}\) that are better than the EIs of \(RMC_{t-1}\)
22 for\(i\) in \(1\) to \(n\)// For all the EIs (n is the number of EIs)
23 if\(PEI[RMC_{t}]_{i}\in DEI_{i}\)// If the \(\hat{\Pi}\) EI of \(RMC_{t}\) is in the desired range of the \(i\)EI
24\(I<I+1\)// Update the number of EIs of \(RMC_{t}\) in \(DEI\)
25 if\(EI[RMC_{t}]_{i}\
ot>EI[best]_{i}\)// If for the \(\hat{\Pi}\) EI\(RMC_{t}\) performs better than or equal to best RMC
26\(J<I+1\)// Update the number of EIs of \(RMC_{t}\) that are better than best RMC
27 if\(EI[RMC_{t}]\
otin DEI_{i}\) and \(EI[RMC_{t}]_{i}\in[DEI]_{i}\gamma^{\gamma\prime}\)// If the \(\hat{\Pi}\) EI of \(RMC_{t}\) violates the desired range of the \(\gamma\)
28\(\hat{\Pi}\)EI within \(\gamma\gamma_{t}\) (in this problem, \(\gamma\gamma\)=30%)
29\(K<K+1\)// Update the number of EIs of \(RMC_{t}\) that violate \(DEI\) within \(\gamma\gamma_{t}\)
30\(E[RMC_{t}]_{i}\
ot>EI[RMC_{t-1}]\)// If for the \(\hat{\Pi}\) EI, \(RMC_{t}\) performs better than or equal to \(RMC_{t-1}\)
31\(L<L+1\)// Update the number of EIs that \(RMC_{t}\) improves versus \(RMC_{t-1}\)
32 if\(I\succeq t\cdot n\)and\(I<K=n\)// If for at least \(t\) (we set it as 2/3) EIs are in \(DEI\), and the violation rate are all within \(\gamma\gamma\)
33 if\(L\succeq\kappa\cdot n\)// If at least \(\kappa\) (we set it as 2/3) EIs are better than previous synthesis cycle
34\(EI[RMC_{t}]\
ot>EI[RMC_{t-1}]\)// We define that \(RMC_{t}\) overall performs better than \(RMC_{t-1}\)
35 if\(J\succeq\kappa\cdot n\)// If at least \(\kappa\) EIs is better than best RMC
36\(best<\)\(RMC_{t}\)// Update best RMC
37\(\text{nupdt}<\)\(\sim\)\(-1\)// Reset no updating pointer \"nupdt\" as \"\(\sim\)1\"
38 if\(K\succeq 1\)// If at least one violation EI
39\(DEI_{i}\prec EI[RMC_{t}]_{i}\) which \(EI[RMC_{t}]_{i}\
otin DEI_{i}\) and \(EI[RMC_{t}]_{i}\in[DEI]_{i}\gamma^{\prime\prime}\)// Update \(DEI_{i-\theta}\)
40\(\text{nupdt}<\)\(\text{nupdt}+1\)// Increase no updating pointer \"nupdt\" by 1
41 else
42\(\text{nupdt}<\)\(\text{nupdt}+1\)// Increase no updating pointer \"nupdt\" by 1
43 if\(\text{nupdt}\geq M\)// If no updating in \(M\) synthesis cycles in a row (the second stop criterion, and here we set M\(<\)-5)
44 Break
45\(\text{t}<\)\(\text{t+1}\)// Move on to the next synthesis cycle
46 Return best// Return the final best RMC as the appropriate RMC
```

**Algorithm 1**_Cont_

## References

* (1) [PERSON]; [PERSON] Derivative-Free Optimization: A Review of Algorithms and Comparison of Software Implementations. _J. Glob. Optim._**2013**_56_, 1247-1293. [CrossRef]
* (2) [PERSON]; [PERSON]; [PERSON] Particle Swarm Optimization for Computing Nash and Stackelberg Equilibria in Energy Markets. _SN Oper. Res. Forum_**2020**_1_, 20. [CrossRef]
* (3) [PERSON]; [PERSON] A Comparison between Memetic Algorithm and Genetic Algorithm for an Integrated Logistics Network with Flexible Delivery Path. _Oper. Res. Forum_**2021**_, 2_, 47. [CrossRef]
* (4) [PERSON]; [PERSON] A Combined Penalty Function and Outer-Approximation Method for MINLP Optimization. _Comput. Chem. Eng._**1990**_14_, 769-782. [CrossRef]* [PERSON] et al. (2023) [PERSON]; [PERSON]; [PERSON] Comparison of Meta-Heuristic Optimization Algorithms for Global Maximum Power Point Tracking of Partially Shaded Solar Photovoltaic Systems. _Algorithms_**2023**, _16_, 376. [CrossRef]
* [PERSON] and [PERSON] (1993) [PERSON]; [PERSON]; [PERSON] Compromise decision support problem and the adaptive linear programming algorithm. _Prog. Astronaut. Aeronaut. Struct. Optim. Status Promise_**1993**, _150_, 251. * [PERSON] (2019) [PERSON]; [PERSON] A FEAST algorithm for the linear response eigenvalue problem. _Algorithms_**2019**, _12_, 181. [CrossRef]
* [PERSON] and [PERSON] (2000) [PERSON]; [PERSON] Engineering design optimization using interior-point algorithms. _AIAA J._**2000**, _38_, 2127-2132. [CrossRef]
* [PERSON] et al.