## 3 The Optimization Algorithm Used in the Identification Process
### Description of the HS-JAYA Algorithm
#### 3.2.3 Step 3: Evaluation of \(\mathbf{X_{TR}}\) Trial Design and Population Update
If \((\mathbf{X_{\text{NTOP-2}}})^{r,\text{new}}\) is better than \((\mathbf{X_{\text{NTOP-2}}})^{r}\), it holds that \(\Omega((\mathbf{X_{\text{NTOP-2}}})^{r,\text{new}})<\Omega((\mathbf{X_{\text{NTOP-2}}} )^{r})\). By subtracting the current optimum value of \(\Omega\) to both terms of this inequality, it holds that \(\Omega((\mathbf{X_{\text{NTOP-2}}})^{r,\text{new}})-\Omega(\mathbf{X_{\text{OPT}}})< \Omega((\mathbf{X_{\text{NTOP-2}}})^{r})-\Omega(\mathbf{X_{\text{OPT}}})\). Since the RHS of the inequality is equal to \(\Delta\Omega_{(\text{NTOP-2}),r}\), the previous expression can be written as \(\Omega((\mathbf{X_{\text{NTOP-2}}})^{r,\text{new}})-\Omega(\mathbf{X_{\text{OPT}}})< \Delta\Omega_{(\text{NTOP-2}),r}\). The \([\Omega((\mathbf{X_{\text{NTOP-2}}})^{r,\text{new}})-\Omega(\mathbf{X_{\text{OPT}}})]\) variation is approximated by the \((\Delta\Omega_{(\text{NTOP-2}),r-\text{new}})^{appr}\) term defined in Equation (27). The inequality can finally be rewritten as \((\Delta\Omega_{(\text{NTOP-2}),r-\text{new}})^{appr}<\Delta\Omega_{(\text{N TOP-2}),r}\). HS\(-\)JAYA adopts the following strategy to decide whether or not \((\mathbf{X_{\text{NTOP-2}}})^{r}\)new should replace \((\mathbf{X_{\text{NTOP-2}}})^{r}\) in the updated population:\[\left\{ \begin{aligned} \left(\mathbf{\Delta}\Omega_{(\text{NFOP}-2),r }\right)^{\text{threshold}}=Min\Big{\{}\Big{(}\mathbf{\Delta}\Omega_{(\text{NFOP}-2 ),r}\Big{)}^{\text{appr}};\Big{(}\mathbf{\Delta}\Omega_{(\text{NFOP}-2),r}\Big{)} \Big{\}}\\ \left(\mathbf{\Delta}\Omega_{(\text{NFOP}-2),r-\text{new}}\right)^{ \text{appr}}<\Big{(}\mathbf{\Delta}\Omega_{(\text{NFOP}-2),r}\Big{)}^{\text{threshold}} \Rightarrow\ Replace\ (\mathbf{X}_{\text{NFOP}-2})^{r}\ with\ (\mathbf{X}_{\text{NFOP}-2})^{r,\text{new}}\\ \left(\mathbf{\Delta}\Omega_{(\text{NFOP}-2),r-\text{new}}\right)^{ \text{appr}}>\Big{(}\mathbf{\Delta}\Omega_{(\text{NFOP}-2),r}\Big{)}^{\text{threshold}} \Rightarrow\ Leave\ (\mathbf{X}_{\text{NFOP}-2})^{r}\ in\ the\ population\end{aligned} \right. \tag{28}\]  
Using the threshold variation \((\Omega_{(\text{NFOP}-2),r})^{\text{threshold}}\) defined in Equation (28) is a conservative way to minimize the variation \(\Delta\Omega_{(\text{NFOP}-2),r-\text{new}}\) of the error functional. In fact, HS-JAYA is forced to include in the updated population only candidate solutions that effectively yield a low error functional penalty with respect to \(\mathbf{X}_{\text{OPT}}\). In summary, the population is updated only if strictly necessary and the number of structural analyses per optimization cycle performed by HS-JAYA becomes significantly smaller than the \(N_{\text{POP}}\) analyses required by the vast majority of currently available HS and JAYA methods. The population is reordered with respect to \(\Omega\) values of the updated solutions and the next harmony is processed. HS-JAYA finally checks for convergence in step 4. The population updating process performed by HS-JAYA may be classified as an exploration phase because it operates on (\(N_{\text{POP}}\)-2) candidate solutions distributed over the whole design space. The JAYA-based strategy stated by Equation (26) inherently realizes exploration, as defining any new candidate solutions \((\mathbf{X}_{\text{NFOP}-2})^{\text{r,new}}\) involves the best and worst designs of the population. Equations (27) and (28) are somehow based on exploitation because they operate on local variations of error functional \(\Omega\) with respect to \(\Omega_{\text{OPT}}\). However, as optimization progresses, the population becomes more and more concentrated near the current best record \(\mathbf{X}_{\text{OPT}}\), and even the JAYA-based Equation (26) actually generates designs in the neighborhood of \(\mathbf{X}_{\text{OPT}}\) turning towards the exploitation phase. Case 2: \(\Omega(\mathbf{X}_{\text{TR}})>\Omega_{\text{OPT}}\)  
A mirroring strategy is attempted by HS-JAYA to turn the non-descent direction \((\mathbf{X}_{\text{TR}}-\mathbf{X}_{\text{OPT}})\) into a descent one.