## 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  
HS-JAYA evaluates the new trial solution \(\mathbf{X_{\text{TR}}}\) defined in step 2 according to two cases: (1) \(\Omega(\mathbf{X_{\text{TR}}})<\Omega_{\text{OPT}}\); and (2) \(\Omega(\mathbf{X_{\text{TR}}})>\Omega_{\text{OPT}}\). This approach is far more effective than only replacing the worst design \(\mathbf{X_{\text{worst}}}\) of [HM] as it is conducted by classical HS. Case 1: \(\Omega(\mathbf{X_{\text{TR}}})<W_{\text{OPT}}\)  
The new trial solution \(\mathbf{X_{\text{TR}}}\) is reset as \(\mathbf{X_{\text{OPT}}}\). \(\mathbf{X_{\text{worst}}}\) is removed from [HM]. The former optimum is downgraded as the second best solution \(\mathbf{X_{\text{2 ndBEST}}}\) of population, while the second worst solution of the previous population becomes the worst design \(\mathbf{X_{\text{worst}}}\) of the new population. The other (\(N_{\text{TOP}}-2\)) solutions of the population are updated by HS-JAYA as:  
\[ \begin{array}{ll}(\mathbf{X_{\text{NTOP-2}}})^{r,new}&=(\mathbf{X_{\text{NTOP-2}}} )^{r}+\mathbf{\omega_{1}}(\mathbf{X_{\text{OPT}}}-(\mathbf{X_{\text{NTOP-2}}})^{r})\\ &-\mathbf{\omega_{2}}(\mathbf{X_{\text{worst}}}-(\mathbf{X_{\text{NTOP-2}}})^{r})\end{array}  \tag{26}\]  
In Equation (26), \(r\in(N_{\text{TOP}}-2)\); \(\mathbf{\omega_{1}}\) and \(\mathbf{\omega_{2}}\) define two vectors of \(NDV\) random numbers in the \([0,1]\) interval: in particular, \(\omega_{1,\text{j}}\) and \(\omega_{2,\text{j}}\) are defined for the jth variable of processed harmony. Since the \((\mathbf{X_{\text{NTOP-2}}})^{r}\) harmony has to be improved, HS-JAYA explores the descent direction \((\mathbf{X_{\text{OPT}}}-(\mathbf{X_{\text{NTOP-2}}})^{r})\) defined with respect to \((\mathbf{X_{\text{NTOP-2}}})^{r}\) and attempts to move away from the worst design \(\mathbf{X_{\text{worst}}}\) of [HM], which certainly would never improve \((\mathbf{X_{\text{NTOP2}}})^{r}\). If \(\Omega((\mathbf{X_{\text{NTOP-2}}})^{r,\text{new}})<\Omega((\mathbf{X_{\text{NTOP-2}}} )^{r})\), the new harmony \((\mathbf{X_{\text{NTOP-2}}})^{r\text{new}}\) should obviously replace the old harmony \((\mathbf{X_{\text{NTOP-2}}})^{r}\) in [HM]. However, since evaluating the error functional \(\Omega\) always entails a new structural analysis, HS-JAYA adopts an alternative strategy to limit the computational cost of metaheuristic optimization. The variations (i.e., increments) of error functional \((\Delta\Omega_{(\text{NTOP-2},r)})^{appr}\) and \((\Delta\Omega_{(\text{NTOP-2},r),\text{new}})^{appr}\) that would occur if the solution is perturbed from \(\mathbf{X_{\text{OPT}}}\) to the harmony \((\mathbf{X_{\text{NTOP-2}}})^{r}\) or \((\mathbf{X_{\text{NTOP-2}}})^{r,\text{new}}\) may be estimated by two scalar products as:  
\[ \begin{array}{l}\left(\Delta\Omega_{(\text{NTOP-2}),r}\right)^{appr}=\overline {\
abla}\Omega(\mathbf{X_{\text{OPT}}})\cdot\left[(\mathbf{X_{\text{NTOP-2}}})^{r}- \mathbf{X_{\text{OPT}}}\right]\\ \left(\Delta\Omega_{(\text{NTOP-2}),r-\text{new}}\right)^{appr}=\overline{ \
abla}\Omega(\mathbf{X_{\text{OPT}}})\cdot\left[(\mathbf{X_{\text{NTOP-2}}})^{r,new}- \mathbf{X_{\text{OPT}}}\right]\end{array}  \tag{27}\]  
The \((\Delta\Omega_{\Omega(\text{NTOP-2}),r-\text{new}})^{appr}\)and \((\Delta\Omega_{(\text{NTOP-2},r)})^{appr}\) terms in Equation (27) express the approximate variations of the error functional \(\Omega\) that would be obtained if \(\Omega\) is linearized. The real variation \(\Delta\Omega_{(\text{NTOP-2}),r}\) of error functional for \((\mathbf{X_{\text{NTOP-2}}})^{r}\) is known, as this candidate solution is stored in the current population: \(\Delta\Omega_{(\text{NTOP-2}),r}=[\Omega((\mathbf{X_{\text{NTOP-2}}})^{r})- \Omega(\mathbf{X_{\text{OPT}}})]\). The real variation \(\Delta\Omega_{(\text{NTOP-2}),r}\) is obviously positive because \((\mathbf{X_{\text{NTOP-2}}})^{r}\) is worse than \(\mathbf{X_{\text{OPT}}}\). In all likelihood, the approximate variation \((\Delta\Omega_{(\text{NTOP-2}),r})^{appr}\) also may be positive.