## 3 The Optimization Algorithm Used in the Identification Process
### Description of the HS-JAYA Algorithm
#### 3.2.2 Step 2: Generation of New Trial Solutions
\[\frac{\text{If }N_{\rm{RNDj}}<HMCR\text{ and }N_{\rm{RNDj}}<PAR\text{, }x_{\rm{TRj}} \text{ (or }x_{\rm{TRj}}\text{) is pitch-adjusted as:}}\]  
\[\Big{(}x_{\rm{TRj}}^{\rm{pitch,adj}}\Big{)}^{\prime}=x_{\rm{TRj}}\;\pm\;N_{ \rm{RNDj}}\frac{\big{|}x_{\rm{TRj}}-x_{\rm{OPTj}}\big{|}}{\text{NG}_{\rm{tot}}} \cdot\text{NG}_{\rm{pitch,adj}}\quad(j=1,\;2,\;\ldots,\;NDV) \tag{19}\]  
where \(NG_{\rm{pitch,adj}}\) and \(NG_{\rm{tot}}\) are the numbers of pitch-adjusted solutions and generated trial solutions, respectively. The latter counter is reset to \((NG_{\rm{pitch,adj}}+1)\) if the new harmony has more pitch-adjusted variables than those perturbed with gradient information. The bandwidth parameter \(bw\) of classical HS hence becomes unnecessary. Equation (19) accounts for optimization history as it evaluates the ratio of pitch-adjusted trial solutions to the total number of trial solutions currently generated. In Equation (19), the \"+\" sign is used if \(\partial\Omega/\partial x_{\rm{j}}<0\) to maximize reduction of error functional \(\bigg{[}\Big{(}x_{\rm{TRj}}^{\rm{pitch,adj}}\Big{)}^{\prime}-x_{\rm{OPTj}} \bigg{]}\cdot\partial\Omega/\partial x_{\rm{j}}\) with respect to the \(j\)th material/structural property, while the \"-\" sign is used if \(\partial\Omega/\partial x_{\rm{j}}>0\). \(HMCR\) and \(PAR\) are automatically adapted by HS-JAYA in each iteration as:  
\[\left\{ \begin{aligned} &\text{HMCR}^{\rm{q}}=\text{ HMCR}_{\rm{extracted}} ^{\rm{q}}\cdot\frac{\Omega_{\rm{aver,int}}\partial^{\rm{q-1}}}{\Omega_{\rm{aver,int}} \partial^{\rm{q-1}}}\cdot\frac{\text{NG}_{\rm{pitch,adj}}}{\text{NG}_{\rm{ gradient}}}\\ &\text{PAR}^{\rm{q}}=\text{ PAR}_{\rm{extracted}}^{\rm{q}}\cdot \frac{\Omega_{\rm{aver,int}}\partial^{\rm{q-1}}}{\Omega_{\rm{aver,int}} \partial^{\rm{q-1}}}\cdot\frac{\big{|}\big{|}\big{|}\mathbf{X_{\rm{OPT,end}}- \mathbf{X_{\rm{WORST,Ent}}}\big{|}}\big{|}^{\rm{q-1}}}\cdot\frac{\text{NG}_{ \rm{pitch,adj}}}{\text{NG}_{\rm{gradient}}}\end{aligned} \right.\end{aligned}\right. \tag{20}\]  
In Equation (20), \(q\) refers to current iteration; \(\Omega_{\rm{aver,int}}\,^{\rm{q-1}}\) and \(\Omega_{\rm{aver,end}}\,^{\rm{q-1}}\), respectively, are the average error functional values of solutions stored in [HM] at the beginning and the end of the previous iteration (rated successful if \(\Omega_{\rm{aver,end}}\,^{\rm{q-1}}/\Omega_{\rm{aver,int}}\,^{\rm{q-1}}<1\)); \(\mathbf{X_{\rm{OPT,int}}}\) and \(\mathbf{X_{\rm{WORST,Ent}}}\), \(\mathbf{X_{\rm{OPT,end}}}\) and \(\mathbf{X_{\rm{WORST,end}}}\), respectively, denote the best and worst solutions  
In all likelihood, there will be between one and three unit descent directions in the neighborhood of \(\mathbf{X_{\text{OPT}}}\). Derivatives of error functional are evaluated as \((j=1,2,\ldots,NDV)\):  
\[\frac{\partial\Omega}{\partial x_{j}}=\text{Min}\left\{ \begin{array}{l}\frac{ \left[\Omega(\mathbf{X_{\text{CR}}}^{(1)})-\Omega_{\text{OPT}}\right]}{\|\mathbf{S_{ \text{BST}}}^{unit}\|}\frac{(\chi_{\text{CR}_{i}}^{(1)}-\chi_{\text{OPT}_{i}}) }{\|\mathbf{S_{\text{BST}}}^{unit}\|},\frac{\left[\Omega(\mathbf{X_{\text{CR}}}^{(2)})- \Omega_{\text{OPT}}\right]}{\|\mathbf{S_{\text{BST}}}^{unit}\|}\frac{(\chi_{\text{ CR}_{i}}^{(2)}-\chi_{\text{OPT}_{i}})}{\|\mathbf{S_{\text{BST}}}^{unit}\|};\\ \frac{\left[\Omega(\mathbf{X_{\text{CR}}}^{(3)})-\Omega_{\text{OPT}}\right]}{\|\mathbf{S_ {\text{BST}}}^{unit}\|}\frac{(\chi_{\text{CR}_{i}}^{(3)}-\chi_{\text{OPT}_{i}}) }{\|\mathbf{S_{\text{FAST}}}^{unit}\|}\end{array} \right\}\right\} \tag{24}\]  
In Equation (24), sensitivity corresponds to the minimum value amongst the three terms defined for \(\mathbf{S_{\text{BST}}}^{unit}\), \(\mathbf{S_{\text{2 ndBST}}}^{unit}\) and \(\mathbf{S_{\text{FAST}}}^{unit}\). This allows us to minimize the increment or maximize the reduction of \(\Omega\) functional in the neighborhood of \(\mathbf{X_{\text{OPT}}}\). The approximate gradient evaluation strategy of HS-JAYA limits computational cost of the identification process to a great extent.