## 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  
Let \(\mathbf{X_{\text{OPT}}}=\{x_{\text{OPT},1},\ x_{\text{OPT},2},\ \dots,\ x_{\text{OPT}, \text{NDV}}\}\) be the best element of population and \(\overline{\
abla}\Omega(\mathbf{X_{\text{OPT}}})\) the gradient vector of the error functional \(\Omega\) evaluated at \(\mathbf{X_{\text{OPT}}}\). A random number \(N_{\text{RNDJ}}\) is extracted from the (0, 1) interval for each design variable. A recursive cycle is completed for each unknown material/structural property from 1 to \(NDV\). If \(N_{\text{RNDJ}}\)j _HMCR_, the new value \(x_{\text{TRJ}}\) randomly defined for the jth material/structural property (\(j=1,\ 2,\dots,NDV\)) currently perturbed is:  
\[\left\{\begin{array}{l}\frac{\partial\Omega}{\partial x_{j}}<0\ \&\left(x_{\text{OPT}\text{J}}-x_{\text{I}}^{\text{L}}\right)>\ \left(x_{j}^{\text{U}}-x_{\text{OPT  
If \(N_{\rm{RNDj}}<HMCR\), for any \(PAR\) values, HS-JAYA defines the interval limited by \(x_{\rm{TRj}}^{\rm{HM,less}}\) and \(x_{\rm{TRj}}^{\rm{FIM,more}}\), adjacent to \(x_{\rm{OPTJ}}\) coordinate of \(\mathbf{X_{\rm{OPT}}}\), sorting values as \(x_{\rm{TRj}}^{\rm{FIM,less}}\) \(<x_{\rm{OPTJ}}<x_{\rm{TRj}}^{\rm{FIM,more}}\). The jth material/structural property is updated as (\(j=1,2,\ldots,NDV\)):  
\[x_{\rm{TRj}}=x_{\rm{OPTJ}}+(N_{\rm{RNDj}}-0.5)\cdot\text{Max}\;[(x_{\rm{OPTJ}} -\hat{x}_{\rm{TRj}}^{\rm{HM,less}})\;;(\hat{x}_{\rm{TRj}}^{\rm{HM,more}}-x_{ \rm{OPTj}})] \tag{17}\]  
If \((x_{\rm{TRj}}-x_{\rm{OPTJ}})\cdot\partial\Omega/\partial x_{\rm{j}}<0\), the \(x_{\rm{TRj}}\) trial value of Equation (17) is checked for pitch adjustment. Otherwise, if \((x_{\rm{TRj}}-x_{\rm{OPTJ}})\cdot\partial\Omega/\partial x_{\rm{j}}>0\), JAYA's characteristic Equation (13) becomes:  
\[x_{\rm{TRj}}^{\prime}=x_{\rm{OPTJ}}+\beta_{1,j}\Big{(}\hat{x}_{\rm{TRj}}^{\rm{ best}}-x_{\rm{OPTJ}}\Big{)}-\beta_{2,j}\Big{(}\hat{x}_{\rm{TRj}}^{\rm{worst}}-x_{ \rm{OPTj}}\Big{)} \tag{18}\]  
In Equation (18), \(\beta_{1,j}\) and \(\beta_{2,j}\) are random numbers in the [0, 1] interval; \(\hat{x}_{\rm{TRj}}^{\rm{worst}}=\text{Min}\Big{[}\hat{x}_{\rm{TRj}}^{\rm{HM, more}};x_{\rm{TRj}}\Big{]}\); \(\hat{x}_{\rm{TRj}}^{\rm{best}}=\text{Min}\Big{[}\hat{x}_{\rm{TRj}}^{\rm{HM,less }};\big{(}2x_{\rm{TRj}}-x_{\rm{OPTJ}}\big{)}\Big{]}\). The JAYA-based strategy implemented here limits search in the \([\hat{x}_{\rm{TRj}}^{\rm{best}},\hat{x}_{\rm{TRj}}^{\rm{worst}}]\) neighborhood of current best value \(x_{\rm{OPTJ}}\) for the jth material/structural property, thus exploring descent directions \((\hat{x}_{\rm{TRj}}^{\rm{best}}-x_{\rm{OPTj}})\) and \(-(\hat{x}_{\rm{TRj}}^{\rm{worst}}-x_{\rm{OPTj}})\). HS-JAYA always searches good trial designs in the neighborhood of current best design. This eltiti strategy is implemented in Equation (17) and further in Equation (18), defining descent directions for the JAYA operator. Interestingly, HS-JAYA mixes exploration and exploitation to define the new trial design \(\mathbf{X_{\rm{TR}}}\). Such a combination is made at single variable level. For example, Equation (15) is an exploration scheme because perturbations of each design variable of \(\mathbf{X_{\rm{TR}}}\) depend on the distance between \(\mathbf{X_{\rm{OPT}}}\) and variable space boundaries. Conversely, Equations (16) through (18) are exploitation schemes because they perturb single optimization variables within a neighborhood of \(\mathbf{X_{\rm{OPT}}}\), either selecting a \"mirror\" direction of a non-descent direction (i.e., Equation (16)), or using an interval of adjacent values \(x_{\rm{TRj}}^{\rm{HM,less}}\) and \(x_{\rm{TRj}}^{\rm{HM,more}}\) to the currently optimum value \(x_{\rm{OPTJ}}\) of the \(j\)th perturbed variable (i.e., Equation (17)), or using descent directions defined by \(x_{\rm{OPTJ}}\) and values \(\hat{x}_{\rm{TRj}}^{\rm{best}}\) and \(\hat{x}_{\rm{TRj}}^{\rm{worst}}\) close to \(\hat{x}_{\rm{TRj}}^{\rm{HM,less}}\) and \(\hat{x}_{\rm{TRj}}^{\rm{FIM,more}}\) (i.e., Equation (18)).