## 3 The Optimization Algorithm Used in the Identification Process
### Background  
Harmony search optimization is a population-based metaheuristic algorithm that mimics the search of a perfect harmony state made by jazz players. Three improvisation options can be selected by a skilled musician: (1) play any famous music (i.e., a series of pitches in HS) from his/her memory; (2) play something similar to known music (i.e., pitch adjusting in HS); and (3) compose new or random notes. [PERSON] et al. [50,51] formalized these options into the formulation of an HS algorithm. N\({}_{\text{POP}}\) randomly generated solutions are stored in the harmony memory [HM] matrix. New trial designs are extracted from [HM] or randomly selected based on harmony memory considering rate (HMCR) parameter values. Design variables may be modified based on pitch adjustment rate (PAR) parameter values; the bandwidth parameter (bw) defines the range of variation of a given variable in the pitch adjustment phase. Here, the basic HS formulation was adapted to the implicit formulation of the error functional \(\Omega\), and approximate line searches were performed in order to quickly approach the global optimum.  
The JAYA method is another population-based metaheuristic algorithm developed by [PERSON] [52]. It implements the simplest search strategy of metaheuristic optimization: trial solutions are always defined approaching the best design and moving away from the worst design of population. JAYA includes only two very standard control parameters: population size and limit number of iterations. Its straightforward formulation as well as its inherently efficient search strategy made JAYA become one of the most exploited and studied metaheuristic algorithms. JAYA's formulation was modified in this study to reduce the computational cost of the identification process.  
As mentioned before, JAYA's implementation is very simple because there is only one equation to perturb design. Let \(X_{j,k,it}\) be the value of the \(j\)th material/structural property (\(j=1,\ldots,\) NDV) for the \(k\)th candidate solution included in the population (\(k=1,\ldots,\) N\({}_{\text{POP}}\)) in the present \(i\)th iteration. The \(X_{j,k,it}\) value is perturbed as:  
\[X_{j,k,it}^{new}=X_{j,k,it}+r_{1,j,it}\Big{(}X_{j,best,it}-\left|X_{j,k,it} \right|\Big{)}-r_{2,j,it}\Big{(}X_{j,worst,it}-\left|X_{j,k,it}\right|\Big{)} \tag{13}\]  
In Equation (13), \(X_{j,k,it}^{new}\) denotes the perturbed material/structural property with respect to the \(X_{j,k,it}\) current value; \(r_{1,j,it}\) and \(r_{2,j,it}\) are two random numbers extracted from the [0, 1] interval for the \(j\)th material/structural property; and \(X_{j,best,it}\) and \(X_{j,worst,it}\), respectively, denote the \(j\)th material/structural property values stored in the best and worst elements \(\mathbf{X}_{\text{best,it}}\) and \(\mathbf{X}_{\text{worst,it}}\) of current population.  
Term \(r_{1,j,it}\Big{(}X_{j,best,it}-\left|X_{j,k,it}\right|\Big{)}\) represents JAYA's tendency to approach best design \(\mathbf{X}_{\text{best,it}}\), while term \(-r_{2,j,it}\Big{(}X_{j,worst,it}-\left|X_{j,k,it}\right|\Big{)}\) portrays JAYA's tendency to move away from worst design \(\mathbf{X}_{\text{worst,it}}\). Random factors \(r_{1}\) and \(r_{2}\) guarantee good exploration of search space. The absolute value \(\left|X_{j,k,it}\right|\) enhances JAYA's exploration capability [52].  
The new trial design \(\mathbf{X}_{k}^{\text{new}}\) defined by Equation (13) is compared with the corresponding element \(\mathbf{X}_{k}^{\text{pre}}\) of population. If \(\mathbf{X}_{k}^{\text{new}}\) improves \(\mathbf{X}_{k}^{\text{pre}}\), it replaces \(\mathbf{X}_{k}^{\text{pre}}\) in the updated population. The JAYA search stops when the limit number of iterations/analyses is reached.