## 3 Whale Optimization Algorithm  
The Whale Optimization Algorithm (WOA) is a swarm-based metaheuristic recently proposed by [PERSON] and [PERSON] in [56], who developed the algorithm by taking inspiration from the bubble-net hunting strategy typically employed by humpback whales. In the last few years, other researchers have applied multiple modifications to the algorithm in order to solve various engineering optimization problems. For instance, [PERSON] and [PERSON] [61] studied the sizing optimization of skeletal structures, [PERSON] et al. [62] solved bar truss design and I-beam design problems, and [PERSON] et al. [63] optimized a fuzzy controller for seismically excited nonlinear structures. WOA strategy first defines a population of whales and evaluates a set of random solutions according to both the position of each whale and a certain objective function. By comparing the primary random solutions, WOA selects an initial best search agent. In each iteration, whales update their position with respect to either a randomly chosen search agent or the best solution obtained so far. Then, the new positions are calculated according to Equations (12) and (13):  
\[\Delta=|\Gamma\cdot\mathbf{X}^{*}(w)-\mathbf{X}(w)| \tag{12}\]  
\[\mathbf{X}(w+1)=\mathbf{X}^{*}(w)-\Psi\cdot\Delta \tag{13}\]  
where \(w\) is the current whale generation, and \(\mathbf{X}^{*}\) is the updated position vector of the best solution. In addition, the coefficient vectors \(\Psi\) and \(\Gamma\) are calculated as follows:  
\[\Psi=2a\cdot\mathbf{r}-a \tag{14}\]  
\[\Gamma=2\mathbf{r} \tag{15}\]  
with \(\mathbf{r}\) as a random value in the domain (0, 1). Besides, \(a\) denotes a sequentially decreasing number from 2 to 0 as whales generation \(w\) increases, to provide exploration and exploitation. According to [PERSON] and [PERSON] [56], a random search agent is chosen when \(\mid\Psi\mid\geq 1\), while the best solution is selected when \(\mid\Psi\mid<1\) for updating the position of the search agents. The bubble-net feeding behavior of humpback whales serves as an inspiration to update the position of the search agents. It is assumed that there is the same probability of selecting between either a shrinking encircling mechanism or a spiral model, to improve the position of the search agents during the optimization procedure and get closer to the optimal solution. This concept is mathematically formulated as:  
\[\mathbf{X}(w+1)=\left\{ \begin{array}{ll}\mathbf{X}^{*}(w)-\Psi\cdot\Delta& \text{if }p<\ 0.5\\ \Delta^{\prime}\cdot e^{bl}\cdot\cos(2\pi l)+\mathbf{X}^{*}(w)&\text{if }p\ \geq 0.5 \end{array} \right. \tag{16}\]  
where \(b\) is a constant that defines the shape of the logarithmic spiral; \(l\) is a random number in the space \((-1,1)\); \(p\) is a random probability in \((0,1)\). In the adaptation of the algorithm, the constant \(b\) was set to 1 following the recommendations provided in [61; 62; 63]. On the other hand, the \(l\) and \(p\) parameters are chosen randomly in their respective domain for every generation \(w\), in order to avoid convergence at local minimums and to improve the efficiency of the algorithm. Finally, \(\Delta^{\prime}\) defines the distance of the \(i\)th whale to the best solution obtained so far, which could be calculated as:  
\[\Delta^{\prime}=|\mathbf{X}^{*}(w)-\mathbf{X}(w)| \tag{17}\]Six objective functions are proposed in the optimization process to diminish the dynamic response of the structure when it is subjected to seismic excitations. These functions are associated with the reduction of the maximum horizontal peak of displacement of the structure, and the reduction of the root mean square (RMS) values of displacements. Certainly, there are other critical parameters to be analyzed in a robust structural design, such as inter-story drifts and peak floor accelerations, which are closely related to structural damage.