## 3 Optimization Formulation  
Addressing the main aim of this study, the mathematical formulation of the optimization process for end-plate connections is developed, considering the key nonlinear characteristics of structural behavior. The proposed equations are integrated into the design framework to provide an effective solution for beam-to-column assemblies. The primary objective of the formulation is to minimize the total volume of the connection components--including the end-plate (\(V_{ep}\)), bolts (\(V_{b}\)), and various types of stiffeners (\(V_{s}\))--while ensuring that the equivalent plastic strain (\(\overline{\varepsilon}^{pl}\)), joint rotation (\(\theta\)), and critical buckling load factor (\(\overline{\lambda}\)) remain within predefined threshold values. This enablesthe development of a structurally safe and materially efficient design. Accordingly, the optimization problem is formulated as follows:  
\[min.:fitness=f(V)+p_{1}\Big{(}\overline{\varepsilon}^{pl}\Big{)}+p_{2}(\theta)+p _{3}(\overline{\lambda}) \tag{13}\]  
_Subject to_ :  
\[f(V)=\frac{V_{ep}+V_{b}+V_{s}}{V_{max}} \tag{14}\]  
\[p_{1}\Big{(}\overline{\varepsilon}^{pl}\Big{)}= \begin{cases}\quad\quad 0,if\ \overline{ \varepsilon}^{pl}\leq\overline{\varepsilon}^{pl}_{max}\\ 1-\frac{\overline{\varepsilon}^{pl}_{max}}{\overline{\varepsilon}^{pl}},\ \ if\ \overline{ \varepsilon}^{pl}>\overline{\varepsilon}^{pl}_{max}\end{cases}  \tag{15}\]  
\[p_{2}(\theta)= \begin{cases}\quad\quad 0,\ if\ \theta\leq\theta_{max}\\ 1-\frac{\theta_{max}}{\theta},\ \ if\ \theta>\theta_{max}\end{cases}  \tag{16}\]  
\[p_{3}(\overline{\lambda})= \begin{cases}\quad\quad 0,\ if\ \overline{\lambda}\geq \lambda_{stab}\\ 1-\frac{\overline{\lambda}}{\lambda_{stab}},\ \ if\ \overline{\lambda}<\lambda_{ stab}\end{cases}  \tag{17}\]  
\[e\geq 1.2d_{b}\ \text{and}\ p\geq 2.4d_{b} \tag{18}\]  
\[\mathbf{X}=\mathbf{X}_{0}+\Delta\mathbf{x}\ \text{where}\ \Delta\mathbf{x}=\sum_{j=1}^{N}\mathbf{ \phi}_{j}\omega_{j} \tag{19}\]  
In this context, Equation (13) defines the objective function, where the fitness value--used to assess the quality of a potential solution--is composed of four components. The term \(f(V)\) represents the normalized volume of the connection elements, specifically the end-plate (\(V_{ep}\)), bolts (\(V_{b}\)), and stiffeners (\(V_{s}\)). During the optimization, a maximum allowable volume \(V_{max}\) is prescribed to guide the search toward material-efficient solutions. The first structural criterion, defined by Equation (15), imposes a constraint on the maximum equivalent plastic strain. Specifically, if the maximum equivalent plastic strain \(\overline{\varepsilon}^{pl}\) of a given configuration remains below the allowable limit \(\overline{\varepsilon}^{pl}_{max}\), the corresponding penalty function \(p_{1}\Big{(}\overline{\varepsilon}^{pl}\Big{)}\) remains zero. Conversely, if the strain exceeds this limit, the penalty function is activated, with a value ranging between 0 and 1, depending on the extent to which \(\overline{\varepsilon}^{pl}\) surpasses \(\overline{\varepsilon}^{pl}_{max}\), as defined by the penalty formulation. In a similar manner, Equation (16) introduces a criterion related to the maximum allowable joint rotation, denoted as \(\theta_{max}\). If the joint rotation \(\theta\) of a given configuration remains less than or equal to this threshold, the corresponding penalty function \(p_{2}(\theta)\) remains inactive. Otherwise, its value is determined based on the ratio between \(\theta\) and \(\theta_{max}\). Additionally, a stability-related constraint is introduced in the optimization process through Equation (17). To clarify, the critical buckling load factor \(\overline{\lambda}\) is obtained by performing a linear buckling analysis (LBA), during which multiple buckling mode shapes \(\mathbf{\phi}\) and their corresponding eigenvalues are computed. Among these, the first eigenvalue, associated with the primary mode shape \(\mathbf{\phi}_{j=1}\), is used to define \(\overline{\lambda}\).