## 3 Design Methodology
### Generation of Trade-Off Designs  
The presented bi-objective optimization algorithm involves two stages: (i) generation of the extreme (or corner) designs and (ii) sequential determination of the Pareto set using constrained optimizations. The first corner solution is obtained by solving the following minimization problem:  
\[{\mathbf{x}_{1.d_{n}}}^{*}=\arg\min_{{\mathbf{x}_{d_{n}}}\in\mathbf{X}}F_{1}({\mathbf{x}_{d_{n}}}) \tag{4}\]  
Note that Equation (4) involves optimization of the structure in regard to a single objective. The second design is obtained through constrained optimization, oriented toward minimization of the second performance criterion:  
\[{\mathbf{x}_{2.d_{n}}}^{*}=\arg\min_{{\mathbf{x}_{d_{n}}}\in\mathbf{X}}F_{2}({\mathbf{x}_{d_{n }}}) \tag{5}\]  
The goal of Equation (5) is to minimize \(F_{2}\) while ensuring that the response of the optimized design \({\mathbf{x}_{2.d_{n}}}^{*}\) does not violate the defined threshold \(F_{1.\max}\). The reasoning behind the introduction of a design constraint is to permit minimization of the second objective while maintaining the response regarding \(F_{1}({\mathbf{x}_{d_{n}}})\) at an acceptable level. Generation of the extreme points is conceptually illustrated in Figure 3.  
In the second stage, the algorithm sequentially identifies the Pareto designs. In each step, the new points are obtained between the already existing optimal solutions. Let \({\mathbf{x}_{p.d_{m}}}^{*}\) (\(p=1,\,\ldots,\,l_{k};\,l_{k}=2^{k-1}+1\)) represent the Pareto designs obtained in the \(k-1\) iteration, where \(k=1,\,\ldots,\,K\). It should be noted that \(l_{0}=2\) refers to two corner designs--\({\mathbf{x}_{1.d_{m}}}^{*}\), \({\mathbf{x}_{2.d_{n}}}^{*}\) --found in the first design stage. The candidate solutions are determined as follows:  
\[{\mathbf{x}_{c.d_{m}}^{(0)}}=\Big{[}{\mathbf{x}_{p.d_{m}}^{*}+{\mathbf{x}_{p+1.d_{m}}}^{*} }\Big{]}/2 \tag{6}\]  
where \(c=l_{k}+b\) and \(b=1,\,\ldots,\,l_{k}-1\). It should be noted that the reference designs in Equation (6) are approximated using \(d_{m}\) variables. Hence, the \(c\)th candidate design can be generated even when using \({\mathbf{x}_{p.d_{m}}^{*}}\) and \({\mathbf{x}_{p+1.d_{m}}^{*}}\)\(d_{n}\
eq d_{o}\) (\(o=1,\,\ldots,\,N\)). This is ensured by interpolating the reference designs' dimensionality as \({\mathbf{x}_{p.d_{m}}}^{(0)}\)\(\leftarrow\)\({\it sp}({\mathbf{x}_{p.d_{m}}}^{*},d_{m})\), \({\mathbf{x}_{p+1.d_{m}}}^{(0)}\)\(\leftarrow\)\({\it sp}({\mathbf{x}_{p+1.d_{o}}}^{*},d_{m})\) (see Section 2). The obtained candidate solutions are utilized as starting points for constrained optimization:  
\[{\mathbf{x}_{c.d_{n}}}^{*}=\arg\min_{{\mathbf{x}_{c.d_{n}}}\in\mathbf{X}}F_{1}({\mathbf{x}_{c.d_{n}}}) \tag{7}\]  
Figure 2: Conceptual illustration of the feature space for the bi-objective problem, characterized by a gradual increase of dimensionality. (**a**) \(d_{1}\), (**b**) \(d_{2}>d_{1}\), and (**c**) \(d_{3}>d_{2}\). The red line represents the Pareto front [39]. Note that the solutions that are attainable for the structure represented using, for example, \(d_{1}\) variables, represent a subset of the responses that can be found when the dimensionality is increased. The goal of Equation (7) is to minimize the objective function \(F_{1}\) while maintaining the structure response with respect to the second objective below the specified threshold. The latter is set to \(F_{2.c.\max}=0.5(F_{2}(x_{p,dn})+F_{2}(x_{p+1.dn}))\). In order to facilitate generation of the Pareto set, the constraint on \(F_{2}\) is implemented in the form of a penalty coefficient included with the objective function. The process in Equation (7) is performed in a nested optimization loop as shown in Section 3.2. As already mentioned, such an approach allows one to gradually increase the quality of the Pareto designs by appropriate adjustment of the antenna dimensionality.