## 3 Design Methodology
### Generation of Trade-Off Designs
Finally, the designs \(\{x_{p,dn}\}_{p=1,\ldots,\,l}\) and \(\{x_{c,dn}\}_{c=\,lk\,+\,(1,\ldots,\,l)}\) are concatenated as \(\{x_{1,d}\,,x_{1+1,dn}\,,x_{2,dn}\,,x_{2+2,dn}\,,\ldots,x_{lk-1,dn}\,,x_{2k-1,dn}\,,x_{lk,dn}\}\) and reindexed to \(\{x_{1,d}\,,x_{2,dn}\,,x_{3,dn}\,,x_{4,dn}\,,\ldots,x_{2 lk-3,dn}\,,x_{2 lk-2,dn }\,,x_{2 lk-1,dn}\,\}\). The discussed optimization routine (see Figure 4 for a block diagram) can be summarized as follows:  
1. Set the initial designs \(x_{1,d1}\)\({}^{(0)}\) and \(x_{2,d1}\)\({}^{(0)}\); set \(d=[d_{1}\,d_{2}\,\ldots\,d_{m}\,\ldots\,d_{n}\,\ldots\,d_{N}]^{T}\);
2. Obtain \(x_{1,dn}\)\({}^{*}\) and \(x_{2,d0}\)\({}^{*}\) (\(n\leq N\), \(o\leq N\)) by solving Equations (4) and (5) in a nested loop (see Section 3.2);
3. Set \(k=1\);
4. Set \(p=1\);
5. Set \(x_{p,dm}\)\({}^{*}\gets sp(x_{p,dn}\,,d_{m}),x_{p+1,dm}\)\({}^{*}\gets sp(x_{p+1,d0}\,,d_{m})\) (see Section 2), and then calculate \(x_{c,dm}\)\({}^{(0)}\) and set \(F_{2,c.\max}\);  
Figure 3: Generation of the extreme Pareto designs, with a conceptual illustration of the feature (**left**) and design (**right**) spaces; optimization w.r.t. (**a**) first and (**b**) second design objective. Note that the design space is represented in the form of three-dimensional projections (hence, the solutions for \(d_{1}=3\) are shown only on the black plot, whereas the designs with \(d_{n}\) and \(d_{N}\) parameters are shown on two and three plots at a time). The optimal geometry obtained for the structure represented using \(d_{n-1}\) dimensional design is used as a starting point for optimization of the \(d_{n}\) dimensional antenna. * Generate \(\mathbf{x}_{c.dtn}\)' by solving Equation (7) in a nested optimization loop;
* If \(p=l_{k}\), go to 7; otherwise set \(p=p+1\) and go to 5;
* Concatenate the designs as \(\{\mathbf{x}_{1,d}\)', \(\mathbf{x}_{b+1,dn}\)', \(\mathbf{x}_{2,dn}\)', \(\mathbf{x}_{b+2,dn}\)',..., \(\mathbf{x}_{k-1,dn}\)', \(\mathbf{x}_{2k-1,dn}\)', \(\mathbf{x}_{2k,dn}\)'}, and then reindex the resulting set;
* If \(k=K\), END; otherwise, set \(k=k+1\) and go to 4. In the first stage of the presented framework, the initial designs \(\mathbf{x}_{1,d1}\)\({}^{(0)}\) and \(\mathbf{x}_{2,d1}\)\({}^{(0)}\) are obtained using a greedy search-based topology adjustment. The goal of the procedure is to find reasonable starting points for the optimization process. Upon identification of the initial designs, the generation of the Pareto set is handled by the above-outlined routine. It is worth mentioning that the concept of generating intermediate designs is similar to the one presented in [47]. Here, however, the bisections are used only for the generation of candidate designs, which are further refined by means of constrained optimization. Furthermore, the algorithm of [47] does not support increasing the dimensionality of the problem in the course of structure optimization. Owing to the identification of new designs between the already existing ones, the algorithm allows for obtaining a Pareto set characterized by relatively even separation between the consecutive solutions (in the objective function space). Generation of the Pareto designs using the discussed algorithm is conceptually illustrated in Figure 5.  
Figure 4: Block diagram of the proposed bi-objective optimization algorithm.