# Large-Scale Truss-Sizing Optimization with Enhanced Hybrid HS Algorithm
## 4 Test Problems and Optimization Results
### Spatial 3586-Bar Truss Tower
Interestingly, optimal cross-sectional areas of LSSO-HHJSJA and HFSA [53] are much closer to the SQP's area distribution than those of the hybrid HS/BBBC variants [51; 53]. The linearly increasing areas realize the uniform stiffness profile required by the second loading condition acting on the tower. Convergence curves for this optimization problem are shown in Figure 8, where the plot is limited to the first 16,000 structural analyses and the 200-3000 ton weight range is represented for the \(Y\)-axis. The figure does not include the optimization histories of improved/parameterless JAYA [54; 55; 56] and BBBC-UBS [57] algorithms as these curves lie well above the one relative to hybrid BBBC with line search of Ref. [51]. This is a direct consequence of the low ratio (i.e., less than 0.1) between population size and number of optimization variables determined for \(N_{\text{POP}}=20\). Although convergence behavior of improved/parameterless JAYA was proven to be insensitive to population size in many studies, including the results presented here, it is a matter of fact that this algorithm (as well as BBBC-UBS) attempts to replace candidate designs one by one without discharging \(\mathbf{X_{\text{worst}}}\) before all of the \(N_{\text{POP}}=20\) designs were perturbed in the current iteration. Such a strategy may lead to too many analyses being performed, especially when there is a large set of optimization variables that drive the search process. For example, the bottom segments \"4\" or \"5\" include between 27% and 30% of the total number of design variables and count by 50% of the structural weights of the two towers. Figure 7: Optimized cross-sectional areas of the 3586-bar tower. It can be seen that LSSO-HHSJA again generated better intermediate designs than hybrid HS with line search [51,53] throughout optimization process. The significant improvement in convergence behavior seen for the hybrid HS and hybrid BBBC variants of Ref. [53] over the formulations of [51] was due both to having enhanced the line search strategy and carried out independent optimization runs to have statistically significant results. HFSA [53] was competitive with LSSO-HHSJA over the first 6400 structural analyses (see the inset of Figure 8 showing the detail of convergence curves in the weight range from 300 to 475 ton) and the convergence curves of the two algorithms repeatedly crossed each other before this turning point. Interestingly, the convergence trends of SA-based variants (i.e., CMLPSA and HFSA) almost overlapped with the SQP's trend in the early optimization iterations. This can be explained with the informal argument that simulated annealing develops one design at a time like SLP/SQP and CMLPSA/HFSA form their trial designs by perturbing the current best record in the fashion \(\mathbf{X_{\mathrm{TR}}}=\mathbf{X_{\mathrm{OPT}}}+\overline{\
abla}\mathbf{W}^{\mathrm{ T}}(\mathbf{X_{\mathrm{OPT}}})(\mathbf{\rho}*\delta\mathbf{X})\), which practically corresponds to a linearization with random coefficients; the \"\(*\)\" symbol denotes a new vector is defined by multiplying term by term one vector of random numbers and one perturbation vector with respect to \(\mathbf{X_{\mathrm{OPT}}}\). This similarity becomes more evident as the initial design of the gradient-based optimizer is close to the starting point or the best design included in the initial population of the metaheuristic algorithm. The SQP's convergence curve shown in Figure 8 presents the typical steps corresponding to the transition from MATLAB to DOT optimization routines, which made it possible to reach a monotonic convergence behavior. As usual, the gradient-based optimizer reduced the structural weight by a great extent in the early optimization cycles and then stepped in order to recover the constraint violation. Conversely, LSSO-HHSJA operates  
Figure 8: Convergence curves obtained for the 3586-bar tower design example. on a set of descent directions and can select the best path to remain always in the feasible search space.