# Large-Scale Truss-Sizing Optimization with Enhanced Hybrid HS Algorithm
## 3 The LSSO-HHSJA Algorithm
### Step 2: Evaluation of Trial Design \(\mathbf{X_{TR}}\) and Population Updating
Nevertheless, the present algorithm utilizes a mirroring strategy to transform the non-descent direction \((\mathbf{X_{\text{TR}}}-\mathbf{X_{\text{OPT}}})\) into the descent direction \((\mathbf{X_{\text{TR}}}^{\text{mirr}}-\mathbf{X_{\text{OPT}}})\). Furthermore, \((\mathbf{X_{\text{TR}}}-\mathbf{X_{\text{OPT}}})\) is also an unfeasible direction and hence LSSO-HHJJA tries to move along \((\mathbf{X_{\text{TR}}}^{\text{mirr}}-\mathbf{X_{\text{OPT}}})\) to recover such a gap. The new trial design \(\mathbf{X_{\text{TR}}}^{\text{mirr}}\) is defined as:  
\[\mathbf{X_{\text{TR}}}^{\text{mirr}}=(1+\eta_{\text{mirr}})\mathbf{X_{\text{OPT}}}- \eta_{\text{mirr}}\mathbf{X_{\text{TR}}}\ \ \ (16(\text{a})\ \text{rep.})\]  
The new harmony \(\mathbf{X_{\text{TR}}}^{\text{mirr}}\) is evaluated. The best record and the population are updated as explained above if case (1) or case (2) or case (3) occurs. Convergence check is then done in Step 3. If \(\mathbf{X_{\text{TR}}}\) and \(\mathbf{X_{\text{TR}}}^{\text{mirr}}\) are infeasible and \(Min\{W(\mathbf{X_{\text{TR}}})\); \(W(\mathbf{X_{\text{TR}}}^{\text{mirr}})\}>W_{\text{OPT}}\), their positions are updated by reducing distances from \(\mathbf{X_{\text{OPT}}}\) and in all likelihood constraint violations. That is:  
\[\left\{ \begin{array}{c}\left(\mathbf{X_{\text{TR}}}^{\text{mirr}}\right)^{\prime}= \mathbf{X_{\text{OPT}}}+\left(\mathbf{X_{\text{TR}}}^{\text{mirr}}-\mathbf{X_{\text{OPT}}} \right)\left(\frac{G_{\text{IM}}}{Max\left\{G_{\text{K}}\right\}}\right)\\ \left(\mathbf{X_{\text{TR}}}\right)^{\prime}=\mathbf{X_{\text{OPT}}}+\left(\mathbf{X_{\text{ TR}}}-\mathbf{X_{\text{OPT}}}\right)\left(\frac{G_{\text{IM}}}{Max\left\{G_{\text{K}} \right\}}\right)\end{array} \right. \tag{18}\]  
where: k = 1,..., \(NC_{\text{act}}\) is the number of violated constraints; \(Max\left\{G_{\text{K}}\right\}\) is the largest nodal displacement or element stress (including buckling) computed from structural analysis and \(G_{\text{LIM}}\) is the corresponding allowable limit. In Ref. [53], Equation (18) was re-iterated until at least one trial point among \((\mathbf{X_{\text{TR}}})^{\prime}\) and \((\mathbf{X_{\text{TR}}}^{\text{mir}})^{\prime}\) became feasible and, ultimately, a 1-D simulated annealing search in the neighborhood of \(\mathbf{X_{\text{OPT}}}\) was carried out to locally improve designs included in [HM]. A different strategy is adopted by LSSO-HHSIA to reduce the number of structural analyses entailed by this phase. If \((\mathbf{X_{\text{TR}}})^{\prime}\) or \((\mathbf{X_{\text{TR}}}^{\text{mir}})^{\prime}\) is feasible, the new design is ranked with respect to the current population as described for cases (1) and (2), where the latter case (i.e., \(W(\mathbf{X_{\text{TR}}})>W_{OPT}\)) is much more likely to occur. If both \((\mathbf{X_{\text{TR}}})^{\prime}\) and \((\mathbf{X_{\text{TR}}}^{\text{mir}})^{\prime}\) are infeasible, the best design with the lowest constraint violation amongst \(\mathbf{X_{\text{TR}}}\), \((\mathbf{X_{\text{TR}}})^{\prime}\) and \((\mathbf{X_{\text{TR}}}^{\text{mir}})^{\prime}\) is set as \(\mathbf{X_{\text{TR,WORST}}}\) and a JAYA-based equation is used for generating yet another trial design:  
\[(\mathbf{X_{\text{TR}}})^{\text{JAYA}}=\mathbf{X_{\text{TR}}}+\mathbf{\omega_{1}}(\mathbf{X_{ \text{OPT}}}-\mathbf{X_{\text{TR}}})-\mathbf{\omega_{2}}(\mathbf{X_{\text{TR,WORST}}}-\bm {X_{\text{TR}}}) \tag{19}\]  
where random vectors \(\mathbf{\omega_{1}}\) and \(\mathbf{\omega_{2}}\) are similar to those of Equations (14) and (16). Basically, LSSO-HHSIA attempts first to simultaneously reduce constraint violation and cost function with Equations (16) and (18). Should this not work, Equation (19) moves \(\mathbf{X_{\text{TR}}}\) towards \(\mathbf{X_{\text{OPT}}}\) by escaping from the infeasible region of search space.