# 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
[53] re-iterated the search by increasing the order of polynomial approximation up to 10 and eventually performed a 1-D probabilistic search based on simulated annealing. However, this process may require too many structural analyses in large-scale optimization. For this reason, the LSSO-HHSJA algorithm developed in this study combines a mirroring strategy with a JAYA-basedstrategy to turn \(\mathbf{X_{\text{TR}}}\) feasible. For that purpose, two trial designs \((\mathbf{X_{\text{TR}}})^{\prime}\) and \((\mathbf{X_{\text{TR}}})^{\prime}\) are defined as follows:  
\[\left\{ \begin{array}{c}\left(\mathbf{X_{\text{TR}}}^{\text{mirr}}\right)^{\prime}= (1+\eta_{\text{mirr}})\mathbf{X_{\text{OPT}}}-\eta_{\text{mirr}}\mathbf{X_{\text{TR}}} \\ \left(\mathbf{X_{\text{TR}}}^{\text{mirr}}\right)^{\
u}=\mathbf{X_{\text{TR}}}+\mathbf{ \omega_{1}}(\mathbf{X_{\text{OPT}}}-\mathbf{X_{\text{TR}}})-\mathbf{\omega_{2}}(\mathbf{X_{ \text{2 ndBEST}}}-\mathbf{X_{\text{TR}}})\end{array} \right. \tag{16}\]  
where \(\eta_{\text{MIRR}}\) is a random number in the interval (0,1) while random vectors \(\mathbf{\omega_{1}}\) and \(\mathbf{\omega_{2}}\) are similar to those of Equation (14). The mirroring strategy used for generating \(\left(\mathbf{X_{\text{TR}}}^{\text{mirr}}\right)^{\prime}\), Equation (16) (i.e., the first relationship of Equation (16)), steers the search in the opposite direction \((\mathbf{X_{\text{OPT}}}-\mathbf{X_{\text{TR}}})\) to the infeasible direction \((\mathbf{X_{\text{TR}}}-\mathbf{X_{\text{OPT}}})\); however, \((\mathbf{X_{\text{OPT}}}-\mathbf{X_{\text{TR}}})\) is a non-descent direction. Hence, the random number \(\eta_{\text{MIRR}}\) has two functions: (i) to reduce step size in order not to increase cost function too much; (ii) if \(\mathbf{X_{\text{OPT}}}\) lies on or is very close to the boundary of feasible search domain, it may be good to limit step size in order to reduce the probability of generating another infeasible trial design. The JAYA-based strategy used for generating \(\left(\mathbf{X_{\text{TR}}}^{\text{mirr}}\right)^{\prime\prime}\), Equation (16) (i.e. the second relationship of Equation (16)), tries to approach \(\mathbf{X_{\text{TR}}}\) to the current best record by moving along \((\mathbf{X_{\text{OPT}}}-\mathbf{X_{\text{TR}}})\), which is opposite to the infeasible direction \((\mathbf{X_{\text{TR}}}-\mathbf{X_{\text{OPT}}})\). Furthermore, the search is \"confined\" between the best two candidate solutions currently stored in [HM], where it may be easier to define high quality trial designs. This would allow at least to minimize the increase in cost function. The new trial designs \(\left(\mathbf{X_{\text{TR}}}^{\text{mirr}}\right)^{\prime}\) and \(\left(\mathbf{X_{\text{TR}}}^{\text{mirr}}\right)^{\prime\prime}\) are evaluated. If both designs are feasible, they are checked against \(\mathbf{X_{\text{OPT}}}\) according to case (1) and case (2) described above. The same is done if only one between \(\left(\mathbf{X_{\text{TR}}}^{\text{mirr}}\right)^{\prime}\) and \(\left(\mathbf{X_{\text{TR}}}^{\text{mirr}}\right)^{\prime\prime}\) is feasible. If no feasible solution is obtained, a trial design between \(\mathbf{X_{\text{OPT}}}\) and \(\mathbf{X_{\text{2 ndBEST}}}\) is generated as follows:  
\[\mathbf{X_{\text{TR}}}=\mathbf{X_{\text{OPT}}}+\alpha(\mathbf{X_{\text{OPT}}}-\mathbf{X_{ \text{2 ndBEST}}}) \tag{17}\]  
where \(\alpha\) is a random number between 0 and 1. The new trial design will be in all likelihood feasible if both \(\mathbf{X_{\text{OPT}}}\) and \(\mathbf{X_{\text{2 ndBEST}}}\) are feasible and hence it will be evaluated according to case (1) or case (2). 2.4 Case 4: \(\mathbf{X_{\text{TR}}}\) Infeasible and \(W(\mathbf{X_{\text{TR}}})>W_{\text{OPT}}\)  
This is the worst possible case, although very unlikely to occur because LSSO-HHJJA forms trial designs by perturbing variables so as to move along descent directions.