# An Efficient and Fast Hybrid GWO-JAYA Algorithm for Design Optimization
## 3 Test Problems and Optimization Results
### Calibration of Nonlinear Hydrologic Models
The computational efficiency of FHGWJA operators generating new trial solutions in the search process is confirmed by the fact that the \(W(\vec{X_{i,tr}})\leq 1.1W(\vec{X_{best}})\) elitist strategy allowed mAHS, mBBBC-UBS, and MsinDE algorithms to converge to the global optimum value SSQ = 36.760 and reduce computational cost by 9.1%, 11%, and 35.6% with respect to the original AHS, BBBC-UBS, and sinDE algorithms of Refs. [94; 95; 96]. Interestingly, mAHS and mBBBC-UBS became very competitive with FHGWJA in terms of average computational cost (respectively, 2368 and 2344 function evaluations vs. only 2005 of FHGWJA) and peak computational speed of the fastest optimization run (respectively, 1670 and 1711 vs. only 1637 of FHGWJA). The hybrid HS/BBBC/SA algorithms of Ref. [81] that utilized gradient information and approximate line searches could even complete their fastest optimization runs within less function evaluations than FHGWJA (i.e., between 1281 and 1331 vs. 1637 of FHGWJA) but converged to higher values of SSQ than the global optimum 37.760 found with FHGWJA. Furthermore, hybrid HS/BBBC/SA required on average more function evaluations than FHGWJA (i.e., between 2468 and 2755 vs. only 2005 of FHGWJA). Again, in a highly nonlinear problem like the hydrologic model calibration, the elitist strategies implemented in FHGWJA were extremely efficient in generating high-quality trial solutions over the whole search process. This happened because generation of trial solutions in FHGWJA is not biased to any extent by the quality of gradient information that may instead drive hybrid HS/BBBC/SA towards poor trial solutions should the cost function gradient not be well approximated. However, looking at Table 5, it appears that computational cost was not an issue for this test problem: HS [102] and COBSA [104], respectively, required 20,000 and 100,000 function evaluations vs. only 2000 of FHGWJA while no information on the effective number of function evaluations were given in Refs. [103; 104; 105; 106] for metaheuristic (i.e., IICSA, GA, DE, and PSO) as well as gradient-based algorithms (BFGS) and zero-order heuristic methods (NMS). Figure 11 confirms the superiority of FHGWJA over its competitors in terms of convergence behavior also in this test problem. The figure compares the optimization histories of the best runs for the algorithms listed in Table 5; the average FHGWJA convergence curve is also shown in the figure. Since FHGWJA always converged to the global optimum of 36.760 in all optimization runs, its best run corresponds to the fastest one. The plot is limited to the first 4500 function evaluations of optimization history and to the 35-70 cost function interval for the sake of clarity. It can be seen from the figure that the best and average optimization runs' convergence curves for FHGWJA lie below those of the other algorithms practically over the whole search history for this test problem. FHGWJA started its best optimization run from the very large cost of 1520.145 (i.e., about 41.3 times the target optimum of 36.760 quoted in Table 5) while the initial cost for all other optimizers ranged between 181.683 and 1122.263. However, the present algorithm immediately recovered the initial gap in cost function with respect to its competitors reducing cost to 37.5 (just 2% more than the target optimum) within only 280 function evaluations while all other algorithms required at least 520 function evaluations to reach the same intermediate cost function value. The hybrid harmony search algorithm of Ref. [81] was competitive with FHGWJA only for the first 150 function evaluations, while the hybrid simulated annealing algorithm (HFSA) of Ref. [81] went close to the average convergence curve of FHGWJA at about 325 function evaluations. The \(W(\vec{X_{i,tr}})\leq 1.1W(\vec{X_{opt}})\) elitist strategy of FHGWJA allowed mAHS, mBBBC-UBS, and MsinDE to approach the cost function reduction rate of FHGWJA from 400 to 700 function evaluations, while the hybrid big bang-big crunch algorithm of Ref.