# Adaptive Multi-Level Search for Global Optimization:
## 5 Benchmark Problem
### Mathematical Problem G10  
This problem was first described in [47] and then was considered one of the benchmark problems at the 2006 IEEE Congress on Evolutionary Computation [48]. In this optimization example, there are eight variables and six inequality constraints (three linear and three nonlinear). The mathematical formulations are shown below. \[ \begin{split} min\ f(\mathbf{x})&=x_{1}+x_{2}+x_{3}\\ s.t.\ g_{1}(\mathbf{x})&=-1+0.0025(x_{4}+x_{6})\leq 0\\ g_{2}(\mathbf{x})&=-1+0.0025(x_{5}+x_{7}-x_{4})\leq 0 \\ g_{3}(\mathbf{x})&=-1+0.01(x_{8}-x_{5})\leq 0\\ g_{4}(\mathbf{x})&=-x_{1}x_{6}+833.33252x_{4}+100x_{1}- 83,333.333\leq 0\\ g_{5}(\mathbf{x})&=-x_{2}x_{7}+1250x_{5}+x_{2}x_{4}-1 250x_{4}\leq 0\\ g_{6}(\mathbf{x})&=-x_{3}x_{8}+1,\!250,\!000+x_{3}x_{ 5}-2500x_{5}\leq 0\\ \text{where }100\leq x_{1}\leq 10,\!000,1000\leq x_{i}\leq 10,\!000 (i=2,\,3),\\ 10\leq x_{i}\leq 1000(i=4,\ldots,\,8)\end{split}  \tag{20}\]  
The optimal solutions found by ASIM and MAM are given in Table 9 as well as the known optimum. In Table 10, nine independent experiments have been performed and each experiment includes 40 parallel runs of MAM. Although each run by MAM is initialized with a random starting point, there is no guarantee that the converged global optimum can be achieved. As there has a very small feasible region \((0.0010\%)\) in this challenging example, limited runs by MAM could not find a feasible solution, and normally, a bad design with a very large value of the fitness function (up to 100,000) is obtained. However, a feasible solution could be achieved within 20,000 function evaluations. Applying the developed ASIM, the capability of the adaptive multi-level search for the global optimum was significantly improved, and the statistical results are shown in Table 11. Using the same parameter settings in the previous example, the worst solution found by the particles is about 7361, which is only 4.42% higher than the global optimum 7049.248. In the mean time, all nine independent experiments of ASIM could find a decent global optimum, which is slightly \(10^{-5}\) higher than the global optimum even in the worst case (number 5 in Table 11). In Figure 7, it shows how 10 independent runs initialized with a total of 50 particles converge to the global optimum by ASIM. It is noted that the initial design varies dramatically for each particle, and finally, all particles succeed in finding the global optimum. It is concluded that the PSO process applied in ASIM remarkably boosts the exploration capability. Owing to advantages such as the guidance of personal memory for the best position and social cognition in addition to the stochastic search behavior, ASIM is a robust and efficient algorithm for solving such a challenging problem. \begin{table}
\begin{tabular}{c l l l} \hline \hline
**Description** & **Solution**\([\mathbf{x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}}]\) & **Objective Value** \\ \hline \multirow{4}{*}{Known optimum [48]} & [579.3066850, 1359.9706780, 5109.970657, \\  & 182.0176996, 295.6011737, 217.98230036, 286.4165259, \\  & 395.6011737] \\ \hline \multirow{2}{*}{ASIM} & [579.0697378, 1360.029849, 5110.148583, 181.9979046, \\  & 295.5940579, 218.0020914, 286.4038427, 395.5940579] \\ \hline \multirow{2}{*}{MAM} & [579.2439615, 1360.814966, 5109.189094, 182.0124631, \\  & 295.6324343, 217.9875329,286.380036, 395.6324333] \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Optimal solutions of G10 found by ASIM and MAM. Recently, other algorithms including evolutionary optimization by approximate ranking and surrogate models (EOAS) [49], constraint optimization via particle swarm optimization (COPSO) [50], league championship algorithm (LCA) [46], hybrid cuckoo search (HCS) [23], and surrogate-assisted differential evolution (SADE) [51] have also solved this optimization problem.