# A Multi-Level Optimization Method for Elastic Constants Identification of Composite Laminates
## 3 Numerical Examples of Global Minimum Determination  
The proposed MLOM was first used to find the global minima of the following well-known functions. \[f=0.5+\frac{sin^{2}(x^{2}-y^{2})-0.5}{\left[1+0.001(x^{2}+y^{2})\right]^{2}} \tag{9}\]  
\[ \begin{array}{c}f=100\sqrt{\left|y-0.01x^{2}\right|}+0.01|x+10|\\ \text{for }-15\leqq\text{x}\leqq-5,\;-3\leqq\text{y}\leqq\;3\\ \text{or for }-100\leqq\text{x}\leqq 100,\;-100\leqq\text{y}\leqq\;100\end{array}  \tag{10}\]  
The function \(f\) in Equation (9) or (10) that was minimized can be expressed in the form of Equation (1). The proposed multi-level optimization technique, as described in Section 2, was used to search for the global minimum of the function. At the first level where MGST was applied to solve the unconstrained minimization problem, as stated in Equation (2), ten starting points were randomly generated from a uniform distribution to determine the quasi-global minima. For example, the contracted feasible region, adopted for the next level, was determined as \(\left[\mu_{i}\pm q_{i}\sigma_{i}\right]\), where \(q_{i}=1\). At the second level, where PSOT was applied to solve the optimization problem, five sets, of which each set contained 10 randomly generated starting points, were used to search for the quasi-global minima. The contracted feasible region, adopted for the next level, was also determined as \(\left[\mu_{i}\pm\sigma_{i}\right]\). The iteration between the two levels continued until the standard deviations of the quasi-true design variables were less than the prescribed value (\(\sigma_{\zeta}=10^{-7}\)). Herein, three levels were used to search for the global minima of the functions. Using the Bukin function (I) as an example, the initial and reduced feasible regions at different levels of optimization are shown in Figure 2. It is noted that, at any level, the reduced feasible region was significantly smaller than the initial one. Figure 1: The schematic diagram of the particle swarm optimization technique. The solutions of the functions, including the statistics of the identified design variables, are listed in Table 1. It is noted that the present method can very accurately predict results with less than or equal to 0.01% errors. For comparison purposes, other global minimization techniques, for instance, genetic algorithm (GA) and particle swarm (PS), available in MATLAB and the Statistical Multi-start Global Optimization Method (MSGO) [25], were also used directly to search for the global minima of the functions. The GA and PS are available in the global optimization toolbox of MATLAB [26]. Regarding GA, ten sets of randomly generated starting points were used to find the global minimum. For each set, the number of starting points (population size) was 10,000, the maximum number of generations was 10,000, the crossover fraction was 0.8, and the mutation rate was \(10^{-5}\). Here, the true global minimum was treated as the lowest value of the quasi-global minima, which was identified from the ten sets of randomly generated starting points. Regarding PS, ten sets, of which each set had 100 randomly generated particles, were chosen to find the global minimum with the use of the following coefficients: The coefficient of inertia 0.01, the cognitive coefficient 0.1, and the social coefficient 0.01. Again, the true global minimum was treated as the lowest value of the quasi-global minima identified from the ten sets. Regarding MSGO, 100 starting points were randomly generated to search for the global minimum  
Figure 2: Solution of the Bukin function (I). with the acceptance probability equal to 0.98. The identified design variables predicted using different methods are listed in Table 2 for comparison. It is noted that the overall performance of the present method was superior to the other methods. For instance, among the adopted optimization methods, only PS and MLOM could produce the exact values for the design variables of the Bukin function (I) with small bounds of the design variables.