# Adaptive Multi-Level Search for Global Optimization:
## 2 Brief Review of the Multipoint Approximation Method (MAM)
Here, the selection of weighting factors \(\omega_{i}\) should reflect the quality of the objective function and the location  
where \(\alpha,\beta>0\) are user-defined constants, where, here, \(\alpha=4\) and \(\beta=1.5\) are used; \(x^{k}\) is the starting point in the \(k\)th iteration; and \(x_{i}\) is the \(i\)th design point in the fitting points. With this definition, a point with a larger objective function has a smaller weighting coefficient component \(w_{i}^{\alpha}\). For a constraint function \(g(\mathbf{x})\), a point that is much closer to the boundary of the feasible region of \(g(\mathbf{x})\) is given a larger weighting coefficient component \(w_{i}^{c}\). For building a surrogate of the objective function \(f(\mathbf{x})\), the weighting coefficient \(w_{i}\) only considers the component \(w_{i}^{\alpha}\). However, for building a surrogate of the constraint function \(g(\mathbf{x})\), the weighting coefficient \(w_{i}\) will also take the constraint component \(w_{i}^{c}\) into consideration. It should be noted here that, in MAM, both the objective and constraint functions will be approximated by Equation (3). The simplest case of \(\varphi_{l}\) is the first-order polynomial metamodel, and more complex ones are intrinsically linear functions (ILF) that have been successfully applied for solving various design optimization problems [25,28,29]. ILFs are nonlinear but they can be led to linear ones by simple transformations. Currently, five functions are considered in the regressor pool \(\{\varphi_{l}(\mathbf{x})\}\):  
\[\varphi_{1}(\mathbf{x}) =a_{0}+\sum_{i=1}^{N}a_{i}x_{i}\] \[\varphi_{2}(\mathbf{x}) =a_{0}+\sum_{i=1}^{N}a_{i}x_{i}^{2}\] \[\varphi_{3}(\mathbf{x}) =a_{0}+\sum_{i=1}^{N}a_{i}/\,x_{i} \tag{7}\] \[\varphi_{4}(\mathbf{x}) =a_{0}+\sum_{i=1}^{N}a_{i}/\,x_{i}^{2}\] \[\varphi_{5}(\mathbf{x}) =a_{0}\prod_{i=1}^{N}x_{i}^{a_{i}}\]  
In the second stage, for each function (\(f(\mathbf{x})\) or \(g(\mathbf{x})\)), different surrogates are assembled into one metamodel:  
\[\tilde{F}(\mathbf{x})=\sum_{l=1}^{n_{l}}b_{l}\varphi_{l}(\mathbf{x}) \tag{8}\]  
where \(n_{l}\) is the number of surrogates applied in the model bank \(\{\varphi_{l}(\mathbf{x})\}\) and \(b_{l}\) is the regression coefficient corresponding to each surrogate \(\varphi_{l}(\mathbf{x})\), which reflects the quality of the individual \(\varphi_{l}(\mathbf{x})\) on the set of validation points. Similar to Equation (3), \(b_{l}\) can be determined in the same manner:  
\[min\quad\sum_{i=1}^{n}\omega_{i}\big{[}F(\mathbf{x}_{i})-\tilde{F}(\mathbf{x}_{i}, \mathbf{b})\big{]}^{2} \tag{9}\]  
It should be noted that, in the process of metamodel building, the design of experiments (DOE) is fixed, i.e., \(\omega_{i}\) remains unchanged across the aforementioned stages. Figure 1 illustrates the main steps in MAM. Note that, once the metamodels for the objective and constraint functions have been built, the constrained optimization subproblem formulated in the trust region (Equation (2)) could be solved by any existing optimizers. In this paper, the sequential quadratic programming (SQP) method [31] is applied to solve the constrained optimization subproblem for the optimal solution. Since numerical optimization solvers such as SQP are deterministic, the quality of the obtained solution is highly sensitive to the initial point. In other words, MAM could not perform the global search very well. To address this issue, the ASIM framework in Section 4 has been proposed to integrate the stochastic nature with the exploratory search ability of PSO for the global optimal solution.