# Adaptive Multi-Level Search for Global Optimization:
## 2 Brief Review of the Multipoint Approximation Method (MAM)  
The MAM [25; 26] was proposed to tackle black-box optimization problems and has gained continuous development in recent years, e.g., [PERSON] [27] enhanced MAM to solve large-scale optimization problems, one of which is the optimization of transonic axial compressor rotor blades; [PERSON] [28] implemented discrete capability into MAM. Recently, [PERSON] [29] has applied MAM to solve a multi-objective problem. Based on a response surface methodology, multipoint approximation method (MAM) aims to construct mid-range approximations and is suitable to solve complex optimization problems owing to (1) producing better-quality approximations that are sufficiently accurate in a current trust region and (2) affordability in terms of computational costs required for their building. These approximation functions have a relatively small number (\(N+1\) where \(N\) is number of design variables) of regression coefficients to be determinedm and the corresponding least squares problem can be solved easily [25]. In general, a black-box optimization problem can be formulated as follows:  
\[ \begin{split}& min\ f(\mathbf{x})\\ & s.t.\ g_{j}(\mathbf{x})\leq 1\ (j=1,\ldots,M)\\ &\ A_{i}\leq x_{i}\leq B_{i}\ (i=1,\ldots,N)\end{split}  \tag{1}\]  
where \(\mathbf{x}\) refers to the vector of design variables; \(A_{i}\) and \(B_{i}\) are the given lower and upper bounds of the design variable \(x_{i}\); \(N\) is the total number of design variables; \(f(\mathbf{x})\) is the objective function; \(g_{j}(\mathbf{x})\) is the \(j\)th constraint function and \(M\) is the total number of the constraint functions. In order to represent the detailed physical model using the response functions and to reduce the number of calls for the response function evaluations, the MAM replaces the optimization problem with a sequence of approximate optimization problems as follows:  
\[ \begin{split}& min\ \vec{f}^{k}(\mathbf{x})\\ & s.t.\ \tilde{g}^{k}_{j}(\mathbf{x})\leq 1\ (j=1,\ldots,M)\\ &\ A_{i}\leq A_{i}^{k}\leq x_{i}\leq B_{i}^{k}\leq B_{i}\ (i=1,\ldots,N)\end{split}  \tag{2}\]  
where \(\vec{f}^{k}(\mathbf{x})\) and \(\tilde{g}^{k}_{j}(\mathbf{x})\) are the functions which approximate the functions \(f(\mathbf{x})\) and \(g_{j}(\mathbf{x})\) defined in Equation (1); \(A_{i}^{k}\) and \(B_{i}^{k}\) are the side constraints of a trust sub-region; and \(k\) is the iteration number. Compared with the time spent by evaluation of the actual response functions \(g_{j}(\mathbf{x})\), the selected form of approximate functions \(\tilde{g}^{k}_{j}(\mathbf{x})\) (\(j=0,\ldots M\)) remarkably reduces the computational expense and adequately improves the accuracy in a current trust region. This is achieved by appropriate planning of numerical experiments and use of the trust region defined by the side constraints \(A_{i}^{k}\) and \(B_{i}^{k}\). Once the current suboptimization problem is solved, the suboptimal solution becomes the starting point for the next step. Meanwhile, the move limits are modified and the trust region is resized [25; 26]. Based on this information, the metamodel is updated in the next iteration until eventually the optimum is reached. The process of metamodel building in MAM can be described as an assembly of multiple surrogates into one single metamodel using linear regression. Therefore, there are two stages of metamodel building. In the first stage, the parameter \(\mathbf{a}_{l}\) of an individual surrogate \(\varphi_{l}\) is determined by solving a weighted least squares problem involving \(n\) fitting points as  
\[ \begin{split} min&\sum_{i=1}^{n}\omega_{i}[F(\mathbf{x }_{i})-\varphi_{l}(\mathbf{x}_{i},\mathbf{a}_{l})]^{2}\end{split}  \tag{3}\]  
where \(\omega_{i}\) denotes the weighting parameters and \(F\) is the original function that needs to be approximated.