# Adaptive Multi-Level Search for Global Optimization:
## 4 Adaptively Integrated Swarm Intelligence-Metamodelling Framework
### Model Management
#### 4.2.2 Modified Trust Region Strategy in MAM  
The aim of the trust region strategy in MAM is to control the quality of a metamodel constructed. When the approximation gets better, the trust region is further reduced for the  
Figure 2: Flow chart of ASIM Framework. optimal solution. The track of the trust regions also indicates a path of the direction from the initial starting point to the optimum over the entire search domain. At each iteration, a new trust region must be updated, i.e., its new size and its location have to be specified. Several indicators are formulated to support the control of the trust region and to facilitate the search process. The basic knowledge about these indicators was introduced in [38]. The first indicator is to evaluate the quality of the metamodel and focuses on the accuracy of the constraint approximations at the obtained suboptimal point \(\mathbf{x}^{k+1}\). This is based on the following equation:  
\[E^{k}=Max\left(|\frac{\tilde{g}\Big{(}\mathbf{x}^{k+1}\Big{)}-g\Big{(}\mathbf{x}^{k+1 }\Big{)}}{g\big{(}\mathbf{x}^{k+1}\Big{)}}|\right) \tag{13}\]  
where \(\tilde{g}\Big{(}\mathbf{x}^{k+1}\Big{)}\) and \(g\Big{(}\mathbf{x}^{k+1}\Big{)}\) are normalized functions of the approximate and true constraints at the suboptimal point \(\mathbf{x}^{k+1}\), respectively. In this way, a single maximal error quantity between explicit approximation and implicit simulation is defined. Then, the quality of metamodel can be labeled as \"bad\", \"reasonable\", or \"good\" shown below. \[E^{k}\Rightarrow \begin{cases}\geq 0.25\times S^{k}&\Rightarrow\text{`Bad'}\\ \leq 0.01\times S^{k}&\Rightarrow\text{`Good'}\\ Else&\Rightarrow\text{`Reasonable'}\end{cases}  \tag{14}\]  
where \(S^{k}\) represents the maximum ratio of the dimension length between the present trust region, and the entire design space is defined by  
\[S^{k}=Max\left(\frac{B_{i}^{k}-A_{i}^{k}}{B_{i}-A_{i}}\right)\quad(i=1,\dots,d) \tag{15}\]  
The second indicator is for indicating the location of the current iterate \(\mathbf{x}^{k+1}\) in the present search subregion. For each dimension, if none of the current move limits \((\mathbf{A}^{k},\mathbf{B}^{k})\) is active, this solution is regarded as \"internal\"; otherwise, it is viewed as \"External\". The third and fourth indicators reflect the movement history for the entire optimization process. For this purpose, the angle between the last two move vectors is calculated. The formulation of this measure \(\theta^{k}\) is given below:  
\[\theta^{k}=\frac{\mathbf{x}^{k+1}-\mathbf{x}^{k}}{\|\mathbf{x}^{k+1}-\mathbf{x}^{k}\|}\times \frac{\mathbf{x}^{k}-\mathbf{x}^{k-1}}{\|\mathbf{x}^{k}-\mathbf{x}^{k-1}\|} \tag{16}\]  
If \(\theta^{k}>0\) holds, the movement will be denoted as \"forward\", while \(\theta^{k}\leq 0\) is denoted as moving \"backward\". Moreover, if \(\theta^{k}\leq 0.3\), the convergence history is labelled as \"curved\"; otherwise, it is \"Straight\". The fifth indicator in MAM, as a termination criterion, is the size of the current search subregion. It can be marked as \"small\" or \"large\" according to the quality of the metamodel determined by the first indicator. When the approximations are \"bad\" and \(S^{k}\leq 0.0005\), the present search subregion is considered \"small\". When the approximations are \"reasonable\" or \"good\", the trust region is denoted as \"small\" if \(S^{k}\leq 0.001\). The sixth indicator is based on the most active constraint. It is considered \"close\" to the boundary between the feasible and infeasible design space if \(g_{max}(\mathbf{x}^{k+1})\in[-0.1,0.1]\); otherwise, it is denoted as \"far\". Both reduction and enlargement of the trust region is executed using  
\[B_{i}^{k+1}-A_{i}^{k+1}=\frac{1}{\tau}\left(B_{i}^{k}-A_{i}^{k}\right)(i=1, \dots,d) \tag{17}\]  
where \(\tau\) is the resizing parameter. When the approximations are \"bad\" and the trust region is \"small\", the current trust region is considered too small for any further reduction to achieve reasonable approximations and the process is aborted.