# A Multi-Level Optimization Method for Elastic Constants Identification of Composite Laminates
## 2 Multi-Level Optimization Method  
The following constrained minimization problem should be considered, which is stated in terms of \(N\) design variables \(\mathbf{x}=\) = \([x_{1},x_{2},\ldots,x_{N}]\), the objective function \(F(\mathbf{x})\), and a set of initially prescribed side constraints \(g(\mathbf{x})=[g_{1}(\mathbf{x}),g_{2}(\mathbf{x}),\ldots,g_{2N}(\mathbf{x}) ]^{T}\leq 0\), with the superscript \(T\) denoting the transpose of the row vector. \[ \begin{array}{c}\text{Minimize }F(\mathbf{x})\\ \text{Subject to }g_{k}(\mathbf{x})\leq 0;k=1,2,\ldots,2N\end{array}  \tag{1}\]  
In general, a number of local minima in the feasible region of the above minimization problem may exist, especially in the case of elastic constants identification of composite materials. Herein, a multi-level optimization method (MLOM) in which several level-wise optimization problems were solved sequentially for determining the global minimum was proposed to deal with the above constraint minimization problem. In the proposed method, for any level of optimization, a set of starting points, randomly generated from a uniform distribution in the level-wise feasible region, were used to determine the quasi-global minima when solving the level-wise minimization problem. The statistics (means and standard deviations) of the quasi-optimal design variables associated with the quasi-global minima were determined to see if the true global minimum was attained, and if not, whether it was necessary to construct the new feasible region of the next level of optimization. The search for the true global minimum at different levels was terminated when the standard deviations of the quasi-optimal design variables were less than the prescribed value. Without the loss of generality, the commonly used global minimization algorithms such as the multi-start global search technique (MGST) [20] and particle swarm optimization technique (PSOT) [21] were adopted, respectively, at two consecutive levels to determine the quasi-global minima. Hence, in the proposed MLOM, the MGST was used to solve the first level optimization problem. The means and standard deviations of the quasi-optimal design variables obtained at this level were used to produce a contracted/reduced level-wise feasible region to establish the second level optimization problem. Following this line, the MGST and PSOT took turns to solve the subsequent levels of optimization problems. The advantage of the proposed solution strategy was that the adoption of a series of contracted level-wise feasible regions at different levels enhanced the chance of attaining the true global minimum. Herein, at the first-level of optimization, the constrained minimization problem of Equation (1) was converted to the following unconstrained minimization problem by introducing the general augmented Lagrangian [22]. \[\overline{\psi}(x,\alpha,\eta,r_{p})=F(x)+\sum_{j=1}^{N}\Bigl{[}\alpha jz_{j}+r_{p} z_{j}^{2}+\eta_{j}\phi_{j}+r_{p}\phi_{j}^{2}\Bigr{]} \tag{2}\]  
with  
\[ \begin{array}{c}z_{j}=\max\left[g_{j}\bigl{(}x_{j}\bigr{)},\frac{-a_{j}}{2r_{ p}}\right]\\ g_{j}\Bigl{(}\overline{x}_{j1}\Bigr{)}=x_{j}-x_{j}^{L}\leq 0\\ \phi_{j}=\max\left[H_{j}\bigl{(}\overline{x}_{j1}\bigr{)},\frac{-\eta_{j}}{2r_ {p}}\right]\\ H_{j}\Bigl{(}x_{j}\Bigr{)}=x_{j}^{L}-x_{j}\leq 0;j=1-N\end{array}  \tag{3}\]  
where the superscript \(U\) and subscript \(L\) denote the upper and lower bounds, respectively; \(\alpha_{j}\), \(\eta_{j}\), and \(r_{p}\) are multipliers; and \(\max\) [\",\"] takes on the maximum value of the numbers in the bracket. The updated formulas for the multipliers \(\alpha_{j}\), \(\eta_{j}\), and \(r_{p}\) are  
\[ \begin{array}{c}\alpha_{j}^{n+1}=\alpha_{j}^{n}+2r_{p}^{n}z_{j}^{n}\\ \eta_{j}^{n+1}=\eta_{j}^{n}+2r_{p}^{n}\phi_{j}^{n}\ ;\ j=1-N\\ r_{p}^{n+1}=\left\{\begin{array}{ccc}\gamma r_{p}^{n}&\text{if}&r_{p}^{n+1} \geq r_{p}^{\max}\\ r_{p}^{\max}&\text{if}&r_{p}^{n+1}\geq r_{p}^{\max}\end{array}\right.\end{array}