## 3 Proposed Algorithm
### Offspring Generation Using TCA  
In recent years, more and more attention has been paid to transfer learning, which can effectively solve issues between different problems by conducting knowledge transfer [37; 38]. Domain adaption is an important branch of transfer learning and transfer component analysis (TCA) is commonly used to reuse the knowledge across problems [39; 40]. In this paper, TCA is applied to EGT to minimize the two different probability distributionsfor source and target domains in latent space [41]. In the reproducing kernel Hilbert space (RKHS), the distance between two different distributions can be calculated using a kernel function. Based on the RKHS, the maximum mean discrepancy (MMD) is employed to minimize the discrepancy between two distributions [41]. To retain the key properties and reduce computational complexity, we construct the kernel function \(\kappa\) from the data to obtain a low-rank transformation matrix \(W\).  
The TCA algorithm for solving optimization problems can be presented as follows.  
\[\begin{split}\underset{W}{\arg\min}\ \mu\cdot\text{tr}\left(W^{T}W \right)+\text{tr}\left(W^{T}KLKW\right)\\ \text{subject to}\ W^{T}KHKW=\mathbf{I}\end{split} \tag{12}\]  
where \(\mathbf{H}=\mathbf{I}-(1/[m+n])\mathbf{1}^{T}\) and \(\mathbf{I}\in\mathbf{R}^{(m\times n)\times(m\times n)}\). \(m\) and \(n\) denote the numbers of sampled solutions for source and target data. The solution of Equation (12) can be solved by the well-known generalized eigenvalue decomposition [42].  
In the following algorithm, Tr-OG algorithm (provide pseudo code in Algorithm 2), taking sampled solutions in \(F_{\text{s}}(X_{s})\) and \(F_{\text{f}}(Y_{t})\) as the input to the TCA, the transformation matrix \(W\) can be naturally obtained. The latent space (LS) is constructed with the \(W\). Further, a decision variable \(x\) is searched in LS according to \(\|\varphi(F_{\text{f}}(x))-l\|\) (line 6), which is used to generate the offspring population for \(F_{\text{f}}(Y_{t})\).  
```
0:\(F_{\text{s}}(.)\), the optimization function of source domain; \(F_{\text{f}}(.)\), the optimization function of target domain; \(POF_{g}\), the POF of the function \(F_{\text{s}}(.)\); \(\kappa(\cdot,\cdot)\), kernel function.
0: A offspring for \(F_{\text{f}}(.)\).
1: Initialization.
2: Randomly sample two sets of the solutions with respect to \(F_{\text{s}}(.)\) and \(F_{\text{f}}(.)\), defined by \(X_{\text{s}}\) and \(Y_{\text{f}}\), respectively.
3: Calculate \(F_{\text{s}}(X_{s})\) and \(F_{\text{f}}(Y_{t})\) of two different optimization problems.
4: Obtain a transformation matrix \(W\) through TCA \(\{F_{\text{s}}(X_{s})\), \(F_{\text{f}}(Y_{t})\), \(\kappa(\cdot,\cdot)\}\).
5: Construct the LS by using a mapping function \(\varphi(.)\).
6: Generate a offspring for \(F_{\text{f}}(.)\) via \(\underset{x}{\arg\min}\|\varphi(F_{\text{f}}(x))-l\|\), \(l\in\text{LS}\).
```  
**Algorithm 2** Framework of Tr-OG  
For example, the Tr-EGT algorithm is used to solve the optimization problem \(OP_{\text{l}}\), which contains two objectives of yield and production capacity utilization. Suppose we have obtained the POF of \(OP_{\text{l}}\) problem, \(POF_{g}\) and \(p\in POF_{g}\). A mapping function \(\varphi(.)\) is obtained via the TCA algorithm, which is used to map the \(p\) into a high-dimension LS. Then a solution \(x\) for the \(OP_{\text{2}}\) problem at generation g+1 will be found by using the Tr-EGT algorithm, and this solution \(x\) meets the requirements which is closest to \(l\) in the LS. This solution \(x\) will be output as one of the offspring with regard to the initial population, which can help the \(OP_{\text{2}}\) problem at generation g+1 solve faster. At the same time, when we get the POF of the \(OP_{\text{2}}\) problem, the above method can also speed up the solution of \(OP_{\text{1}}\) faster, thereby achieving an efficient solution of the problem between the two tasks.