## 3 Multi-Objective Optimization
### SVR Regression Model  
After the data of 45 test cases are obtained by numerical simulation, it is necessary to fit them to obtain the regression models of heat transfer quantity, oil pressure drop and oil outlet temperature with respect to fin angle, oil flow rate and water flow rate. These regression models will be used as objective functions of multi-objective optimization. Support vector regression (SVR) is an extension of support vector machine (SVM), which is widely used for regression of engineering problems. The given data set is set as \(D=\{(x_{1},y_{1}),\)\((x_{2},y_{2}),\)\(...,\)\((x_{n},y_{n})\}\), where \(x_{i}\in R_{n}\) is the vector of input variables and \(y_{i}\in R_{n}\) is the corresponding scalar output (target) value. The target of SVR is to minimize the \"distance\" to the farthest sample point of the fitted hyperplane, so as to accurately predict the target \(\{y_{i}\}\) corresponding to a set of input samples \(\{x_{i}\}\). SVR creates a \"spacing band\" on both sides of the hyperplane with a spacing of \(\varepsilon\) (tolerance bias), and does not calculate the loss for all samples that fall into the spacing band. For the linear problem, SVR constructs the following linear model [31; 32]:  
\[f(x_{i})\ =\ \omega x_{i}+b \tag{7}\]  
Thus, its optimization objective is:  
\[ \begin{cases}\min\frac{1}{2}||\omega||^{2}\\ s.t.|y_{i}-(\omega x_{i}+b)|\leq\varepsilon,\ \forall\text{i}\end{cases}  \tag{8}\]  
SVR allows for the presence of samples outside the spacing band, but the losses should be as small as possible so that the optimization objective of SVR can be formalized as:  
\[\min\frac{1}{2}||\omega||^{2}+C\sum\limits_{i=1}^{m}L_{c}(f(x_{i}),y_{i}) \tag{9}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Simulation Serial** & **Oil-Side Flow Rate** & **Water-Side Flow Rate** & **Fin Angle (Degree)** \\
**No.** & **(L/min)** & **(L/min)** & **Fin Angle (Degree)** \\ \hline I & 5 & 5 & 90 \\ II & 10 & 10 & 90 \\ III & 15 & 15 & 90 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Three simulated cases for validation via \(\eta\)-NTU. where \(C\sum\limits_{i=\;1}^{m}L_{c}(f(x_{i}),y_{i})\) is the empirical error term of SVR, consisting of a loss function \(L_{C}\) (\(f\left(x_{i}\right),y_{i}\)) and a factor \(C\). The loss function is expressed as:  
\[L_{C}(f(x_{i}),y_{i})\ =\ \left\{ \begin{matrix}0,\ |y_{i}-f(x_{i})|\leq\varepsilon\\ |y_{i}-f(x_{i})|-\varepsilon,\ |y_{i}-f(x_{i})|>\varepsilon\end{matrix} \right. \tag{10}\]  
and the factor \(C\) represents the weights. To explain the error beyond the limit \(\varepsilon\), the relaxation variables \(\xi\) and \(\xi^{*}\) are introduced. At this point, all sample data meet the condition:  
\[|y_{i}-(\omega x_{i}+b)|\leq\varepsilon+\xi,\ \forall i \tag{11}\]  
The relaxation variable transforms the SVR problem into a dual optimization problem, and its optimization objectives are:  
\[\left\{ \begin{matrix}\min\limits_{\omega,b_{i},\xi_{i}}\frac{1}{2}||\omega \,||^{2}+C\sum\limits_{i=\;1}^{m}\left(\xi_{i},\ \xi_{i}^{*}\right)\\ s.t.y_{i}-f(x_{i})\leq\varepsilon+\xi_{i}\\ f(x_{i})-y_{i}\leq\varepsilon+\xi_{i}^{*}\\ \xi_{i},\xi_{i}^{*}\geq 0,i\ =\ 1,2,\ldots m\end{matrix} \right. \tag{12}\]  
The optimization problem with constraints can be transformed into an unconstrained optimization problem by Lagrange multiplier method. With the introduction of Lagrange multipliers \(\alpha_{i}\) and \(\alpha_{i}^{*}\), the linear model can be rewritten as:  
\[f(x)\ =\ \omega x+b\ =\ \sum\limits_{i=\;1}^{m}\left(\alpha_{i}-\alpha_{i}^{* }\right)x_{i}^{T}x+b \tag{13}\]  
For nonlinear problems, the nonlinear problem can be mapped to a linear problem by boosting the sample dimension. However, when the data dimension itself is large, boosting can make the computational effort increase dramatically. To overcome the contradiction between high-dimensional feature space and computational complexity, SVR will define appropriate kernel functions [33].