# Midspan Deflection Prediction of Long-Span Cable-Stayed Bridge Based on DIWPSO-SVM Algorithm
## 2 DIWPSO-SVM Algorithm  
### SVM Algorithm  
Support vector machine (SVM) is a new learning machine algorithm based on statistical learning and structural risk theory. Its basic model is defined as a linear classifier with maximum intervals on the feature space, and the essential work is finding a separating hyperplane by dividing the sample data [29]. Suppose the dataset on the feature space is \(T=\{(x_{1},y_{1}),(x_{2},y_{2}),\ldots,(x_{n},y_{n})\}\) when the training dataset is linearly divisible, the process of finding the maximum separating hyperplane for the training dataset \(T\) is as follows:  
As shown in Figure 1, for a given training dataset \(T\) in the feature space and hyperplane (\(wx+b=0\)) (in which \(w\) is the weight vector and \(b\) is the fitting deviation), the geometric spacing of the hyperplane to the sample point (\(x_{i},y_{i}\)) is defined as shown in Equation (1), and the minimum of the geometrical interval of all the hyperplanes about the sample point is shown in Equation (2):  
\[\gamma_{i}=y_{i}(\frac{w}{\|w\|}x_{i}+\frac{b}{\|w\|}) \tag{1}\]  
\[\gamma=min\{\gamma_{i}\},i=1,2,\ldots,n \tag{2}\]According to Equations (3) and (4), it can be seen that the process of finding the maximum separating hyperplane can be transformed into an optimization problem of maximizing \(\gamma\) under the constraint of minimizing all geometric intervals. This problem is equivalent to finding the minimization \(\|w\|^{2}/2\), such that \(\gamma=1/\|w\|\); the objective function and constraints of this objective function are shown in Equations (3) and (4):  
\[\min\frac{1}{2}\|w\|^{2} \tag{3}\]  
\[s.t.\;\;y_{i}(wx_{i}+b)\geq 1\;\;,i=1,2,\ldots,n \tag{4}\]  
The convex quadratic programming problem with constraints is transformed into a Lagrangian dyadic problem without constraints using the Lagrange multiplier method by attaching a Lagrange multiplier \(\alpha\) to the constraints, and the newly constructed function is shown in Equation (5):  
\[L(w,b,\alpha)=\frac{1}{2}\|w\|^{2}-\sum_{i=1}^{n}\alpha_{i}[y_{i}(wx_{i}+b)-1] \tag{5}\]  
The \(L(w,\,b,\,\alpha)\) is, respectively, a partial derivative of \(w\) and \(b\) to \(0\), and its result is brought back to the original function to obtain a new objective function and constraints, which can be used to find the maximum-margin hyperplane of the training dataset, i.e., the \"decision plane\". However, the midspan deflection monitoring data of the main bridge girder cannot be attained in a linearly differentiated manner. Thus, the objective function and constraints under the nonlinearly differentiated dataset are shown in Equations (6) and (7), respectively, through the introduction of the kernel function (\(K(x_{i},x_{j})\)) and the penalty parameter (C) to allow specific data points to fail to meet the constraint conditions.  
\[\min_{\alpha}\frac{1}{2}\sum_{i=1}^{n}\sum_{j=1}^{n}\alpha_{i}x_{j}y_{i}y_{j}K \big{(}x_{i},x_{j}\big{)}-\sum_{i=1}^{n}\alpha_{i} \tag{6}\]  
\[s.t.\;\;\sum_{i=1}^{n}\alpha_{i}y_{i}=0\;;0\leq\alpha_{i}\leq C\;,i=1,2,\ldots,n \tag{7}\]  
Figure 1: Schematic of the decision boundary.  
The optimal \(\alpha^{*}=(\alpha_{1}^{*},\alpha_{2}^{*},\ldots,\alpha_{n}^{*})^{T}\) for the dataset under nonlinearly differentiable conditions can be obtained by the above transformation.