## 2 Introduction to Optimized Method
### Multi-Objective Programming Model for Cable Force Optimization
#### 2.2.3 Establishment of Optimization Model  
According to Equation (3), a multi-objective programming model can be formulated as follows:  
\[\min f(\Delta\mathbf{N})=\mathbf{C}\Delta\mathbf{N}-\mathbf{d}, \tag{4}\]  
where \(\mathbf{f}=\begin{bmatrix}\mathbf{M}(\Delta\mathbf{N})&\mathbf{T}(\Delta\mathbf{N})\end{bmatrix}^{ \mathsf{T}}\) represents the difference in each control parameter between the target value and the value after cable tension adjustment. \(\mathbf{C}=\begin{bmatrix}\mathbf{C}_{\mathsf{M}}&\mathbf{C}_{\mathsf{T}}\end{bmatrix}^{ \mathsf{T}}\) represents the influence matrix of each control parameter under unit cable force, and \(\mathbf{d}=\begin{bmatrix}\Delta\mathbf{M}&\Delta\mathbf{T}\end{bmatrix}^{\mathsf{T}}\) represents the difference in each control parameter between the target value and the value before cable tension adjustment.  
Convert the Multi-Objective Model to an Unconstrained Quadratic Model Through Weighted Sum of Squares Method  
For the multi-objective programming problem presented in Equation (4):  
\[\min F(\Delta\mathbf{N})=\begin{pmatrix}f_{1}(\Delta\mathbf{N}),f_{2}(\Delta\mathbf{N}), \cdots,f_{\mathsf{P}}(\Delta\mathbf{N})\end{pmatrix}^{\mathsf{T}}, \tag{5}\]  
The solution approach involves first transforming the multi-objective programming model into a single-objective programming model and then solving the resulting single-objective model. The weighted sum of squares method, a commonly used approach in the evaluation function method, is an effective way to achieve this. The basic principle is to provide a set of number \(f_{1}^{0},f_{2}^{0},\cdots,f_{\mathsf{P}}^{0}\), which represents the lower limit of each single objective \(\min f_{\mathsf{i}}(\Delta\mathbf{N})(i=1,2,\cdots,p)\). In other words, if \(\min f_{\mathsf{i}}(\Delta\mathbf{N})(i=1,2,\cdots,p)\) is valid, the evaluation function can be formulated as follows:  
\[h[F(\Delta\mathbf{N})]=\sum_{\mathsf{i}=1}^{\mathsf{P}}\rho_{\mathsf{i}}\Big{[}f_ {\mathsf{i}}(\Delta\mathbf{N})-f_{\mathsf{i}}^{0}\Big{]}^{2}, \tag{6}\]  
where \(\rho_{\mathsf{i}}\) represents the weight coefficient, the value of which depends on the relative importance of each objective function \(f_{\mathsf{i}}(\Delta\mathbf{N})\).  
Let \(f_{\mathsf{i}}^{0}=0(i=1,2,\cdots,p)\); then, Equation (6) can be converted into a single-objective programming model as follows:  
\[\min h[F(\Delta\mathbf{N})]=\sum_{\mathsf{i}=1}^{\mathsf{P}}\lambda_{\mathsf{i}} [f_{\mathsf{i}}(\Delta\mathbf{N})]^{2}, \tag{7}\]  
Substituting Equation (4) into Equation (7), Equation (8) is given:  
\[\min h[F(\Delta\mathbf{N})] =(\mathbf{C}\Delta\mathbf{N}-\mathbf{d})^{\mathsf{T}}\mathbf{\rho}(\mathbf{C}\Delta \mathbf{N}-\mathbf{d}) \tag{8}\] \[=\Delta\mathbf{N}^{\mathsf{T}}\mathbf{C}^{\mathsf{T}}\mathbf{\rho}\mathbf{C}\Delta \mathbf{N}-2\mathbf{d}^{\mathsf{T}}\mathbf{\rho}\mathbf{C}\Delta\mathbf{N}+\mathbf{d}^{\mathsf{T}}\mathbf{ \rho}\mathbf{d}^{\prime}\]  
where \(\mathbf{\rho}=\begin{pmatrix}\rho_{1},\rho_{2},\cdots,\rho_{\mathsf{P}}\end{pmatrix} ^{\mathsf{T}}\) represents the weight coefficient vector for each optimization objective.  
Omit the constant term \(\mathbf{d}^{\mathsf{T}}\mathbf{\rho}\mathbf{d}\), and then an unconstrained single-objective quadratic programming model can be gained as follows:  
\[\min f(\Delta\mathbf{N})=\frac{1}{2}\Delta\mathbf{N}^{\mathsf{T}}\mathbf{H}\Delta\mathbf{N}+ \mathbf{c}^{\mathsf{T}}\Delta\mathbf{N}, \tag{9}\]  
where \(\mathbf{H}=2\mathbf{C}^{\mathsf{T}}\mathbf{\rho}\mathbf{C}\) represents the influence of unit cable force on the optimization objectives, and \(\mathbf{c}=-\mathbf{C}^{\mathsf{T}}\mathbf{\rho}\mathbf{d}\) represents the influence of difference between each control parameter's target value and current value, as well as the unit cable force on the optimization objectives.