# Impact Load Sparse Recognition Method Based on Mc Penalty Function
## 3 Impact Load Sparse Identification Model
### Implementation of MC Penalty Function Impact Load Sparse Identification Model Algorithm  
Because the MC sparse penalty function impact load identification model of penalty function is convex, the regularization model does not have explicit solutions and thus needs to rely on optimization algorithms [35]. It usually cannot use singular value decomposition methods to solve the model, so the convexity for non-convex penalty function is more difficult to solve the sparse model; often it can only get the local optimal solution or approximate optimal solution. Therefore, the solution of MC penalty function impact load identification model can be roughly divided into two steps: (1) Pre-processing the objective function. The local approximation method is used to pre-process the objective function. (2) Using the improved primal-dual interior point method to solve the pre-processed objective function. Finally, the selection of the regularization parameter \(\lambda\) of the sparse model is determined by the L-curve criterion (or cross test criterion) method. In the implementation process of the improved primal-dual interior point method, [PERSON]'s duality principle is firstly determined according to the minimum multiplication solution of the linear equation, and the dual feasibility variables are constructed to simplify the linear optimization model. \[\min\!\!\!\!\!  
The corresponding Lagrangian duality formula is:  
\[\underset{f}{\text{minimize}}\ \text{G}(\mathbf{v})=-\frac{1}{4}\mathbf{v}^{\text{T}}\mathbf{H} \mathbf{H}^{\text{T}}\mathbf{v}-\mathbf{v}^{\text{T}}\mathbf{y},\ \text{subject to}\ \left(\mathbf{H}^{\text{T}}\mathbf{v}\right)_{i}\leq\lambda \tag{32}\]  
In the formula, \(i=1,\ldots n\); \(G(\mathbf{v})\) represents a Lagrange dual function, it is a necessary condition for the existence of the problem of minimum Lagrange function for all variables. The partial derivative of the multiplier is zero, because the system of equations for the linear equation of the conjugate gradient method can be used to solve. According to the necessary condition for the existence of extremal by Equation (32) and the corresponding Lagrangian dual function for the eta, the differences between the dual gap \(\eta\) its formula is as follows:  
\[\eta=\left|\left|\mathbf{H}\mathbf{f}-\mathbf{y}\right|\right|_{2}^{2}+\lambda\sum_{i=1}^{ m}\phi(\mathbf{f}\text{a})-G(\mathbf{v}) \tag{33}\]  
\[u=\frac{\eta}{2m} \tag{34}\]  
\[\psi(\mathbf{f},\mathbf{\phi})=\sum_{i=1}^{n}\ ln(u_{i}+\mathbf{f}_{i})+\sum_{i=1}^{n}\ ln(u_{i}-\mathbf{f}_{i}) \tag{35}\]  
where \(u_{i}\) is the slack variable. In this way, the corrected center path trajectory can be obtained. On this basis, the interior point method is introduced to solve the constraint function \(\psi(\mathbf{f},\mathbf{\phi})\), which represents the relaxation variable. In this way, the modified center path trajectory can be obtained. \[\underset{(\mathbf{f},\mathbf{\phi})}{\text{minimize}}(\mathbf{f},\mathbf{\phi})=\left|\left| \mathbf{H}\mathbf{f}-\mathbf{y}\right|\right|_{2}^{2}+\lambda\sum_{i=1}^{m}\mathbf{\phi}(\mathbf{f }:a)-\frac{1}{t}\psi(\mathbf{f},\mathbf{\phi}) \tag{36}\]  
where \(t\) represents a small positive scalar, known as the barrier parameter. According to the trajectory of the center path in Equation (36), the preconditioned conjugate gradient (PCG) method can be used to iteratively search according to the specified center trajectory. The calculation process of PCG is shown in Algorithm 1. ```
1:While iteration \(t\)do
2: Calculation gradient \[\text{g}:=\frac{1}{m}\sum\limits_{i=1}^{m}\
abla_{\theta}\mathcal{L}\left(f \left(x^{(i)};\theta\right),y^{(i)}\right)\]
3: Computational search: \(d_{t}=g+\psi(f,\phi)\)
4: Perform line search: \[\alpha^{*}=\text{argmin}_{\alpha}\frac{1}{m}\sum\limits_{i=1}^{m}\left( \mathcal{L}\left(f\left(x^{(i)};\gamma\right)0,y^{(i)}\right)\right)\] where: \(\gamma=\theta-\alpha d_{t}\)
5: Parameter update \(\theta:=\theta-\alpha^{*}d_{t}\)
6: The judgment condition satisfies \(t>k\), where, \(k\) represents the sparsity parameter.