## 6 Observability of the graph
### Constrained Nonlinear Programming Model
First-order and second-order conditions are able to work together for non-convex optimization problems by determining local minimum points strongly relying on the selection of the initial point [19; 20; 21; 22; 23; 24; 25]. We can use local algorithms such as interior-point methods (IPMs) and sequential quadratic programming (SQP) methods to find locality, and if we adopt multi-start initialization techniques [41], a global solution can be reached with high probability [19; 20; 21; 22; 23; 24; 25]. SQP algorithms solve a sequence of optimization sub-problems, each of which optimizes a quadratic model of the objective under a linearization of the constraint function. The nonlinear algorithms can reach a local solution in a reasonable running time [19; 20; 21]. The other option is to adopt a global optimizer routine to determine a global minimum point but at a slower time [19; 20; 21; 22; 23; 24; 25; 37; 38]. Furthermore, derivative-free optimizers (DFOs) are adopted to handle such a kind of nonlinear programming model [22; 41; 70; 71]. DRO is a totally free direct-search algorithm that calculates the gradient of the objective and constraint functions per iteration [22; 41; 70; 71]. Direct-search algorithms guarantee global convergence to a local solution with optimality conditions fully satisfied [22; 41; 70; 71]. For that convergence, the Clarke calculus is adopted for handling non-smooth functions [22; 41; 70; 71]. The generalized pattern search algorithm (GPSA) is a direct-search algorithm perfectly suited to PMU allocation problem solving [22; 41; 70; 71]. The search methodology is implemented to estimate a limited number of trial points [22; 41; 70; 71]. GPSA executes a global exploration of the feasible set and permits us to optimize the programming model [22; 41; 70; 71]. The aim is to find a local solution point [22]. In the case of failing to find any improvement, the poll suggests trial points in the neighborhood of the incumbent solution relying on suitable optimality conditions [22; 41]. The poll methodology guarantees that the algorithm has a global convergence property and that a local minimum can be reached [22; 41; 70; 71]. This optimization problem is smooth but many local minimum points can be found by gradient-based algorithms or derivative-free optimizers [20; 41; 70; 71]. The nonlinear optimization algorithms usually present the general iterative scheme for the optimization problem of the form in Algorithm 5[19; 20; 21; 22; 23; 82]:  
\[min\{J(x):x\epsilon\Omega\} \tag{26}\]  
```
Step 0: initial estimate \(x_{0}\epsilon\Omega\), we set the iterative number equal to zero, \(i=0\) Step 1: Computation of the search direction \(h_{i}=h(x_{i})\) Step 2: Computation of step-length \(\lambda_{i}\) such as: \(z(x_{i}+\lambda_{i}\times h_{i})=min_{\lambda}\{\{z(x_{i}+\lambda\times h_{i}) \}:\lambda\geq 0\}\), \(x_{i}+\lambda_{i}\times h_{i}\epsilon\Omega\) Step 3: \(x_{i+1}=x_{i}+\lambda_{i}h_{i}\) Step 4: if \(\|x_{i+1}-x_{i}\|\leq\epsilon\), then stop. Otherwise, set \(i=i+1\) and go to step \(1\). ```  
**Algorithm 5** Steps of nonlinear algorithms  
\(J^{*}=min\{J^{*}=J(y):x\epsilon\Omega\}\) is solved either with SQP or IPMs [19; 20; 21]. The objective function is declared in a quadratic form and the minimization problem is formulated as follows [19; 20; 21; 22; 23; 24; 25; 41]:  
\[minJ(y)=y^{T}\times W\times y,\ W\in I_{m\times m} \tag{27}\]  
\[s.t.f(y)=0 \tag{28}\]  
\[y\geq 0 \tag{29}\]  
where \(J:R^{m}\to R,y:R^{m}\to R^{n}\) are differentiable functions. Furthermore, nonlinear inequality constraints can be formed in this formulation by utilizing slack variables [19; 20; 21]. The IPM strategy achieves a solution by approximately solving a sequence of barrier problems of the following format [19; 20; 21; 24; 25; 41; 72; 73]:  
\[min\varphi_{p}(y)=J(y)-\mu\sum_{i=1}^{m}ln(y_{i}) \tag{30}\]  
\[s.t.f(y)=0 \tag{31}\]where \(\upmu\) is a barrier parameter that is reducing and converges to zero [19; 20; 21].