# UAV-BS Site Planning Based on Circular Coverage Strategy
## 3 Theory
### Optimization Algorithms  
(1) Interior Point Method: The interior point method transforms a linear programming problem into a nonlinear programming problem with a convex objective function. The constraints of this problem include the constraints of the original linear programming problem and an additional \"interior point constraint\", which restricts the solution set to a specific convex set. The interior point method solves this nonlinear programming problem iteratively. Each iteration solves an \"interior point problem\", whose solution is close to the solution of the previous iteration. The closer the solution of the interior point problem is to the optimal solution, the closer the interior point method is to the optimal solution of the linear programming problem. Suppose the linear programming problem is as follows:  
\[ \begin{array}{rl}\min_{x}&c^{T}x\\ \text{s.t.}&Ax\leq b,\end{array}  \tag{7}\]  
Let \(x\in\mathbb{R}^{n}\) be the decision variable, \(c\in\mathbb{R}^{n}\) be the coefficients of the objective function, \(A\in\mathbb{R}^{m\times n}\) be the coefficient matrix of dimensions \(m\times n\), and \(b\in\mathbb{R}^{m}\) be the vector on the right-hand side of the constraints. The iterative formula for the interior point method is as follows:  
\[ \begin{array}{rl}\min_{x}&c^{T}x-\mu\sum_{i=1}^{m}\ln(b_{i}-a_{i}^{T}x)\\ \text{s.t.}&Ax\leq b,\end{array}  \tag{8}\]  
Here, \(\mu\) is a parameter greater than zero, known as the \"central path parameter\". In each iteration, the value of \(\mu\) is gradually decreased, causing the interior point to approach the feasible region of the linear programming problem. As \(\mu\) approaches zero, the solution of the interior point method gradually converges to the optimal solution of the linear programming problem. (2) Gradient Descent Method: The gradient descent method minimizes the value of the objective function by continuously performing gradient iterations. Specifically, the algorithm calculates the gradient of the objective function at the current point and moves the current point a certain step length in the opposite direction of the gradient, causing the objective function to gradually approach its optimal value. The general formula for the gradient descent method is as follows:  
\[\theta_{i+1}=\theta_{i}-\alpha\
abla J(\theta_{i}), \tag{9}\]  
Here, \(\theta_{i}\) represents the parameter value at the \(i\)-th iteration, \(\alpha\) is the learning rate, and \(\
abla J(\theta_{i})\) denotes the gradient of the loss function at \(\theta_{i}\). The gradient descent method can also be divided into two forms: batch gradient descent and stochastic gradient descent. Batch gradient descent calculates the gradient using all training data in each iteration, which is slower but yields more accurate results. Stochastic gradient descent, on the other hand, randomly selects sample data to calculate the gradient during iterations, consuming fewer computational resources. However, the issue is that the optimal result may not be a global optimum but a local one. The convergence rate and final outcome of the gradient descent method largely depend on the choice of step size. If the step size is too large, it may lead to the algorithm failing to converge or converging to an unstable result. If the step size is too small, the convergence speed of the algorithm will slow down. Therefore, it is often necessary to choose an appropriate step size strategy based on the actual problem, such as a fixed step size or a dynamic step size. (3) [PERSON]'s Method: [PERSON]'s method is an iterative algorithm used to solve unconstrained optimization problems. Its basic idea is to approximate the objective function using a second-order Taylor expansion and then solve the equation where the first derivative is zero to obtain the direction of descent.