Читать книгу Mathematical Programming for Power Systems Operation - Alejandro Garcés Ruiz - Страница 45

The optimal conditions for both constrained and unconstrained problems constitute a set of algebraic equations S(x) for the first case and S(x, λ) for the second case. This set can be solved using Newton’s method⁴.

Consider a set of algebraic equations S(x) = 0 where S : ⁿ → ⁿ is continuous and differentiable. Thus, the following approximation is made:

(2.52)

where is the Jacobian matrix of S and Δx = x − x₀. This constitutes the first-order approximation of S around a point x₀; finding the zero of S means to approximate successively the solution by using the following iteration:

(2.53)

(2.54)

This iteration is the primary Newton’s method. Compared to the gradient method, this method is faster since it includes information from the second derivative⁵. In addition, Newton’s method does not require defining a step t as in the gradient method. However, each iteration of Newton’s method is computationally expensive since it implies the formulation of a jacobian and solves a linear system in each iteration.