A line search is a crucial technique used in algoritmos de otimização, particularly in iterative methods for minimizing or maximizing a function. The primary goal of a line search is to determine the best step size along a given direction in the search space, which leads to a more efficient convergence towards an solução ótima.
Em termos matemáticos, ao otimizar uma função f(x), where x is a vector of parameters, the line search focuses on a specific direction d (which is typically derived from the gradient of the function). Starting from a current point xk, the line search seeks to minimize the function along the line defined by xk + αd, where α represents the step size. The optimal step size α is the value that minimizes f(xk + αd).
Existem vários métodos para realizar uma busca em linha, incluindo:
- Busca em Linha Exata: This method finds the step size α that exactly minimizes the function along the line. It can be computationally expensive, especially in high-dimensional spaces.
- Busca em Linha Inexata: Instead of finding the exact minimum, this approach looks for a step size that sufficiently reduces the function value, often using criteria such as the Wolfe conditions or Armijo rule.
A busca por linha é comumente usada em algoritmos de otimização como Gradiente Descendente, Newton’s Method, and Conjugate Gradient. By effectively finding the right step size, line search helps improve the speed and efficiency of convergence towards an optimal solution, making it an essential tool in machine learning, inteligência artificial, and various fields requiring optimization.