Elección Eficiente de una Base para el Espacio Nulo de una Matriz

Abstract

The undptprmined linear system Ax = b, and thp problpm of finding a basis of the null space of .4 are considered. This problem appears in many algorithms based on Successive
Quadratic Programming (SQP) Method, for solving optimization problems with equality constraints. The efficiency of the algorithm for solving the quadratic subproblem depends
strongly on the choice of the basis of the tangent space of the constraints. Avoiding factorization of the matrix, the computational cost is reduced. Therefore, the objective of this
contribution is to present an efficient method such that at each iteration it solves a least squares problem with constraints: find a matrix. with block triangular structure with no singular diagonal blocks nearest, in the Frobenius norm, to a square submatrix of t.he given matrix A.
The strategy used is based on the Alternate Projection Method for closed convex cones suggested by Dykstra. An algorithm is dpduced modifying Dykstra's method according to the
constraints of the least squares problem, which appear from the structure of the matrix.
Convergence properties are stated. Preliminaries numerical experiments showing how this technique works are presented.