### 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.

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.

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Güemes 3450

S3000GLN Santa Fe, Argentina

Phone: 54-342-4511594 / 4511595 Int. 1006

Fax: 54-342-4511169

E-mail: amca(at)santafe-conicet.gov.ar

**Asociación Argentina de Mecánica Computacional**Güemes 3450

S3000GLN Santa Fe, Argentina

Phone: 54-342-4511594 / 4511595 Int. 1006

Fax: 54-342-4511169

E-mail: amca(at)santafe-conicet.gov.ar

**ISSN 2591-3522**