The eigenvalues for preconditioned system in interior points method for quadratic programming

الملخص EN

Krylov subspace methods are one of the strongest iterative approach, which are used to solve large scale linear systems.

These methods are based on a general type of projection process.

As at each iteration of interior points method (IPM) at least one linear system has to be solved, where the main computational effort of IPMs consist in the computational of these linear systems.

Therefore, we suggest to use Krylov subspace methods to solve these linear systems.

In the paper (Al-Jeiroudi, 2011), it has been introduced a new preconditioner for the augmented system, which arises from IPM for quadratic programming.

Since, in most cases Krylov subspace methods require preconditioner to improve the global convergence.

The eigenvalues play critical role in the convergence analysis of Krylov subspace methods.

Therefore, in this paper we are going to focus our attention to study the eigenvalues of the preconditioned augmented system, which are preconditioned by the previous preconditioner.

In this paper, we introduced new theorem, which studies the eigenvalues behaviour of the preconditioned system, and we show that, the eigenvalues are bounded away from zero.

In addition, they are equal to one exactly or one plus a positive number.

Moreover, they are well clustered and consequently this play an important role in the convergence speed