# A Preconditioner for Linear Systems Arising From Interior Point Optimization Methods

@article{Rees2007APF, title={A Preconditioner for Linear Systems Arising From Interior Point Optimization Methods}, author={T. Rees and Chen Greif}, journal={SIAM J. Sci. Comput.}, year={2007}, volume={29}, pages={1992-2007} }

We explore a preconditioning technique applied to the problem of solving linear systems arising from primal-dual interior point algorithms in linear and quadratic programming. The preconditioner has the attractive property of improved eigenvalue clustering with increased ill-conditioning of the (1,1) block of the saddle point matrix. It fits well into the optimization framework since the interior point iterates yield increasingly ill-conditioned linear systems as the solution is approached. We…

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## References

SHOWING 1-10 OF 38 REFERENCES

### Augmentation preconditioning for saddle point systems arising from interior point methods

- Computer Science
- 2006

This work investigates a preconditioning technique applied to the problem of solving linear systems arising from primal-dual interior point algorithms in linear and quadratic programming and shows that for stabilized saddle point systems a minimum residual Krylov method will converge in just two iterations.

### Inexact constraint preconditioners for linear systems arising in interior point methods

- Computer ScienceComput. Optim. Appl.
- 2007

Issues of indefinite preconditioning of reduced Newton systems arising in optimization with interior point methods are addressed and an approximate constraint preconditionser is considered in which sparse approximation of the Jacobian is used instead of the complete matrix.

### Preconditioning Indefinite Systems in Interior Point Methods for Optimization

- Computer ScienceComput. Optim. Appl.
- 2004

Two types of preconditioners which use some form of incomplete Cholesky factorization for indefinite systems are proposed in this paper, and it is revealed that the solution times for such problems on a modern PC are measured in minutes when direct methods are used and drop to seconds when iterative methods with appropriate preconditionsers are used.

### Implicit-Factorization Preconditioning and Iterative Solvers for Regularized Saddle-Point Systems

- Computer Science, MathematicsSIAM J. Matrix Anal. Appl.
- 2006

A number of families of implicit factorizations are constructed that are capable of reproducing the required sub-blocks and (some) of the remainder of the remaining blocks in the conjugate-gradient like methods for solving block symmetric indefinite linear systems that arise from saddle-point problems or, in particular, regularizations thereof.

### A new class of preconditioners for large-scale linear systems from interior point methods for linear programming

- Computer Science, Mathematics
- 2005

### Constraint Preconditioning for Indefinite Linear Systems

- Computer ScienceSIAM J. Matrix Anal. Appl.
- 2000

The problem of finding good preconditioners for the numerical solution of indefinite linear systems is considered. Special emphasis is put on preconditioners that have a 2 × 2 block structure and…

### Preconditioners for saddle point linear systems with highly singular blocks.

- Computer Science, Mathematics
- 2006

A new preconditioning technique is introduced for the iterative solution of saddle point linear systems with (1,1) blocks that have a high nullity, using symmetric positive definite weight matrices.

### Convergence of a Class of Inexact Interior-Point Algorithms for Linear Programs

- MathematicsMath. Oper. Res.
- 1999

This work presents a convergence analysis for a class of inexact infeasible-interior-point methods for solving linear programs and allows that these linear systems are only solved to a moderate accuracy in the residual, but no assumptions are made on the accuracy of the search direction in the search space.

### Approximate Factorization Constraint Preconditioners for Saddle-Point Matrices

- Computer ScienceSIAM J. Sci. Comput.
- 2006

We consider the application of the conjugate gradient method to the solution of large, symmetric indefinite linear systems. Special emphasis is put on the use of constraint preconditioners and a new…

### Iterative Solution of Augmented Systems Arising in Interior Methods

- Computer ScienceSIAM J. Optim.
- 2007

A family of constraint preconditioners is proposed that provably eliminates the inherent ill-conditioning in the augmented system of linear equations that arise in interior methods for general nonlinear optimization.