A new polynomial-time algorithm for linear programming
Combinatorica
A primal-dual interior point algorithm for linear programming
Progress in Mathematical Programming Interior-point and related methods
Theoretical convergence of large-step primal-dual interior point algorithms for linear programming
Mathematical Programming: Series A and B
A primal-dual infeasible-interior-point algorithm for linear programming
Mathematical Programming: Series A and B
Presolving in linear programming
Mathematical Programming: Series A and B
Stability of Augmented System Factorizations in Interior-Point Methods
SIAM Journal on Matrix Analysis and Applications
Modified Cholesky Factorizations in Interior-Point Algorithms for Linear Programming
SIAM Journal on Optimization
Sparse Matrix Ordering Methods for Interior Point Linear Programming
INFORMS Journal on Computing
On Numerical Issues of Interior Point Methods
SIAM Journal on Matrix Analysis and Applications
The cholesky factorization in interior point methods
Computers & Mathematics with Applications
On the implementation of interior point methods for dual-core platforms
Optimization Methods & Software - The 2nd Veszprem Optimization Conference: Advanced Algorithms (VOCAL), 13-15 December 2006, Veszprem, Hungary
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Regularization techniques, i.e., modifications on the diagonal elements of the scaling matrix, are considered to be important methods in interior point implementations. So far, regularization in interior point methods has been described for linear programming problems, in which case the scaling matrix is diagonal. It was shown that by regularization, free variables can be handled in a numerically stable way by avoiding column splitting that makes the set of optimal solutions unbounded. Regularization also proved to be efficient for increasing the numerical stability of the computations during the solutions of ill-posed linear programming problems. In this paper, we study the factorization of the augmented system arising in interior point methods. In our investigation, we generalize the methods developed and used in linear programming to the case when the scaling matrix is positive semidefinite, but not diagonal. We show that regularization techniques may be applied beyond the linear programming case.