Direct methods for sparse matrices
Direct methods for sparse matrices
Preconditioners for indefinite systems arising in optimization
SIAM Journal on Matrix Analysis and Applications
A preconditioned iterative method for saddlepoint problems
SIAM Journal on Matrix Analysis and Applications
A primal-dual infeasible-interior-point algorithm for linear programming
Mathematical Programming: Series A and B
Fast iterative solution of stabilised Stokes systems part II: using general block preconditioners
SIAM Journal on Numerical Analysis
On the Stability of Cholesky Factorization For Symmetric Quasidefinite Systems
SIAM Journal on Matrix Analysis and Applications
A QMR-based interior-point algorithm for solving linear programs
Mathematical Programming: Series A and B - Special issue: interior point methods in theory and practice
Primal-dual interior-point methods
Primal-dual interior-point methods
Journal of Optimization Theory and Applications
Atomic Decomposition by Basis Pursuit
SIAM Journal on Scientific Computing
LSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares
ACM Transactions on Mathematical Software (TOMS)
On the Solution of Equality Constrained Quadratic Programming Problems Arising in Optimization
SIAM Journal on Scientific Computing
Krylov Subspace Methods for Saddle Point Problems with Indefinite Preconditioning
SIAM Journal on Matrix Analysis and Applications
SIAM Journal on Optimization
Interior-Point Methods for Massive Support Vector Machines
SIAM Journal on Optimization
Preconditioning Indefinite Systems in Interior Point Methods for Optimization
Computational Optimization and Applications
Implicit-Factorization Preconditioning and Iterative Solvers for Regularized Saddle-Point Systems
SIAM Journal on Matrix Analysis and Applications
SIAM Journal on Optimization
Inexact constraint preconditioners for linear systems arising in interior point methods
Computational Optimization and Applications
Exact Regularization of Convex Programs
SIAM Journal on Optimization
Further development of multiple centrality correctors for interior point methods
Computational Optimization and Applications
Computational Optimization and Applications
A Matrix-Free Algorithm for Equality Constrained Optimization Problems with Rank-Deficient Jacobians
SIAM Journal on Optimization
Quadratic regularizations in an interior-point method for primal block-angular problems
Mathematical Programming: Series A and B
A constraint-reduced variant of Mehrotra's predictor-corrector algorithm
Computational Optimization and Applications
GPU acceleration of the matrix-free interior point method
PPAM'11 Proceedings of the 9th international conference on Parallel Processing and Applied Mathematics - Volume Part I
Solving large-scale optimization problems related to Bell's Theorem
Journal of Computational and Applied Mathematics
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In this paper we present a redesign of a linear algebra kernel of an interior point method to avoid the explicit use of problem matrices. The only access to the original problem data needed are the matrix-vector multiplications with the Hessian and Jacobian matrices. Such a redesign requires the use of suitably preconditioned iterative methods and imposes restrictions on the way the preconditioner is computed. A two-step approach is used to design a preconditioner. First, the Newton equation system is regularized to guarantee better numerical properties and then it is preconditioned. The preconditioner is implicit, that is, its computation requires only matrix-vector multiplications with the original problem data. The method is therefore well-suited to problems in which matrices are not explicitly available and/or are too large to be stored in computer memory. Numerical properties of the approach are studied including the analysis of the conditioning of the regularized system and that of the preconditioned regularized system. The method has been implemented and preliminary computational results for small problems limited to 1 million of variables and 10 million of nonzero elements demonstrate the feasibility of the approach.