A new polynomial-time algorithm for linear programming
Combinatorica
An algorithm for solving linear programming programs in O(n3L) operations
Progress in Mathematical Programming Interior-point and related methods
A primal-dual interior point algorithm for linear programming
on Progress in Mathematical Programming: Interior-Point and Related Methods
A polynomial-time algorithm for a class of linear complementary problems
Mathematical Programming: Series A and B
Interior path following primal-dual algorithms. Part I: Linear programming
Mathematical Programming: Series A and B
A primal-dual infeasible-interior-point algorithm for linear programming
Mathematical Programming: Series A and B
Iterative methods for solving linear systems
Iterative methods for solving linear systems
Convergence of a Class of Inexact Interior-Point Algorithms for Linear Programs
Mathematics of Operations Research
Stabilization of Interior-Point Methods for Linear Programming
Computational Optimization and Applications
SIAM Journal on Optimization
Linear Programming in O([n3/ln n]L) Operations
SIAM Journal on Optimization
Condition Numbers, the Barrier Method, and the Conjugate-Gradient Method
SIAM Journal on Optimization
Preconditioning Indefinite Systems in Interior Point Methods for Optimization
Computational Optimization and Applications
Polynomiality of an inexact infeasible interior point algorithm for semidefinite programming
Mathematical Programming: Series A and B
Uniform Boundedness of a Preconditioned Normal Matrix Used in Interior-Point Methods
SIAM Journal on Optimization
SIAM Journal on Optimization
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In this paper, we present a long-step infeasible primal-dual path-following algorithm for convex quadratic programming (CQP) whose search directions are computed by means of a preconditioned iterative linear solver. In contrast to the authors' previous paper [Z. Lu, R.D.C. Monteiro, and J.W. O'Neal. An iterative solver-based infeasible primal-dual path-following algorithm for convex quadratic programming, SIAM J. Optim. 17(1) (2006), pp. 287-310], we propose a new linear system, which we refer to as the hybrid augmented normal equation (HANE), to determine the primal-dual search directions. Since the iterative linear solver can only generate an approximate solution to the HANE, this solution does not yield a primal-dual search direction satisfying all equations of the primal-dual Newton system. We propose a recipe to compute an inexact primal-dual search direction, based on a suitable approximate solution to the HANE. The second difference between this paper and [Z. Lu, R.D.C. Monteiro, and J.W. O'Neal. An iterative solver-based infeasible primal-dual path-following algorithm for convex quadratic programming, SIAM J. Optim. 17(1)(2006), pp. 287-310] is that, instead of using the maximum weight basis (MWB) preconditioner in the aforesaid recipe for constructing the inexact search direction, this paper proposes the use of any member of a whole class of preconditioners, of which the MWB preconditioner is just a special case. The proposed recipe allows us to: (i) establish a polynomial bound on the number of iterations performed by our path-following algorithm and (ii) establish a uniform bound, depending on the quality of the preconditioner, on the number of iterations performed by the iterative solver.