On the Convergence of an Inexact Primal-Dual Interior Point Method for Linear Programming

Baryamureeba, Venansius; Steihaug, Trond

doi:10.1007/11666806_72

Cited by 21 publications

(23 citation statements)

References 9 publications

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“…In this subsection, we present two examples of matrices F , which are (λ L , λ U )-approximations ofD 2 , and an estimation of their corresponding constants λ L and λ U . As a consequence, we will obtain specific expressions for the iteration complexity developed in Theorem 3.1 when the iterative solver used to obtain an approximate solution to the HANE is the PCG method with preconditioner given byÂFÂ T .…”

Section: Specific Applicationsmentioning

confidence: 99%

“…The use of inexact search directions in interior-point methods has been investigated in the context of conic programming problems (see, e.g. [1,2,5,13,17,21,26,29]). For feasibility problems of the form {x ∈ H 1 : Ax = b, x ∈ C}, where H 1 and H 2 are Hilbert spaces, C ⊆ H 1 is a closed convex cone satisfying some mild assumptions, and A : H 1 → H 2 is a continuous linear operator.…”

Section: Introductionmentioning

confidence: 99%

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An iterative solver-based long-step infeasible primal-dual path-following algorithm for convex QP based on a class of preconditioners

Monteiro

O'Neal

2009

Optimization Methods and Software

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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 (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.

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Section: Specific Applicationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

An iterative solver-based long-step infeasible primal-dual path-following algorithm for convex QP based on a class of preconditioners

Monteiro

O'Neal

2009

Optimization Methods and Software

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“…Inequality (32) gives the possibility of relating the accuracy of the approximate solution u of the KKT system to the quality of the current IP iterate, thus allowing to reduce the computational cost of the solution of the system, and hence the overall cost of the IP method [3,14,52]. The idea is to use adaptive inner iteration stopping criteria that require low accuracy when the outer IP iterate is far from the optimal solution and to require higher accuracy as soon as the IP iterate approaches the solution.…”

Section: Termination Controlmentioning

confidence: 99%

On mutual impact of numerical linear algebra and large-scale optimization with focus on interior point methods

2008

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The solution of KKT systems is ubiquitous in optimization methods and often dominates the computation time, especially when large-scale problems are considered. Thus, the effective implementation of such methods is highly dependent on the availability of effective linear algebra algorithms and software, that are able, in turn, to take into account specific needs of optimization. In this paper we discuss the mutual impact of linear algebra and optimization, focusing on interior point methods and on the iterative solution of the KKT system. Three critical issues are addressed: preconditioning, termination control for the inner iterations, and inertia control.

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“…The use of inexact Newton methods in interior point methods for LP was investigated in [2,3,6,8,14,15]. In [2] the convergence of the infeasible interior point algorithm of Kojima, Megiddo, and Mizuno is proved under the assumption that the iterates are bounded.…”

Section: Introductionmentioning

confidence: 99%

“…In [2] the convergence of the infeasible interior point algorithm of Kojima, Megiddo, and Mizuno is proved under the assumption that the iterates are bounded. Monteiro…”

Section: Introductionmentioning

confidence: 99%

Convergence Analysis of the Inexact Infeasible Interior-Point Method for Linear Optimization

Al-Jeiroudi

Gondzio

2008

J Optim Theory Appl

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In this paper we present the convergence analysis of the inexact infeasible path-following (IIPF) interior point algorithm. In this algorithm the preconditioned conjugate gradient method is used to solve the reduced KKT system (the augmented system). The augmented system is preconditioned by using a block triangular matrix.The KKT system is solved approximately. Therefore, it becomes necessary to study the convergence of interior point method for this specific inexact case. We present the convergence analysis of the inexact infeasible path-following (IIPF) algorithm, prove the global convergence of this method and provide complexity analysis.

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On the Convergence of an Inexact Primal-Dual Interior Point Method for Linear Programming

Cited by 21 publications

References 9 publications

An iterative solver-based long-step infeasible primal-dual path-following algorithm for convex QP based on a class of preconditioners

An iterative solver-based long-step infeasible primal-dual path-following algorithm for convex QP based on a class of preconditioners

On mutual impact of numerical linear algebra and large-scale optimization with focus on interior point methods

Convergence Analysis of the Inexact Infeasible Interior-Point Method for Linear Optimization

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