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

D'Apuzzo, Marco; Simone, Valentina De; Serafino, Daniela di

doi:10.1007/s10589-008-9226-1

Cited by 48 publications

(52 citation statements)

References 73 publications

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“…Solving systems with this structure is a vital component in numerous scientific computing algorithms; for example, such systems arise naturally in constrained optimization problems [7,13,15,25], Stokes and Navier-Stokes equations in fluid mechanics [5,9,16], time-harmonic Maxwell equations [26,32], and the application of Kirchhoff's laws in circuit simulation [46,50]. For an overview of solution methods, see the survey paper of Benzi, Golub, and Liesen [4] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Null-Space Preconditioners for Saddle Point Systems

Pestana¹,

Rees²

2016

SIAM J. Matrix Anal. & Appl.

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Abstract. The null-space method is a technique that has been used for many years to reduce a saddle point system to a smaller, easier to solve, symmetric positive definite system. This method can be understood as a block factorization of the system. Here we explore the use of preconditioners based on incomplete versions of a particular null-space factorization and compare their performance with the equivalent Schur complement based preconditioners. We also describe how to apply the nonsymmetric preconditioners proposed using the conjugate gradient method (CG) with a nonstandard inner product. This requires an exact solve with the (1,1) block, and the resulting algorithm is applicable in other cases where Bramble-Pasciak CG is used. We verify the efficiency of the newly proposed preconditioners on a number of test cases from a range of applications.

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Section: Introductionmentioning

confidence: 99%

Null-Space Preconditioners for Saddle Point Systems

Pestana¹,

Rees²

2016

SIAM J. Matrix Anal. & Appl.

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“…There is increased research activity in this area and it is natural to expect that it will produce new interesting developments. It is encouraging that recently there has been a noticeable shift of interest from direct to iterative methods, see the survey of D'Apuzzo et al [24].…”

Section: Linear Algebra Of Ipmsmentioning

confidence: 99%

Interior point methods 25 years later

Gondzio

2012

European Journal of Operational Research

308

259

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Interior point methods for optimization have been around for more than 25 years now. Their presence has shaken up the field of optimization. Interior point methods for linear and (convex) quadratic programming display several features which make them particularly attractive for very large scale optimization. Among the most impressive of them are their lowdegree polynomial worst-case complexity and an unrivalled ability to deliver optimal solutions in an almost constant number of iterations which depends very little, if at all, on the problem dimension. Interior point methods are competitive when dealing with small problems of dimensions below one million constraints and variables and are beyond competition when applied to large problems of dimensions going into millions of constraints and variables.In this survey we will discuss several issues related to interior point methods including the proof of the worst-case complexity result, the reasons for their amazingly fast practical convergence and the features responsible for their ability to solve very large problems. The ever-growing sizes of optimization problems impose new requirements on optimization methods and software. In the final part of this paper we will therefore address a redesign of interior point methods to allow them to work in a matrix-free regime and to make them well-suited to solving even larger problems.

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“…It is also interesting to analyze how the use of inexact directions affects the reduction of the complementarity product (5). The gap at the new point (8) becomes…”

Section: Crash Start Procedures and Its Implementationmentioning

confidence: 99%

“…In this paper we propose a practical method to construct a point which satisfies all three requirements. Recently there has been a major increase in interest in the use of iterative methods to compute Newton directions in IPMs [5,9] and a variety of preconditioners for Krylov subspace methods applied in this context have been proposed. Many preconditioners have already been proposed for the normal equations (Schur complement of the KKT system) [3,4,18] as well as for the indefinite augmented form of the KKT system [7,8,15].…”

Section: Introductionmentioning

confidence: 99%

Crash start of interior point methods

Gondzio

2016

European Journal of Operational Research

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The starting point used by an interior point algorithm for linear and convex quadratic programming may significantly influence the behaviour of the method. A widely used heuristic to construct such a point consists of dropping variable nonnegativity constraints and computing a solution which minimizes the Euclidean norm of the primal (or dual) point while satisfying the appropriate primal (or dual) equality constraints, followed by shifting the variables so that all their components are positive and bounded away from zero. In this Short Communication a new approach for finding a starting point is proposed. It relies on a few inexact Newton steps performed at the start of the solution process. A formal justification of the new heuristic is given and computational results are presented to demonstrate its advantages in practice. Computational experience with a large collection of small-and medium-size test problems reveals that the new starting point is superior to the old one and saves 20-40% of iterations needed by the primal-dual method. For larger and more difficult problems this translates into remarkable savings in the solution time.

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

Cited by 48 publications

References 73 publications

Null-Space Preconditioners for Saddle Point Systems

Null-Space Preconditioners for Saddle Point Systems

Interior point methods 25 years later

Crash start of interior point methods

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