Interior-Point Solver for Large-Scale Quadratic Programming Problems with Bound Constraints

Cafieri, Sonia; D'Apuzzo, Marco; Marino, Marina; Mucherino, Antonio; Toraldo, Gerardo

doi:10.1007/s10957-006-9043-6

Cited by 22 publications

(19 citation statements)

References 26 publications

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“…compute δw k by solving the system (3) compute θ k as in (9) However, in [9,28] it has been observed that, in practice, the following simple choice for the step length,…”

Section: The Potential Reduction Algorithmmentioning

confidence: 99%

“…This suggests that the line search (8) can be replaced by (9). The PR method previously described can be summarized into the algorithmic framework in Fig.…”

Section: The Potential Reduction Algorithmmentioning

confidence: 99%

“…3, the CG algorithm applied to the preconditioned augmented system behaves as if it were applied to suitably preconditioned normal equations, if a proper choice of the initial approximation is made. We analyze the behaviour of the Constraint Preconditioner with the CG algorithm in the context of a Potential Reduction (PR) method, which has already been successfully applied to large-scale QP problems with box constraints [9,12,28]. An adaptive stopping criterion is used to avoid unnecessary CG iterations when the PR iterate is far from the solution of the problem pair (P, D), and thus a high accuracy in the computed Newton step is not required.…”

Section: Introductionmentioning

confidence: 99%

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On the iterative solution of KKT systems in potential reduction software for large-scale quadratic problems

et al. 2007

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Iterative solvers appear to be very promising in the development of efficient software, based on Interior Point methods, for large-scale nonlinear optimization problems. In this paper we focus on the use of preconditioned iterative techniques to solve the KKT system arising at each iteration of a Potential Reduction method for convex Quadratic Programming. We consider the augmented system approach and analyze the behaviour of the Constraint Preconditioner with the Conjugate Gradient algorithm. Comparisons with a direct solution of the augmented system and with MOSEK show the effectiveness of the iterative approach on large-scale sparse problems.

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“…compute δw k by solving the system (3) compute θ k as in (9) However, in [9,28] it has been observed that, in practice, the following simple choice for the step length,…”

Section: The Potential Reduction Algorithmmentioning

confidence: 99%

“…This suggests that the line search (8) can be replaced by (9). The PR method previously described can be summarized into the algorithmic framework in Fig.…”

Section: The Potential Reduction Algorithmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the iterative solution of KKT systems in potential reduction software for large-scale quadratic problems

et al. 2007

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“…This preconditioner allows to specify a priori, through a suitable parameter, the amount of fill-in for the incomplete factorization, without the need of using a drop tolerance; thus it has a predictable memory behaviour, which is a desirable feature in large-scale optimization algorithms. An adaptive choice of the fill-in parameter is performed at each PR step; a suitable diagonal scaling of the condensed matrix is also applied to improve the quality of this matrix [17].…”

Section: Inertia 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 complexity of the problem can be derived from the data structures used and from the mathematical expression of the objective function and the constraints. If the objective function is linear, or convex quadratic, and the problem has box, linear, or convex quadratic constraints, then the optimization problem can be solved efficiently by particular methods, which are tailored to the objective function [6,13,18]. If the objective function and the constraints are nonlinear without any restriction, then more general approaches must be used.…”

Section: Modeling Real-life Problemsmentioning

confidence: 99%

Modeling and Solving Real-Life Global Optimization Problems with Meta-heuristic Methods

Mucherino

Şeref

2008

Springer Optimization and Its Applications

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Many real-life problems can be modeled as global optimization problems. There are many examples that come from agriculture, chemistry, biology, and other fields. Meta-heuristic methods for global optimization are flexible and easy to implement and they can provide high-quality solutions. In this chapter, we give a brief review of the frequently used heuristic methods for global optimization. We also provide examples of real-life problems modeled as global optimization problems and solved by meta-heuristic methods, with the aim of analyzing the heuristic approach that is implemented.

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Interior-Point Solver for Large-Scale Quadratic Programming Problems with Bound Constraints

Cited by 22 publications

References 26 publications

On the iterative solution of KKT systems in potential reduction software for large-scale quadratic problems

On the iterative solution of KKT systems in potential reduction software for large-scale quadratic problems

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

Modeling and Solving Real-Life Global Optimization Problems with Meta-heuristic Methods

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