An Algorithm for Large-Scale Quadratic Programming

Gould, Nicholas I. M.

doi:10.1093/imanum/11.3.299

Cited by 37 publications

(26 citation statements)

References 38 publications

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“…Various properties of inertia-controlling methods have been proved by Fletcher and others (see, e.g., [12,13,18,29]). In this section, we use the Schur-complement results of Section 3 to analyze certain sequences of iterates in an inertia-controlling method.…”

Section: Intermediate Iterationsmentioning

confidence: 97%

“…The methods of Gill and Murray [18] and of QPSOL [22] are inertia-controlling methods in which the search direction is obtained from the Cholesky factorization of the reduced Hessian matrix. Gould [29] proposes an inertia-controlling method for sparse problems, based on updating certain LU factorizations. Finally, the Schur-complement QP methods of Gill et al [21,25] are designed mainly for sparse problems, particularly those associated with applying Newton-based sequential quadratic programming (SQP) methods to large nonlinearly constrained problems.…”

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confidence: 99%

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Inertia-Controlling Methods for General Quadratic Programming

et al. 1991

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Abstract. Active-set quadratic programming (QP) methods use a working set to define the search direction and multiplier estimates. In the method proposed by Fletcher in 1971, and in several subsequent mathematically equivalent methods, the working set is chosen to control the inertia of the reduced Hessian, which is never permitted to have more than one nonpositive eigenvalue. (We call such methods inertia-controlling.) This paper presents an overview of a generic inertia-controlling QP method, including the equations satisfied by the search direction when the reduced Hessian is positive definite, singular and indefinite. Recurrence relations are derived that define the search direction and Lagrange multiplier vector through equations related to the Karush-Kuhn-Tucker system. We also discuss connections with inertia-controlling methods that maintain an explicit factorization of the reduced Hessian matrix.

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Section: Intermediate Iterationsmentioning

confidence: 97%

mentioning

confidence: 99%

Inertia-Controlling Methods for General Quadratic Programming

et al. 1991

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“…These problems arise in the optimal placement of nodes in a scheme for solving ordinary differential equations with given boundary values [14]. We solve these problems for different values of κ.…”

Section: Numerical Experimentsmentioning

confidence: 99%

Spectral Gradient Methods for Linearly Constrained Optimization

Martínez¹,

Pilotta²,

Raydan³

2005

J Optim Theory Appl

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Linearly constrained optimization problems with simple bounds are considered in the present work. First, a preconditioned spectral gradient method is defined for the case in which no simple bounds are present. This algorithm can be viewed as a quasiNewton method in which the approximate Hessians satisfy a weak secant equation. The spectral choice of steplength is embedded into the Hessian approximation, and the whole process is combined with a nonmonotone line search strategy. The simple bounds are then taken into account by placing them in an exponential penalty term that modifies the objective function. The exponential penalty scheme defines the outer iterations of the process. Each outer iteration involves the application of the previously defined preconditioned spectral gradient method for linear equality constrained problems. Therefore, an equality constrained convex quadratic programming problem needs to be solved at every inner iteration. The associated extended KKT matrix remains constant unless the process is reinitiated. In ordinary inner iterations, only the right hand side of the KKT system changes. Therefore, suitable sparse factorization techniques can be effectively applied and exploited. Encouraging numerical experiments are presented.

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“…For a detailed discussion of the properties of the KKT equations in this context, see Gould [9,Theorem 2.3].…”

Section: A Proof Of the Optimality Conditionsmentioning

confidence: 99%

“…A unique feature of ICQP methods is that constraint deletions are restricted so as to control the inertia of the reduced Hessian, which is never permitted to have more than one nonpositive eigenvalue. Fletcher [4] proposed the first ICQP method, and various methods within this class have also been proposed, see for example Gill et al [5] and Gould [9].…”

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confidence: 99%

On the Identification of Local Minimizers in Inertia-Controlling Methods for Quadratic Programming

Forsgren

Gill

Murray

1991

SIAM J. Matrix Anal. & Appl.

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Abstract. The verification of a local minimizer of a general (i.e., nonconvex) quadratic program is in general an NP-hard problem. The difficulty concerns the optimality of certain points (which we call dead points) at which the first-order necessary conditions for optimality are satisfied, but strict complementarity does not hold. One important class of methods for solving general quadratic programming problems are called inertia-controlling quadratic programming (ICQP) methods. We derive a computational scheme for proceeding at a dead point that is appropriate for a general ICQP method.

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An Algorithm for Large-Scale Quadratic Programming

Cited by 37 publications

References 38 publications

Inertia-Controlling Methods for General Quadratic Programming

Inertia-Controlling Methods for General Quadratic Programming

Spectral Gradient Methods for Linearly Constrained Optimization

On the Identification of Local Minimizers in Inertia-Controlling Methods for Quadratic Programming

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