### Abstract

Range-space methods for convex quadratic programming improve in efficiency as the number of constraints active at the solution decreases. In this paper we describe a range-space method based upon updating a weighted Gram-Schmidt factorization of the constraints in the active set. The updating methods described are applicable to both primal and dual quadratic programming algorithms that use an active-set strategy. Many quadratic programming problems include simple bounds on all the variables as well as general linear constraints. A feature of the proposed method is that it is able to exploit the structure of simple bound constraints. This allows the method to retain efficiency when the number of general constraints active at the solution is small. Furthermore, the efficiency of the method improves as the number of active bound constraints increases.

Original language | English (US) |
---|---|

Pages (from-to) | 176-195 |

Number of pages | 20 |

Journal | Mathematical Programming |

Volume | 30 |

Issue number | 2 |

DOIs | |

State | Published - Oct 1984 |

### Fingerprint

### Keywords

- Active-Set Methods
- Bound Constraints
- Convex Quadratic Programming
- Range-Space Methods
- Updated Orthogonal Factorizations

### ASJC Scopus subject areas

- Mathematics(all)
- Software
- Engineering(all)

### Cite this

*Mathematical Programming*,

*30*(2), 176-195. https://doi.org/10.1007/BF02591884

**A weighted gram-schmidt method for convex quadratic programming.** / Gill, Philip E.; Gould, Nicholas I M; Murray, Walter; Saunders, Michael A.; Wright, Margaret.

Research output: Contribution to journal › Article

*Mathematical Programming*, vol. 30, no. 2, pp. 176-195. https://doi.org/10.1007/BF02591884

}

TY - JOUR

T1 - A weighted gram-schmidt method for convex quadratic programming

AU - Gill, Philip E.

AU - Gould, Nicholas I M

AU - Murray, Walter

AU - Saunders, Michael A.

AU - Wright, Margaret

PY - 1984/10

Y1 - 1984/10

N2 - Range-space methods for convex quadratic programming improve in efficiency as the number of constraints active at the solution decreases. In this paper we describe a range-space method based upon updating a weighted Gram-Schmidt factorization of the constraints in the active set. The updating methods described are applicable to both primal and dual quadratic programming algorithms that use an active-set strategy. Many quadratic programming problems include simple bounds on all the variables as well as general linear constraints. A feature of the proposed method is that it is able to exploit the structure of simple bound constraints. This allows the method to retain efficiency when the number of general constraints active at the solution is small. Furthermore, the efficiency of the method improves as the number of active bound constraints increases.

AB - Range-space methods for convex quadratic programming improve in efficiency as the number of constraints active at the solution decreases. In this paper we describe a range-space method based upon updating a weighted Gram-Schmidt factorization of the constraints in the active set. The updating methods described are applicable to both primal and dual quadratic programming algorithms that use an active-set strategy. Many quadratic programming problems include simple bounds on all the variables as well as general linear constraints. A feature of the proposed method is that it is able to exploit the structure of simple bound constraints. This allows the method to retain efficiency when the number of general constraints active at the solution is small. Furthermore, the efficiency of the method improves as the number of active bound constraints increases.

KW - Active-Set Methods

KW - Bound Constraints

KW - Convex Quadratic Programming

KW - Range-Space Methods

KW - Updated Orthogonal Factorizations

UR - http://www.scopus.com/inward/record.url?scp=0021504719&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0021504719&partnerID=8YFLogxK

U2 - 10.1007/BF02591884

DO - 10.1007/BF02591884

M3 - Article

VL - 30

SP - 176

EP - 195

JO - Mathematical Programming

JF - Mathematical Programming

SN - 0025-5610

IS - 2

ER -