### Abstract

We propose a gradient-based method for quadratic programming problems with a single linear constraint and bounds on the variables. Inspired by the gradient projection conjugate gradient (GPCG) algorithm for bound-constrained convex quadratic programming [J. J. Moré and G. Toraldo, SIAM J. Optim., 1 (1991), pp. 93-113], our approach alternates between two phases until convergence: an identification phase, which performs gradient projection iterations until either a candidate active set is identified or no reasonable progress is made, and an unconstrained minimization phase, which reduces the objective function in a suitable space defined by the identification phase, by applying either the conjugate gradient method or a recently proposed spectral gradient method. However, the algorithm differs from GPCG not only because it deals with a more general class of problems, but mainly for the way it stops the minimization phase. This is based on a comparison between a measure of optimality in the reduced space and a measure of bindingness of the variables that are on the bounds, defined by extending the concept of proportional iterate, which was proposed by some authors for box-constrained problems. If the objective function is bounded, the algorithm converges to a stationary point thanks to a suitable application of the gradient projection method in the identification phase. For strictly convex problems, the algorithm converges to the optimal solution in a finite number of steps even in the case of degeneracy. Extensive numerical experiments show the effectiveness of the proposed approach.

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

Pages (from-to) | 2809-2838 |

Number of pages | 30 |

Journal | SIAM Journal on Optimization |

Volume | 28 |

Issue number | 4 |

DOIs | |

State | Published - Jan 1 2018 |

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### All Science Journal Classification (ASJC) codes

- Software
- Theoretical Computer Science

### Cite this

*SIAM Journal on Optimization*,

*28*(4), 2809-2838. https://doi.org/10.1137/17M1128538

}

*SIAM Journal on Optimization*, vol. 28, no. 4, pp. 2809-2838. https://doi.org/10.1137/17M1128538

**A two-phase gradient method for quadratic programming problems with a single linear constraint and bounds on the variables.** / DI Serafino, Daniela; Toraldo, Gerardo; Viola, Marco; Barlow, Jesse.

Research output: Contribution to journal › Article

TY - JOUR

T1 - A two-phase gradient method for quadratic programming problems with a single linear constraint and bounds on the variables

AU - DI Serafino, Daniela

AU - Toraldo, Gerardo

AU - Viola, Marco

AU - Barlow, Jesse

PY - 2018/1/1

Y1 - 2018/1/1

N2 - We propose a gradient-based method for quadratic programming problems with a single linear constraint and bounds on the variables. Inspired by the gradient projection conjugate gradient (GPCG) algorithm for bound-constrained convex quadratic programming [J. J. Moré and G. Toraldo, SIAM J. Optim., 1 (1991), pp. 93-113], our approach alternates between two phases until convergence: an identification phase, which performs gradient projection iterations until either a candidate active set is identified or no reasonable progress is made, and an unconstrained minimization phase, which reduces the objective function in a suitable space defined by the identification phase, by applying either the conjugate gradient method or a recently proposed spectral gradient method. However, the algorithm differs from GPCG not only because it deals with a more general class of problems, but mainly for the way it stops the minimization phase. This is based on a comparison between a measure of optimality in the reduced space and a measure of bindingness of the variables that are on the bounds, defined by extending the concept of proportional iterate, which was proposed by some authors for box-constrained problems. If the objective function is bounded, the algorithm converges to a stationary point thanks to a suitable application of the gradient projection method in the identification phase. For strictly convex problems, the algorithm converges to the optimal solution in a finite number of steps even in the case of degeneracy. Extensive numerical experiments show the effectiveness of the proposed approach.

AB - We propose a gradient-based method for quadratic programming problems with a single linear constraint and bounds on the variables. Inspired by the gradient projection conjugate gradient (GPCG) algorithm for bound-constrained convex quadratic programming [J. J. Moré and G. Toraldo, SIAM J. Optim., 1 (1991), pp. 93-113], our approach alternates between two phases until convergence: an identification phase, which performs gradient projection iterations until either a candidate active set is identified or no reasonable progress is made, and an unconstrained minimization phase, which reduces the objective function in a suitable space defined by the identification phase, by applying either the conjugate gradient method or a recently proposed spectral gradient method. However, the algorithm differs from GPCG not only because it deals with a more general class of problems, but mainly for the way it stops the minimization phase. This is based on a comparison between a measure of optimality in the reduced space and a measure of bindingness of the variables that are on the bounds, defined by extending the concept of proportional iterate, which was proposed by some authors for box-constrained problems. If the objective function is bounded, the algorithm converges to a stationary point thanks to a suitable application of the gradient projection method in the identification phase. For strictly convex problems, the algorithm converges to the optimal solution in a finite number of steps even in the case of degeneracy. Extensive numerical experiments show the effectiveness of the proposed approach.

UR - http://www.scopus.com/inward/record.url?scp=85050477422&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85050477422&partnerID=8YFLogxK

U2 - 10.1137/17M1128538

DO - 10.1137/17M1128538

M3 - Article

AN - SCOPUS:85050477422

VL - 28

SP - 2809

EP - 2838

JO - SIAM Journal on Optimization

JF - SIAM Journal on Optimization

SN - 1052-6234

IS - 4

ER -