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

Daniela DI Serafino, Gerardo Toraldo, Marco Viola, Jesse Barlow

Research output: Contribution to journalArticle

4 Citations (Scopus)

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 languageEnglish (US)
Pages (from-to)2809-2838
Number of pages30
JournalSIAM Journal on Optimization
Volume28
Issue number4
DOIs
StatePublished - Jan 1 2018

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Gradient methods
Quadratic programming
Gradient Method
Linear Constraints
Quadratic Programming
Gradient Projection
Spectral Gradient Method
Objective function
Conjugate gradient method
Gradient Projection Method
Convex Quadratic Programming
Converge
Active Set
Unconstrained Minimization
Conjugate Gradient Algorithm
Conjugate Gradient
Stationary point
Strictly Convex
Conjugate Gradient Method
Degeneracy

All Science Journal Classification (ASJC) codes

  • Software
  • Theoretical Computer Science

Cite this

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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. Mor{\'e} 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.",
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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.

In: SIAM Journal on Optimization, Vol. 28, No. 4, 01.01.2018, p. 2809-2838.

Research output: Contribution to journalArticle

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