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 journalArticlepeer-review

10 Scopus citations

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

All Science Journal Classification (ASJC) codes

  • Software
  • Theoretical Computer Science

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