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.

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;
2018

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.
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11591/397660
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 20
  • ???jsp.display-item.citation.isi??? 17
social impact