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

Daniela di Serafino (daniela.diserafino***at***unicampania.it)
Gerardo Toraldo (toraldo***at***unina.it)
Marco Viola (marco.viola***at***uniroma1.it)
Jesse Barlow (barlow***at***cse.psu.edu)

Abstract: We propose a gradient-based method for quadratic programming problems with a single linear constraint and bounds on the variables. Inspired by the GPCG algorithm for bound-constrained convex quadratic programming [J.J. More' and G. Toraldo, SIAM J. Optim. 1, 1991], 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 proportioning, 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 case of degeneracy. Extensive numerical experiments show the effectiveness of the proposed approach.

Keywords: quadratic programming, bound and single linear constraints, gradient projection, proportioning.

Category 1: Nonlinear Optimization (Quadratic Programming )


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Entry Submitted: 05/04/2017
Entry Accepted: 05/04/2017
Entry Last Modified: 07/19/2018

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