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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.<\/jats:p>","DOI":"10.1137\/17m1128538","type":"journal-article","created":{"date-parts":[[2018,10,4]],"date-time":"2018-10-04T11:56:19Z","timestamp":1538654179000},"page":"2809-2838","source":"Crossref","is-referenced-by-count":23,"title":["A Two-Phase Gradient Method for Quadratic Programming Problems with a Single Linear Constraint and Bounds on the Variables"],"prefix":"10.1137","volume":"28","author":[{"given":"Daniela","family":"di Serafino","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Gerardo","family":"Toraldo","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Marco","family":"Viola","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jesse","family":"Barlow","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2018,10,4]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1007\/s11222-016-9644-3"},{"key":"atypb2","unstructured":"D. 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Ruggiero, G. Toraldo, and L. Zanni,\n                      A note on spectral properties of some gradient methods\n                      , in Numerical Computations: Theory and Algorithms (NUMTA-2016), AIP Conf. 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