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Projected Barzilai-Borwein methods for large-scale box-constrained quadratic programming. (English) Zbl 1068.65073
Summary: The authors study projected Barzilai-Borwein (PBB) methods [cf. {\it J. Barzilai} and {\it J. M. Borwein}, IMA J. Numer. Anal. 8, No. 1, 141--148 (1988; Zbl 0638.65055)] for large-scale box-constrained quadratic programming. Recent work on this method has modified the PBB method by incorporating the Grippo-Lampariello-Lucidi (GLL) nonmonotone line search [cf. {\it L. Grippo, F. Lampariello}, and {\it S. Lucidi}, SIAM J. Numer. Anal. 23, 707--716 (1986; Zbl 0616.65067)] so as to enable global convergence to be proved. We show by many numerical experiments that the performance of the PBB method deteriorates if the GLL line search is used. We have therefore considered the question of whether the unmodified method is globally convergent, which we show not to be the case, by exhibiting a counter example in which the method cycles. A new projected gradient method (PABB) is then considered that alternately uses the two Barzilai-Borwein steplengths. We also give an example in which this method may cycle, although its practical performance is seen to be superior to the PBB method. With the aim of both ensuring global convergence and preserving the good numerical performance of the unmodified methods, we examine other recent work on nonmonotone line searches, and propose a new adaptive variant with some attractive features. Further numerical experiments show that the PABB method with the adaptive line search is the best PBB-like method in the positive definite case, and it compares reasonably well against the GPCG algorithm of {\it J. J. Moré} and {\it G. Toraldo} [SIAM J. Optim. 1, No. 1, 93--113 (1991; Zbl 0752.90053)]. In the indefinite case, the PBB method with the adaptive line search is shown on some examples to find local minima with better solution values, and hence may be preferred for this reason.

65K05Mathematical programming (numerical methods)
90C20Quadratic programming
90C06Large-scale problems (mathematical programming)
Full Text: DOI
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