Abstract
The authors propose an affine scaling modified gradient path method in association with reduced projective Hessian and nonmonotonic interior backtracking line search techniques for solving the linear equality constrained optimization subject to bounds on variables. By employing the QR decomposition of the constraint matrix and the eigensystem decomposition of reduced projective Hessian matrix in the subproblem, the authors form affine scaling modified gradient curvilinear path very easily. By using interior backtracking line search technique, each iterate switches to trial step of strict interior feasibility. The global convergence and fast local superlinear/quadratical convergence rates of the proposed algorithm are established under some reasonable conditions. A nonmonotonic criterion should bring about speeding up the convergence progress in some ill-conditioned cases. The results of numerical experiments are reported to show the effectiveness of the proposed algorithm.
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T. F. Coleman and Y. Li, An interior trust region approach for minimization subject to bounds, SIAM J. Optimization, 1996, 6(3): 418–445.
D. Zhu, Nonmonotonic back-tracking trust region interior point algorithm for linear constrained optimization, J. of Computational and Applied Mathematics, 2003, 155(2): 285–305.
J. P. Bulteau and J. Ph. Vial, Curvilinear path and trust region in unconstrained optimization, a convergence analysis, Mathematical Programming Study, 1987, 30(1): 82–101.
D. C. Sorensen, Newton’s method with a model trust region modification, SIAM J. Numer. Anal., 1982, 19(3): 409–426
C. B. Gurwitz, Local convergence of a two-piece update of a projected Hessian matrix, SIAM J. Optimization, 1994, 4(3): 461–485.
J. Nocedal and Y. Yuan, Combining trust region and line search techniques, in Advances in Non-linear Programming (ed. by Y. Yuan), Kluwer, Dordrecht, 1998, 153–175.
L. Grippo, F. Lampariello, and S. Lucidi, A nonmonotonic line search technique for Newton’s methods, SIAM Journal on Numerical Analysis, 1986, 23(4): 707–716.
N. Y. Deng, Y. Xiao, and F. J. Zhou, A nonmonotonic trust region algorithm, Journal of Optimization Theory and Applications, 1993, 76(3): 259–285.
D. T. Zhu, Curvilinear paths and trust region methods with nonmonotonic back tracking technique for unconstrained optimization, J. of Computational Mathematics, 2001, 19(3): 241–258.
C. A. Botsaris and D. H. Jacobson, A Newton-type curvilinear search method for optimization, Journal of Mathematical Analysis and Applications, 1976, 54(2): 217–229.
J.P. Vial and I. Zang, Unconstrained optimization by approximation of the gradient path, Mathematics of Operations Research, 1977, 2(3): 253–265.
J. J. Mor’e and D. C. Sorensen, Computing a trust region step, SIAM Journal on Science and Statistical Computing, 1983, 4(4): 553–572.
P. H. Guo and D. T. Zhu, A nonmonotonic interior point algorithn via optimal path for nonlinear optimization with bounds to variables, Journal of Shanghai Normal University, 2004, 33(3): 23–29.
C. A. Floudas and P. M. Pardalos, Handbook of Test Problems in Local and Global Optimization, Kluwer Academic, Dordrecht, 1999, 33.
K. Schittkowski, More Test Examples for Nonlinear Programming Codes, Lecture Notes in Economics and Mathematical Systems 282, Springer-Verlag, Berlin, 1987.
W. Hock and K. Schittkowski, Test Examples for Nonlinear Programming Code, Lecture Notes in Economics and Mathematical Systems 187, Springer-Verlag, Berlin, 1981.
J. R. Bunch and B. N. Parlett, Direct method for solving symmetric indefinite systems of linear equations, SIAM Journal on Numerical Analysis, 1971, 8(4): 639–655.
J. F. Bonnans and C. Pola, A trust region interior point algorithm for linear constrained optimization, SIAM J. Optimization, 1997, 7(3): 717–731.
J. E. Dennis Jr. and R. B. Schnable, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice Hall, New Jersey, 1983.
J. Nocedal and M. L. Overton, Projected Hessian updating algorithms for nonlinearly constrained optimization, SIAM J. Numer. Anal., 1985, 22(4): 821–850.
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The research is partially supported by the National Natural Science Foundation of China under Grant No. 10471094, the Ph.D. Foundation under Grant No. 0527003, the Shanghai Leading Academic Discipline Project (T0401), and the Science Foundation of Shanghai Education Committee under Grant No. 05DZ11.
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Guo, P., Zhu, D. Nonmonotonic Reduced Projected Hessian Method Via an Affine Scaling Interior Modified Gradient Path for Bounded-Constrained Optimization. J. Syst. Sci. Complex. 21, 85–113 (2008). https://doi.org/10.1007/s11424-008-9069-y
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DOI: https://doi.org/10.1007/s11424-008-9069-y