Abstract
We propose a non-interior path following algorithm for convex quadratic programming problems with bound constraints based on Chen-Harker-Kanzow-Smale smoothing technique. Conditions are given under which the algorithm is globally convergent or globally linearly convergent. Preliminary numerical experiments indicate that the method is promising.
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A. Björck, “A direct method for sparse least squares problems with lower and upper bounds,” Numer. Math., vol. 54, pp. 19–32, 1988.
J. Burke, “Decent methods for composite nondifferentiable optimization problems,” Mathematical Programming, vol. 33, pp. 260–279, 1987.
J. Burke and S. Xu, “The global linear convergence of a non-interior path-following algorithm for linear complementarity problem,” Mathematics of Operations Research, vol. 23, pp. 719–734, 1998.
J. Burke and S. Xu, “A non-interior predictor-corrector path-following method for LCP,” in Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, M. Fukushima and L. Qi (Eds.), Kluwer Academic Publishers, 1999, pp. 45–63.
J. Burke and S. Xu, “A non-interior predictor-corrector path following algorithm for the monotone linear complementarity problem,” Mathematical Programming, vol. 87, pp. 113–130, 2000.
B. Chen and X. Chen, “A global and local super-linear continuation method for P 0 + R 0 and monotone NCP,” SIAM J. Optimization, vol. 9, pp. 605–623, 1999.
B. Chen and X. Chen, “A global linear and local quadratic continuation method for variational inequalities with box constraints,” Computational Optimization and Applications, vol. 17, pp. 131–158, 2000.
B. Chen and P.T. Harker, “A non-interior-point continuation method for linear complementarity problems,” SIAM J. Matrix Anal. Appl., vol. 14, pp. 1168–1190, 1993.
B. Chen and N. Xiu, “A global linear and local quadratic non-interior continuation method for nonlinear complementarity problems based on Chen-Mangasarian smoothing functions,” SIAM J. Optimization, vol. 9, pp. 605–623, 1999.
X. Chen, L. Qi, and D. Sun, “Global and superlinear convergence of the smoothing newton method and its application to general box constrained variational inequalities,” Math. Comp., vol. 67, pp. 519–540, 1998.
T.F. Coleman and L.A. Hulbert, “A globally and superlinearly convergent algorithm for convex quadratic programs with simple bounds,” SIAM J. Optim, vol. 3, pp. 298–321, 1993.
R.S. Dembo and U. Tulowitzki, “On the minimization of quadratic functions subject to box constraints,” Technical report, Working paper series B no. 71, School of Organization and Management, Yale University, New Haven, 1983.
S.A. Gabriel and J.J. Moré, Smoothing of mixed complementarity problems, in Complementarity and Variational Problems: State of the Art, M.C. Ferris and J.S. Pang (Eds.), SIAM Publishing, Philadelphia, Pennsylvania, 1997, pp. 105–116.
G.H. Golub and C.F. Van Loan, Matrix Computations, 2nd edn. The John Hopkins University Press, 1989.
R. Glowinski, Numerical Methods for Nonlinear Variational problems. Springer-Verlag, 1984.
K. Hotta and A. Yoshise, “Global convergence of a class of non-interior-point algorithms using Chen-Harker-Kanzow functions for nonlinear complementarity problems,” Mathematical Programming, vol. 86, pp. 105–133, 1999.
C. Kanzow, “Some noninterior continuation methods for linear complementarity problems,” SIAM J. Matrix Anal. Appl., vol. 17, pp. 851–868, 1996.
Y. Lin and C.W. Cryer, “An alternating direction implicit algorithm for the solution of linear complementarity problems arising from free boundary problems,” J. Appl. Math. Optim., vol. 13, pp. 1–17, 1985.
Y.Y. Lin and J.S. Pang, “Iterative methods for large convex quadratic programs: A survey,” SIAM J. Control Optim., vol. 25, pp. 383–411, 1987.
P. Lötstedt, “Solving the minimal least squares problems subject to bound on the variables,” BIT, vol. 24, pp. 206–224, 1984.
J.J. Moré and G. Toraldo, “Algorithms for bound constrained quadratic programming problems,” Numer. Math., vol. 55, pp. 377–400, 1989.
J.J. Moré and G. Toraldo, “On the solution of large quadratic programming problems with bound constraints,” SIAM J. Optimization, vol. 1, pp. 93–113, 1991.
U. Oreborn, “A direct method for sparse nonnegative least squares problems,” Ph.D. Thesis, Department of Mathematics, Thesis 87, Linoping University, Linkoping, Sweden, 1986.
L. Qi and D. Sun, “Improving the convergence of non-interior point algorithms for nonlinear complementarity problems,” Math. Comp., vol. 69, pp. 283–304, 2000.
L. Qi, D. Sun, and G. Zhou, “A new look at smoothing newton methods for non-linear complementarity problems and box constrained variational inequalities,” Mathematical Programming, vol. 87, pp. 1–35, 2000.
S. M. Robinson, “Normal maps induced by linear transformations,” Mathematics of Operations Research, vol. 17, pp. 691–714, 1992.
P. Tseng, “Analysis of a non-interior continuation method based on Chen-Mangasarian smoothing functions for complementarity problems.” in Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, M. Fukushima and L. Qi (Eds.), Kluwer Academic Publishers, pp. 381–404.
S. Xu, “The global linear convergence of an infeasible non-interior path-following algorithm for complementarity problems with uniform P-functions,” Mathematical Programming, vol. 87, pp. 501–517, 2000.
E.K. Yang and J.W. Tolle, “A class of methods for solving large convex quadratic quadratic programs subject to bound constraints,” Technical report, Department of Operations research, University of North Carolina, Chapel Hill, NC, 1988.
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Xu, S. A Non-Interior Path Following Method for Convex Quadratic Programming Problems with Bound Constraints. Computational Optimization and Applications 27, 285–303 (2004). https://doi.org/10.1023/B:COAP.0000013060.16224.31
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DOI: https://doi.org/10.1023/B:COAP.0000013060.16224.31