Abstract
The relationship between the mathematical program with linear complementarity constraints (MPLCC) and its inequality relaxation is studied. Based on this relationship, a new sequential quadratic programming (SQP) method is presented for solving the MPLCC. A certain SQP technique is introduced to deal with the possible infeasibility of quadratic programming subproblems. Global convergence results are derived without assuming the linear independence constraint qualification for MPEC, the nondegeneracy condition, and any feasibility condition of the quadratic programming subproblems. Preliminary numerical results are reported.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. Anitescu, “On solving mathematical programs with complementarity constraints as nonlinear programs,” Preprint ANL/MCS-P864-1200, Argonne National Laboratory, Argonne, IL, 2000.,
M.S. Bazaraa, H.D. Sherali, and C.M. Shetty, Nonlinear Programming, Theory and Algorithms, 2nd edition, John Wiley and Sons: New York, 1993.,
J.V. Burke, “A sequential quadratic programming method for potentially infeasible mathematical programs,” J. Math. Anal. Appl., vol. 139, pp. 319–351, 1989.,
J.V. Burke and S.P. Han, “A robust sequential quadratic programming method,” Math. Prog., vol. 43, pp. 277–303, 1989.,
A.V. De Miguel, M.P. Friedlander, F.J. Nogales and S. Scholtes, “A superlinearly convergent interior-point method for MPECs,” LBS working paper, 2003.,
F. Facchinei, H.Y. Jiang, and L. Qi, “A smoothing method for mathematical programs with equilibrium constraints,” Math. Program., vol. 85, pp. 107–134, 1999.,
R. Fletcher, S. Leyffer, D. Ralph and S. Scholtes, “Local convergence of SQP methods for mathematical programs with equilibrium constraints,” University of Dundee Report NA 209, 2002.,
M. Fukushima, Z.-Q. Luo, and J.-S. Pang, “A globally convergent sequential quadratic programming algorithm for mathematical programs with linear complementarity constraints,” Comput. Optim. Appl., vol. 10, pp. 5–34, 1998.,
M. Fukushima and J.S. Pang, “Some feasibility issues in mathematical programs with equilibrium constraints,” SIAM J. Optim., vol. 8, pp. 673–681, 1998.,
M. Fukushima and J.S. Pang, “Convergence of a smoothing continuation method for mathematical programs with complementarity constraints,” in Ill-posed Variational Problems and Regularization Techniques, M. Theŕa and R. Tichatschke (Eds.), Springer-Verlag: New York, pp. 99–110, 1999.,
M. Fukushima and P. Tseng, “An implementable active-set algorithm for computing a B-stationary point of a mathematical program with linear complementarity constraints,” SIAM J. Optim., vol. 12, pp. 724–739, 2002.,
H.Y. Jiang and D. Ralph, “QPECgen, a MATLAB generator for mathematical programs with quadratic objectives and affine variational inequality constraints,” Computational Optim. Appl., vol. 13, pp. 25–59, 1999.,
H.Y. Jiang and D. Ralph, “Smooth SQP methods for mathematical programs with nonlinear complementarity constraints,” SIAM J. Optim., vol. 10, pp. 779–808, 2000.,
S. Leyffer, “The penalty interior point method fails to converge for mathematical programs with equilibrium constraints,” to appear in a special issue of Optimization Methods and Software.,
S. Leyffer, MacMPEC - www-unix.mcs.anl.gov/ leyffer/MacMPEC/ (2002),
S. Leyffer, “Complementarity constraints as nonlinear equations: Theory and numerical experience,” Preprint, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL 60439, USA, 2003.,
X.W. Liu and Y.X. Yuan, “A robust algorithm for optimization with general equality and inequality constraints,” SIAM J. Sci. Comput., vol. 22, pp. 517–534, 2000.,
X.W. Liu and Y.X. Yuan, “A robust trust region algorithm for solving general nonlinear programming,” J. Comput. Math., vol. 19, pp. 309–322, 2001.,
Z.Q. Luo, J.S. Pang, and D. Ralph, Mathematical Programs with Equilibrium Constraints, Cambridge University Press, 1996.,
A. Raghunathan and L. Biegler, “MPEC formulations and algorithms in process engineering,” Technical report, Carnegie Mellon University, Department of Chemical Engineering, Pittsburgh, PA, 2002.,
D. Ralph, “Sequential quadratic programming for mathematical programs with linear complementarity constraints,” in Computational Techniques and Applications: CTAC95, R.L. May and A.K. Easton (Eds.), World Scientific, 1996, pp. 663–668.,
R.T. Rockafellar, The Theory of Subgradients and its Applications to Problems in Optimization: Convex and Nonoconvex Functions, Helderman Verlag, Berlin, 1981.,
S. Scholtes, “Convergence properties of a regularization scheme for mathematical programs with complementarity constraints,” SIAM J. Optim., vol. 11, pp. 918–936, 2001.,
S. Scholtes and M. Stöhr, “Exact penalization of mathematical programs with equilibrium constraints,” SIAM J. Control Optim., vol. 37, pp. 617–652, 1999.,
S. Scholtes and M. Stöhr, “How stringent is the linear independence assumption for mathematical programs with complementarity constraints,” Math. Oper. Res., vol. 26, pp. 851–863, 2001.,
Y. Yuan, “On the convergence of a new trust region algorithm,” Numer. Math., vol. 70, pp. 515–539, 1995.,
Author information
Authors and Affiliations
Additional information
Research is partially supported by Singapore-MIT Alliance and School of Business, National University of Singapore.
Rights and permissions
About this article
Cite this article
Liu, X., Perakis, G. & Sun, J. A Robust SQP Method for Mathematical Programs with Linear Complementarity Constraints. Comput Optim Applic 34, 5–33 (2006). https://doi.org/10.1007/s10589-005-3075-y
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10589-005-3075-y