Cartesian P-property and its applications to the semidefinite linear complementarity problem

Mathematical Programming

Volumn 106, Issue 1, 2006, Pages 177-201

  Chen, Xin ^a   Qi, Houduo ^b  

^a University of Illinois at Urbana Champaign   (United States)

^b UNIVERSITY OF SOUTHAMPTON   (United Kingdom)

Author keywords

Cartesian P property; Globally unique solvability; Merit functions; Non interior continuation method; SDLCP

Indexed keywords

ITERATIVE METHODS; PROBLEM SOLVING;

CARTESIAN P-PROPERTY; GLOBALLY UNIQUE SOLVABILITY; MERIT FUNCTIONS; SDLPC;

LINEAR PROGRAMMING;

EID: 29144505119     PISSN: 00255610     EISSN: 14364646     Source Type: Journal    
DOI: 10.1007/s10107-005-0601-8     Document Type: Article

References (39)

1. Chen, B., Harker, P.T.: A non-interior-point continuation method for linear complementarity problems. SIAM J. Matrix Anal. Appl. 14, 1168-1190 (1993)
2. Chen, B., Harker, P.T.: A continuation method for monotone variational inequalities. Math. Programming 69, 237-253 (1995)
3. Chen, C., Mangasarian, O.L.: Smoothing methods for convex inequalities and linear complementarity problems. Math. Programming 71, 51-69 (1995)
4. Chen, C., Mangasarian, O.L.: A class of smoothing functions for nonlinear and mixed complementarity problems. Comput. Optim. Appl. 5, 97-138 (1996)
5. Chen, X., Qi, H.-D., Tseng, P.: Analysis of nonsmooth symmetric-matrix functions with applications to semidefinite complementarity problems. SIAM J. Optim. 13, 960-985 (2003)
6. Chen, X., Tseng, P.: Non-interior continuation methods for solving semidefinite complementarity problems. Math. Programming 95, (2003), 431-474 (2003)
7. Chen, X.D., Sun, D., Sun, J.: Complementarity functions and numerical experiments on some smoothing Newton methods for second-order cone complementarity problems. Comput. Optim. Appl. 25, 39-56 (2003)
8. Cottle, R.W., Pang, J.-S., Stone, R. E.: The Linear Complementarity Problem. Academic Press, Boston, 1992
9. De Luca, T., Facchinei, F., Kanzow, C.: A semismooth equation approach to the solution of nonlinear complementarity problems. Math. Programming 75, 407-439, (1996)
10. Facchinei, F., Pang, J.-S.: Finite-dimensional variational inequalities and complementarity problems Vol. I & II. Springer-Verlag, New York, 2003
11. Facchinei, F., Soares, J.: A new merit function for nonlinear complementarity problems and a related algorithm. SIAM J. Optim. 7, 225-247 (1997)
12. Fischer, A.: A special Newton-type optimization method. Optimization 24, 269-284 (1992)
13. Fischer, A.: An NCP-function and its use for the solution of complementarity problems. In: D.-Z. Du, L. Qi, R. Womersley (eds.) Recent Advances in Nonsmooth Optimization, World Scientific, Singapore, 1995, pp. 88-105
14. Gowda, M.S. : On the continuity of the solution map in linear complementarity problems. SLAM J. Optim. 2, 619-634 (1992)
15. Gowda, M.S., Song, Y.: On semidefinite linear complementarity problems. Math. Programming 88, 575-587 (2000)
16. Gowda, M.S., Song, Y.: Some new results for the semidefinite linear complementarity problem. SIAM J. Matrix Anal. Appl. 24, 25-39 (2002)
17. Gowda, M.S., Song, Y.: Semidefinite relaxation of linear complementarity problems. Report, Department of Mathematics and Statistics, University of Maryland, Baltimore County, Baltimore, Maryland, 2002
18. Gowda, M.S., Song, Y., Ravindran, G.: On some interconnections between strict monotonicity, globally uniquely solvable, and P properties in semidefinite linear complementarity problems. Linear Algebra Appl. 370, 355-368 (2003)
19. Jiang, H.: Unconstrained minimization approaches to nonlinear complementarity problems. J. Global Optim. 9, 169-181 (1996)
20. Kanzow, C.: Some non-interior continuation methods for linear complementarity problems. SIAM J. Matrix Anal. Appl. 17, 851-868 (1996)
21. Kanzow, C.: A new approach to continuation methods for complementarity problems with uniform P-functions. Oper. Res. Letters 20, 85-92 (1997)
22. Kanzow, C., Nagel, C.: Semidefinite programs: new search directions, smoothing-type methods, and numerical results. SLAM J. Optim. 13, 1-23 (2002)
23. Kojima, M., Shindoh, S., Hara, S.: Interior-point methods for the monotone semidefinite linear complementarity problem in symmetric matrices. SIAM J. Optim. 7, 86-125 (1997)
24. Lewis, A.S.: Private communication, 2003
25. Löwner, K.: Über monotone matrixfunctionen. Mathematische Zeitschrift 38, 177-216 (1934)
26. Mangasarian, O.L., Solodov, M.V.: Nonlinear complementarity as unconstrained and constrained minimization. Math. Programming 62, 277-297 (1993)
27. Mathias, R., Pang, J.-S.: Error bounds for the linear complementarity problems with a P-matrix. Linear Algebra Appl. 132, 123-136 (1990)
28. Murty, K.G.: On the number of solutions to the complementarity problem and spanning properties of complementary cones. Linear Algebra Appl. 5, 65-108 (1972)
29. Pang, J.-S., Sun, D., Sun, J.: Semismooth homeomorphisms and strong stability of semidefinite and Lorentz complementarity problems. Math. Oper. Res. 28, 39-63 (2003)
30. 2-property, and the GUS property in semidefinite LCPs. Math. Oper. Res. 27, 326-331 (2002)
31. Qi, H.-D., Chen, X.: On stationary points of merit functions for semidefinite complementarity problems. Report, Institute of Computational Mathematics and Scientific/Engineering Computing, Chinese Academy of Sciences, Beijing, 1997
32. Shida, M., Shindoh, S., Kojima, M.: Existence and uniqueness of search directions in interior-point algorithms for the SDP and the monotone SDLCP. SIAM J. Optim. 8 (2), 387-396 (1998)
33. Smale, S. : Algorithms for solving equations. In: Gleason, A. M. (ed.) Proceeding of International Congress of Mathematicians, American Mathematical Society, Providence, Rhode Island, 1987, pp. 172-195
34. Sun, D., Sun, J.: Semismooth matrix-valued functions. Math. Oper. Res. 27, 150-169 (2002)
35. Sun, J., Sun, D., Qi, L.: Quadratic convergence of a smoothing Newton method for nonsmooth matrix equations and its applications in semidefinite optimization problems. SIAM J. Optim. 14, 783-806 (2003)
36. Tseng, P.: Merit functions for semidefinite complementarity problems. Math. Programming 83, 159-185 (1998)
37. Tseng, P. : Search directions and convergence analysis of some infeasible path-following methods for the monotone semi-definite LCP. Optim. Methods Softw. 9 (4), 245-268 (1998)
38. Tseng, P.: Analysis of a non-interior continuation method based on Chen-Mangasarian smoothing functions for complementarity problems. In: Fukushima, M., Qi, L. (ed.) Reformulation - Nonsmooth, Piece-wise Smooth, Semismooth and Smoothing Methods, Kluwer Academic Publishers, Boston, 1999, pp. 381-404
39. Yamashita, N., Fukushima, M.: A new merit function and a descent method for semidefinite complementarity problems. Reformulation: nonsmooth, piecewise smooth, semismooth and smoothing methods (Lausanne, 1997), Appl. Optim., 22, Kluwer Acad. Publ., Dordrecht, 1999, pp. 405-420