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Cartesian P-property and Its Applications to the Semidefinite Linear Complementarity Problem

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Abstract

We introduce a Cartesian P-property for linear transformations between the space of symmetric matrices and present its applications to the semidefinite linear complementarity problem (SDLCP). With this Cartesian P-property, we show that the SDLCP has GUS-property (i.e., globally unique solvability), and the solution map of the SDLCP is locally Lipschitzian with respect to input data. Our Cartesian P-property strengthens the corresponding P-properties of Gowda and Song [15] and allows us to extend several numerical approaches for monotone SDLCPs to solve more general SDLCPs, namely SDLCPs with the Cartesian P-property. In particular, we address important theoretical issues encountered in those numerical approaches, such as issues related to the stationary points in the merit function approach, and the existence of Newton directions and boundedness of iterates in the non-interior continuation method of Chen and Tseng [6].

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References

  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)

    Article  MATH  MathSciNet  Google Scholar 

  2. Chen, B., Harker, P.T.: A continuation method for monotone variational inequalities. Math. Programming 69, 237–253 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  3. Chen, C., Mangasarian, O.L.: Smoothing methods for convex inequalities and linear complementarity problems. Math. Programming 71, 51–69 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  4. Chen, C., Mangasarian, O.L.: A class of smoothing functions for nonlinear and mixed complementarity problems. Comput. Optim. Appl. 5, 97–138 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  6. Chen, X., Tseng, P.: Non-interior continuation methods for solving semidefinite complementarity problems. Math. Programming 95, (2003), 431–474 (2003)

    Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  12. Fischer, A.: A special Newton-type optimization method. Optimization 24, 269–284 (1992)

    MATH  MathSciNet  Google Scholar 

  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. SIAM J. Optim. 2, 619–634 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  15. Gowda, M.S., Song, Y.: On semidefinite linear complementarity problems. Math. Programming 88, 575–587 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  16. Gowda, M.S., Song, Y.: Some new results for the semidefinite linear complementarity problem. SIAM J. Matrix Anal. Appl. 24, 25–39 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  19. Jiang, H.: Unconstrained minimization approaches to nonlinear complementarity problems. J. Global Optim. 9, 169–181 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  20. Kanzow, C.: Some non-interior continuation methods for linear complementarity problems. SIAM J. Matrix Anal. Appl. 17, 851–868 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  21. Kanzow, C.: A new approach to continuation methods for complementarity problems with uniform P-functions. Oper. Res. Letters 20, 85–92 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  22. Kanzow, C., Nagel, C.: Semidefinite programs: new search directions, smoothing-type methods, and numerical results. SIAM J. Optim. 13, 1–23 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  24. Lewis, A.S.: Private communication, 2003

  25. Löwner, K.: Über monotone matrixfunctionen. Mathematische Zeitschrift 38, 177–216 (1934)

    Article  MATH  MathSciNet  Google Scholar 

  26. Mangasarian, O.L., Solodov, M.V.: Nonlinear complementarity as unconstrained and constrained minimization. Math. Programming 62, 277–297 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  27. Mathias, R., Pang, J.-S.: Error bounds for the linear complementarity problems with a P-matrix. Linear Algebra Appl. 132, 123–136 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  30. Parthasarathy, T., Sampangi Raman, D., Sriparna, B.: Relationship between strong monotonicity, P 2-property, and the GUS property in semidefinite LCPs. Math. Oper. Res. 27, 326–331 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  36. Tseng, P.: Merit functions for semidefinite complementarity problems. Math. Programming 83, 159–185 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  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, Piecewise 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

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Authors and Affiliations

  1. Department of Mechanical and Industrial Engineering, University of Illinois Urbana-Champaign 224 Mechanical Engineering Bldg, MC-244 1206 West Green Street, Urbana, IL, 61801, USA

    Xin Chen

  2. School of Mathematics, University of Southampton, Highfield, Southampton, SO17 1BJ, UK

    Houduo Qi

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  1. Xin Chen
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Correspondence to Houduo Qi.

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This work is supported by the annual grant A2004/23 of University of Southampton.

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Chen, X., Qi, H. Cartesian P-property and Its Applications to the Semidefinite Linear Complementarity Problem. Math. Program. 106, 177–201 (2006). https://doi.org/10.1007/s10107-005-0601-8

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  • DOI: https://doi.org/10.1007/s10107-005-0601-8

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