Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
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MERIT FUNCTIONS FOR MATRIX CONE COMPLEMENTARITY PROBLEMS

  • Wang, Li (School of Science, Shenyang Aerospace University) ;
  • Liu, Yong-Jin (School of Science, Shenyang Aerospace University) ;
  • Jiang, Yong (School of Science, Shenyang Aerospace University)
  • Received : 2012.05.09
  • Accepted : 2012.12.10
  • Published : 2013.09.30
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Abstract

The merit function arises from the development of the solution methods for the complementarity problems defined over the cone of non negative real vectors and has been well extended to the complementarity problems defined over the symmetric cones. In this paper, we focus on the extension of the merit functions including the gap function, the regularized gap function, the implicit Lagrangian and others to the complementarity problems defined over the nonsymmetric matrix cone. These theoretical results of this paper suggest new solution methods based on unconstrained and/or simply constrained methods to solve the matrix cone complementarity problems (MCCP).

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References

  1. F. Alizadeh and S. Schmieta, Symmetric cones, potential reduction methods, in Handbook of Semidefinite Programming, H. Wolkowicz, R. Saigal, and L. Vandenberghe, eds., Kluwer, Boston, (2000), 195-233.
  2. G. Auchmuty, Variational principles for variational inequalities, Numerical Functional Analysis and Optimization, 10 (1989), 863-874. https://doi.org/10.1080/01630568908816335
  3. A. Auslender, Optimisation: Methods Numeriques, Masson, Paris, 1976.
  4. E.J. Candes and B. Recht, Exact matrix completion via convex optimization, Foundations of Computational Mathematics, 9 (2008) 717-772.
  5. C. Chen and O.L. Mangasarian, A class of smoothing functions for nonlinear and mixed complementarity problems, Computational Optimization and Applications, 5 (1996), 97-138. https://doi.org/10.1007/BF00249052
  6. J.-S. Chen and P. Tseng, An unconstrained smooth minimization reformulation of the second-order cone complementarity problem, Mathematical Programming, 104 (2005), 297-327.
  7. X. Chen, H.-D. Qi and P. Tseng, Analysis of nonsmooth symmetric matrix functions with applications to semidefinite complementarity problems, SIAM Journal on Optimization, 13 (2003), 960-985. https://doi.org/10.1137/S1052623400380584
  8. C. Ding, D. Sun and K.-C. Toh, An introduction to a class of matrix cone programming, Preprint available at http://www.optimization-online.org/DB HTML/2010/09/2746.html.
  9. F. Facchinei and J.-S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Volumes I and II, Springer-Verlag, New York, 2003.
  10. L. Faybusovich, Euclidean Jordan algebras and interior-point algorithm, Positivity, 1 (1997), 331-357. https://doi.org/10.1023/A:1009701824047
  11. M. Fukushima, Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems, Mathematical Programming, 53 (1992), 99-110. https://doi.org/10.1007/BF01585696
  12. M. Fukushima, Z.-Q. Luo, and P. Tseng, Smoothing functions for second-order cone complementarity problems, SIAM Journal on Optimization, 12 (2002), 436-460. https://doi.org/10.1137/S1052623400380365
  13. D.R. Han, On the coerciveness of some merit functions for complementarity problems over symmetric cones, Journal of Mathematical Analysis and Applications, 336 (2007), 727-737. https://doi.org/10.1016/j.jmaa.2007.03.003
  14. D.W. Hearn, The gap function of a convex program, Operations Research Letters, 1 (1982), 67-71. https://doi.org/10.1016/0167-6377(82)90049-9
  15. J.-B. Hiriart-Urruty and C. Lemarechal, Convex Analysis and Minimization Algorithms, Vols. 1 and 2, Springer-Verlag, Berlin, Heidelberg, New York, 1993.
  16. C. Kanzow, I. Ferenczi and M. Fukushima, On the local convergence of semismooth Newton methods for linear and nonlinear second-order cone programs without strict complementarity, SIAM Journal on Optimization, 20 (2009), 297-320. https://doi.org/10.1137/060657662
  17. L.C. Kong, J. Sun and N.H. Xiu, A regularized smoothing Newton method for symmetric cone complementarity problems, SIAM Journal on Optimization, 19 (2009), 1028-1047.
  18. L.C. Kong, L. Tuncel and N.H. Xiu, Vector-valued implicit Lagrangian for symmetric cone complementarity problems, Asia-Pacific Journal of Operational Research, 26 (2009), 199-233. https://doi.org/10.1142/S0217595909002171
  19. C. Lemarechal and C. Sagastizabal, Practical aspects of the Moreau-Yosida regularization I: theoretical preliminaries, SIAM Journal on Optimization, 7 (1997), 367-385. https://doi.org/10.1137/S1052623494267127
  20. Y.-J. Liu, L.W. Zhang and Y.H. Wang, Some properties of a class of merit functions for symmetric cone complementarity problems, Asia-Pacific Journal of Operational Research, 23 (2006), 473-495. https://doi.org/10.1142/S0217595906000991
  21. M.S. Lobo, L. Vandenberghe, S. Boyd and H. Lebret, Applications of second-order cone programmming, Linear Algebra and its Applications, 284 (1998), 193-228. https://doi.org/10.1016/S0024-3795(98)10032-0
  22. Z.-Q. Luo and P. Tseng, A new class of merit functions for the nonlinear complementarity problem, in: M.C. Ferris, J.-S. Pang (Eds.), Complementarity and Variational Problems: State of the Art, SIAM. Philadephia, 1997, 204-225.
  23. O.L. Mangasarian and M.V. Solodov, Nonlinear complementarity as unconstrained and constrained minimization, Mathematical Programming, 62 (1993), 277-297. https://doi.org/10.1007/BF01585171
  24. F. Meng, D. Sun and G. Zhao, Semismoothness of solutions to generalized equations and the Moreau-Yosida regularization, Mathematical Programming, 104 (2005), 561-581. https://doi.org/10.1007/s10107-005-0629-9
  25. J.J. Moreau, Decomposition orthogonale d'un espace hilbertien selon deux cones mutuellement polaires, Comptes Rendus de l'Academie des Sciences, 255 (1962), 238-240.
  26. J.J. Moreau, Proximite et dualite dans un espace hilbertien, Bulletin de la Societe Math-ematique de France, 93 (1965), 273-299.
  27. R.D.C. Monteiro and T. Tsuchiya, Polynomial convergence of primal-dual algorithms for the second-order cone programs based on the MZ-family of directions, Mathematical Pro-gramming, 88 (2000), 61-83. https://doi.org/10.1007/PL00011378
  28. S.H. Pan, Y.-L. Chang, and J.-S. Chen, Stationary point conditions for the FB merit function associated with symmetric cones, Operations Research Letters, 38 (2010), 372-377. https://doi.org/10.1016/j.orl.2010.07.011
  29. S.-H. Pan and J.-S. Chen, A damped Gauss-Newton method for the second-order cone complementarity problem, Applied Mathematics and Optimization, 59 (2009), 293-318. https://doi.org/10.1007/s00245-008-9054-9
  30. J.-S. Pang, S. Sun, and J. Sun, Semismooth homemorphisms and strong stability of semidefinite and Lorentz complementarity problems, Mathematics of Operations Research, 28 (2003), 39-63. https://doi.org/10.1287/moor.28.1.39.14258
  31. L. Qi, D. Sun and G.L. Zhou, A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities, Mathematical Pro-gramming, 87 (2000), 1-35. https://doi.org/10.1007/s101079900127
  32. S. Schmieta and F. Alizadeh, Associative and Jordan algebras, and polynomial time interior-point algorithms for symmetric cones, Mathematics of Operations Research, 26 (2001), 543-564. https://doi.org/10.1287/moor.26.3.543.10582
  33. D. Sun and J. Sun, Semismooth matrix valued functions, Mathematics of Operations Research, 27 (2002), 150-169. https://doi.org/10.1287/moor.27.1.150.342
  34. D. Sun and J. Sun, Strong semismoothness of Fischer-Burmeister SDC and SOC function, Mathematical Programming, 103 (2005), 575-581. https://doi.org/10.1007/s10107-005-0577-4
  35. T. Tsuchiya, A convergence analysis of the scaling-invariant primal-dual path-following algorithms for second-order cone programming, Optimization Methods and Software, 11 (1999), 141-182. https://doi.org/10.1080/10556789908805750
  36. P. Tseng, Merit functions for semidefinite complementarity problems, Mathematical Programming, 83 (1998), 159-185.
  37. K. Yosida, Functional Analysis, Springer Verlag, Berlin, 1964.
  38. A. Yoshise, Complementarity problems over symmetric cones: a survey of recent developments in several aspects, Handbook on Semidefinite, Conic and Polynomial Optimization, 166 (2012), 339-375. https://doi.org/10.1007/978-1-4614-0769-0_12
  39. E.H. Zarantonello, Projections on convex sets in Hilbert space and spectral theory I and II, Contributions to Nonlinear Functional Analysis (E. H. Zarantonello, ed.), Academic Press, New York, 1971, 237-424.