Vector-valued implicit Lagrangian for symmetric cone complementarity problems

Asia-Pacific Journal of Operational Research

Volumn 26, Issue 2, 2009, Pages 199-233

  Kong, Lingchbn ^a   Tungel, Lbvbnt ^b   Xiu, Naihua ^a  

^a BEIJING JIAOTONG UNIVERSITY   (China)

^b UNIVERSITY OF WATERLOO   (Canada)

Author keywords

C function; Jordan algebra; Symmetric cone complementary problem; Uniform Cartesian P property; Vector valued implicit Lagrangian

Indexed keywords

C-FUNCTION; JORDAN ALGEBRA; SYMMETRIC CONE COMPLEMENTARY PROBLEM; UNIFORM CARTESIAN P-PROPERTY; VECTOR-VALUED IMPLICIT LAGRANGIAN;

ALGEBRA; VECTORS;

LAGRANGE MULTIPLIERS;

EID: 67651069041     PISSN: 02175959     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0217595909002171     Document Type: Article

References (39)

1. Auchmuty, G (1989). Variational principles for variational inequalities. Numerical Functional Analysis and Optimization, 10, 863-874.
2. Chen, B and PT Harker (1997). Smooth approximations to nonlinear complementarity problems. SIAM Journal on Optimization, 7, 403-422.
3. Chen, X and H Qi (2006). Cartesian P-property and its applications to the semidefinite linear complementarity problem, Mathematical Programming, 106, 177-201.
4. Facchinei F and C Kanzow (1997). On unconstrained and constrained stationary points of the Implicit Lagrangian. Journal of Optimization Theory and Applications, 92, 99-115.
5. Facchinei, F and J-S Pang (2003). Finite-Dimensional Variational Inequalities and Complementarity Problems, Volume I and II. New York: Springer-Verlag.
6. Faraut, J and A Koranyi (1994). Analysis on Symmetric Cones. Oxford University Press, New York.
7. Faybusovich, L (1997). Euclidean Jordan algebras and interior-point algorithms. Positivity, 1, 331-357.
8. Fukushima, M (1992). Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems. Mathematical Programming, 53, 99-110.
9. Fukushima, M (1996). Merit functions for variational inequality and complementarity problems. In: Di Pillo, G and F Giannessi (eds.) Nonlinear Optimization and Applications, Plenum Press, New York, 155-170.
10. Gowda, MS, R Sznajder and J Tao (2004). Some P-properties for linear transformations on Euclidean Jordan algebras. Linear Algebra and Its Applications, 393, 203-232.
11. Gowda, MS and R Sznajder (2006). Automorphism invariance of P and GUS properties of linear transformations on Euclidean Jordan algebras. Mathematics of Operations Research, 31, 109-123.
12. Gowda, MS and J Tao (2007). Z-transformation on proper and symmetric cone, Mathematical Programming, Published online: 18 July 2007.
13. Isac, G (2000). Topological Methods in Complementarity Theory. Kluwer Academic Publishers, Dordrecht.
14. Kanzow, C and M Fukushima (1998a). Theoretical and numerical investigation of the D-gap function for box constrained variational inequalities. Mathematical Programming, 83, 55-87.
15. Kanzow, C and M Fukushima (1998b). Solving box constrained variational inequalities by using the natural residual with D-gap function globalization. Operations Research Letters, 23, 45-51.
16. Koecher, M (1999). The Minnesota Notes on Jordan Algebras and Their Applications, edited and annotated by Brieg, A., Walcher, S., Springer, Berlin.
17. Kong, LC and NH Xiu (2007). New smooth C-functions for symmetric cone complementarity problems. Optimization Letters, 1, 391-400.
18. Lin, Y and A Yoshise (2005). A homogeneous model for mixed complementarity problems over symmetric cones, Preprint, University of Tsukuba, Japan.
19. Liu, Y, L Zhang and Y Wang (2006). Some properties of a class of merit functions for symmetric cone complementarity problems. Asia-Pacific Journal of Operational Research, 23, 473-495.
20. Luo, ZQ, OL Mangasarian, J Ren and MV Solodov (1994). New error bounds for the linear complementarity problems. Mathematics of Operations Research, 19, 880-892.
21. Luo, ZQ, J-S Pang and D Ralph (1996). Mathematical Programms with Equilibrium Constraints, Cambridge University Press.
22. Malik, M and SR Mohan (2003). On complementarity problems over symmetric cones, Discussion Paper Series DPS/SQCOR/Delhi/05-2003, Indian Statistical Institute.
23. Malik, M and SR Mohan (2006). Cone complementarity problems with finite solution sets. Operations Research Letters, 34, 121-126.
24. Mangasarian, OL and MV Solodov (1993). Nonlinear complementarity as unconstrained and constrained minimization. Mathematical Programming, 62, 277-297.
25. Peng, JM (1997a). Equivalence of variational inequality problems to unconstrained minimization. Mathematical Programming, 78, 347-355.
26. Peng, JM (1997b). Convexity of the implicit Lagrangian. Journal of Optimization Theory and Applications, 92, 331-341.
27. Peng, JM and M Fukushima (1999). A hybrid Newton method for solving the variational inequality problem via the D-gap function. Mathematical Programming, 86, 367-386.
28. Rockafellar, RT and RJ-B Wets (2004). Variational Analysis, Second Version. Springer, New York.
29. Schmieta, SH and F Alizadeh (2003). Extension of primal-dual interior point algorithms to symmetric cones. Mathematical Programming, 96, 409-438.
30. Solodov, MV and P Tseng (2000). Some methods based on the D-gap function for solving monotone variational inequalities. Computational Optimization and Applications, 17, 255-277.
31. Sun, D, M Fukushima and L Qi (1997). A computable generalized Hessian of the D-gap function and Newton-type methods for variational inequality problems. In: Ferris, M. C., Pang, J.-S. (Eds.), Complementarity and Variational Problems: State of the Art, SIAM, Philadelphia, PA, 452-473.
32. Sun, D and L Qi (1999). On NCP-functions. Computational Optimization and Applications, 13, 201-220.
33. Sun, D and J Sun (2008). Lowners operator and spectral functions on Euclidean Jordan algebras, Mathematics of Operations Research 33.
34. Tao, J and MS Gowda (2005). Some P-properties for nonlinear transformations on Euclidean Jordan algebras. Mathematics of Operations Research, 30, 985-1004.
35. Tseng, P (1998). Merit function for semi-definite complementarity problems. Mathematical Programming, 83, 159-185.
36. Tseng, P, N Yamashita and M Fukushima (1996). Equivalence of complementarity problems to differentiable minimization: A unified approach. SIAM Journal on Optimization, 6, 446-460.
37. Yamashita, N and M Fukushima (1995). On stationary points of the Implicit Lagrangian for nonlinear complementarity problems. Journal of Optimization Theory and Applications, 84, 653-663.
38. Yamashita, N, K Taji and M Fukushima (1997). Unconstrained optimization reformulations of variational inequality problems. Journal of Optimization Theory and Applications, 92, 439-456.
39. Yoshise, A (2006). Interior point trajectories and a homogeneous model for nonlinear complementarity problems over symmetric cones. SIAM Journal on Optimization, 17, 1129-1153.