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The same growth of FB and NR symmetric cone complementarity functions. (English) Zbl 1280.90121

Summary: We establish that the Fischer-Burmeister (FB) complementarity function and the natural residual (NR) complementarity function associated with the symmetric cone have the same growth, in terms of the classification of Euclidean Jordan algebras. This, on the one hand, provides an affirmative answer to the second open question proposed by P. Tseng [J. Optimization Theory Appl. 89, No. 1, 17–37 (1996; Zbl 0866.90127)] for the matrix-valued FB and NR complementarity functions, and on the other hand, extends the third important inequality of Lemma 3.1 in the aforementioned paper to the setting of Euclidean Jordan algebras. It is worthwhile to point out that the proof is surprisingly simple.

MSC:

90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)

Citations:

Zbl 0866.90127

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References:

[1] Chen J.-S., Tseng P.: An unconstrained smooth minimization reformulation of the second-order cone complementarity problem. Math. Program. 104, 293–327 (2005) · Zbl 1093.90063 · doi:10.1007/s10107-005-0617-0
[2] Chen X., Qi H.D.: Cartesian P-property and its applications to the semidefinite linear complementarity problem. Math. Program. 106, 177–201 (2006) · Zbl 1134.90508 · doi:10.1007/s10107-005-0601-8
[3] Faraut, J., Korányi, A.: Analysis on symmetric cones. In: Oxford Mathematical Monographs. Oxford University Press, New York (1994) · Zbl 0841.43002
[4] Faybusovich L.: Euclidean Jordan algebras and interior point algorithms. Positivity 1, 331–357 (1997) · Zbl 0973.90095 · doi:10.1023/A:1009701824047
[5] Facchinei F., Pang J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, vols. I and II. Springer, New York (2003) · Zbl 1062.90002
[6] Gowda M.S., Sznajder R., Tao J.: Some P-properties for linear transformations on Euclidean Jordan algebras. Linear Algebra Appl. 393, 203–232 (2004) · Zbl 1072.15002 · doi:10.1016/j.laa.2004.03.028
[7] Koecher M.: The Minnesota Notes on Jordan Algebras and Their Applications. Edited and annotated by A. Brieg and S. Walcher. Springer, Berlin (1999) · Zbl 1072.17513
[8] Korányi A.: Monotone functions on formally real Jordan algebras. Math. Ann. 269, 73–76 (1984) · Zbl 0552.17014 · doi:10.1007/BF01455996
[9] Kong, L.C., Xiu, N.H.: The Fischer-Burmeister complementarity function on Euclidean Jordan algebras. Pac. J. Optim. (2009, published online) · Zbl 1220.90132
[10] Kong L.C., Sun J., Xiu N.H.: A regularized smoothing Newton method for symmetric cone complementarity problems. SIAM J. Optim. 19, 1028–1047 (2008) · Zbl 1182.65092 · doi:10.1137/060676775
[11] Kong L.C., Tuncel L., Xiu N.H.: Vector-valued implicit Lagrangian for symmetric cone complementarity problems. Asia-Pac. J. Oper. Res. 26, 199–233 (2009) · Zbl 1168.90622 · doi:10.1142/S0217595909002171
[12] Li, Y.M., Wang, X.T., Wei, D.Y.: A new class of complementarity functions for symmetric cone complementarity problems. Optim. Lett. (2010, to appear) · Zbl 1222.90067
[13] Liu Y., Zhang L., Wang Y.: Some propeties of a class of merit functions for symmetric cone complementarity problems. Asia-Pac. J. Oper. Res. 23, 473–496 (2006) · Zbl 1202.90252 · doi:10.1142/S0217595906000991
[14] Pan, S.-H., Chen, J.-S., Li, J.-F.: The same growth of FB-SOC merit function and NR-SOC merit function. Technical report (2009)
[15] Pan S.-H., Chen J.-S.: A one-parametric class of merit functions for the symmetric cone complementarity problem. J. Math. Anal. Appl. 355, 195–215 (2009) · Zbl 1180.90341 · doi:10.1016/j.jmaa.2009.01.064
[16] Pardalos P.M., Ramana M.: Semidefinite programming. In: Terlaky, T. (eds) Interior Point Methods of Mathematical Programming, pp. 369–398. Kluwer, Dordrecht (1996) · Zbl 0874.90131
[17] Qin L.-X., Kong L.C., Han J.-Y.: Sufficiency of linear transformations on Euclidean Jordan algebras. Optim. Lett. 3, 265–276 (2009) · Zbl 1168.15303 · doi:10.1007/s11590-008-0106-5
[18] Schmieta S.H., Alizadeh F.: Extension of primal-dual interior point algorithms for symmetric cones. Math. Program. 96, 409–438 (2003) · Zbl 1023.90083 · doi:10.1007/s10107-003-0380-z
[19] Sun D., Sun J.: Löwner’s operator and spectral functions on Euclidean Jordan algebras. Math. Oper. Res. 33, 421–445 (2008) · Zbl 1218.90197 · doi:10.1287/moor.1070.0300
[20] Tseng P.: Growth behavior of a class of merit functions for the nonlinear complementarity problems. J. Optim. Theory Appl. 89, 17–37 (1996) · Zbl 0866.90127 · doi:10.1007/BF02192639
[21] Tseng P.: Merit function for semidefinite complementarity problems. Math. Program. 83, 159–185 (1998) · Zbl 0920.90135
[22] Yoshise A.: Interior point trajectories and a homogenous model for nonlinear complementarity problems over symmetric cones. SIAM J. Optim. 17, 1129–1153 (2006) · Zbl 1136.90039 · doi:10.1137/04061427X
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