×

A two-parametric class of merit functions for the second-order cone complementarity problem. (English) Zbl 1278.90401

Summary: We propose a two-parametric class of merit functions for the second-order cone complementarity problem (SOCCP) based on the one-parametric class of complementarity functions. By the new class of merit functions, the SOCCP can be reformulated as an unconstrained minimization problem. The new class of merit functions is shown to possess some favorable properties. In particular, it provides a global error bound if \(F\) and \(G\) have the joint uniform Cartesian \(P\)-property. And it has bounded level sets under a weaker condition than the most available conditions. Some preliminary numerical results for solving the SOCCPs show the effectiveness of the merit function method via the new class of merit functions.

MSC:

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

Software:

LBFGS-B; SCCP
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] M. S. Lobo, L. Vandenberghe, S. Boyd, and H. Lebret, “Applications of second-order cone programming,” Linear Algebra and Its Applications, vol. 284, no. 1-3, pp. 193-228, 1998. · Zbl 0946.90050
[2] Y. J. Kuo and H. D. Mittelmann, “Interior point methods for second-order cone programming and OR applications,” Computational Optimization and Applications, vol. 28, no. 3, pp. 255-285, 2004. · Zbl 1084.90046
[3] F. Alizadeh and D. Goldfarb, “Second-order cone programming,” Mathematical Programming B, vol. 95, no. 1, pp. 3-51, 2003. · Zbl 1153.90522
[4] M. Fukushima, Z. Q. Luo, and P. Tseng, “Smoothing functions for second-order-cone complementarity problems,” SIAM Journal on Optimization, vol. 12, no. 2, pp. 436-460, 2002. · Zbl 0995.90094
[5] J. S. Chen and S. Pan, “A survey on SOC complementarity functions and solution methods for SOCPs and SOCCPs,” Pacific Journal of Optimization, vol. 8, pp. 33-74, 2012. · Zbl 1286.90148
[6] R. D. C. Monteiro and T. Tsuchiya, “Polynomial convergence of primal-dual algorithms for the second-order cone program based on the MZ-family of directions,” Mathematical Programming B, vol. 88, no. 1, pp. 61-83, 2000. · Zbl 0967.65077
[7] T. Tsuchiya, “A convergence analysis of the scaling-invariant primal-dual path-following algorithms for second-order cone programming,” Optimization Methods and Software, vol. 11, no. 1, pp. 141-182, 1999. · Zbl 0957.90129
[8] G. Q. Wang and Y. Q. Bai, “A new full Nesterov-Todd step primal-dual path-following interior- point algorithm for symmetric optimization,” Journal of Optimization Theory and Applications, vol. 154, no. 3, pp. 966-985, 2012. · Zbl 1256.90036
[9] L. Qi, D. Sun, and G. Zhou, “A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities,” Mathematical Programming B, vol. 87, no. 1, pp. 1-35, 2000. · Zbl 0989.90124
[10] X. D. Chen, D. Sun, and J. Sun, “Complementarity functions and numerical experiments on some smoothing Newton methods for second-order-cone complementarity problems,” Computational Optimization and Applications, vol. 25, no. 1-3, pp. 39-56, 2003. · Zbl 1038.90084
[11] L. Fang and C. Han, “A new one-step smoothing newton method for the second-order cone complementarity problem,” Mathematical Methods in the Applied Sciences, vol. 34, no. 3, pp. 347-359, 2011. · Zbl 1230.90150
[12] J. Y. Tang, G. P. He, L. Dong, and L. Fang, “A new one-step smoothing Newton method for second- order cone programming,” Applications of Mathematics, vol. 57, no. 4, pp. 311-331, 2012. · Zbl 1265.90229
[13] S. Hayashi, N. Yamashita, and M. Fukushima, “A combined smoothing and regularization method for monotone second-order cone complementarity problems,” SIAM Journal on Optimization, vol. 15, no. 2, pp. 593-615, 2005. · Zbl 1114.90139
[14] J. S. Chen and P. Tseng, “An unconstrained smooth minimization reformulation of the second-order cone complementarity problem,” Mathematical Programming, vol. 104, no. 2-3, pp. 293-327, 2005. · Zbl 1093.90063
[15] N. Lu and Z. H. Huang, “Three classes of merit functions for the complementarity problem over a closed convex cone,” Optimization, vol. 62, no. 4, pp. 545-560, 2013. · Zbl 1273.90148
[16] J. S. Chen and S. Pan, “A one-parametric class of merit functions for the second-order cone complementarity problem,” Computational Optimization and Applications, vol. 45, no. 3, pp. 581-606, 2010. · Zbl 1226.90114
[17] J. S. Chen, “Two classes of merit functions for the second-order cone complementarity problem,” Mathematical Methods of Operations Research, vol. 64, no. 3, pp. 495-519, 2006. · Zbl 1162.90572
[18] U. Faraut and A. Korányi, Analysis on Symmetric Cones, Oxford Mathematical Monographs, Oxford University Press, New York, NY, USA, 1994. · Zbl 0841.43002
[19] J. S. Chen, “A new merit function and its related properties for the second-order cone complementarity problem,” Pacific Journal of Optimization, vol. 2, pp. 167-179, 2006. · Zbl 1178.90324
[20] S. Pan and J. Chen, “A one-parametric class of merit functions for the symmetric cone complementarity problem,” Journal of Mathematical Analysis and Applications, vol. 355, no. 1, pp. 195-215, 2009. · Zbl 1180.90341
[21] L. Kong, L. Tuncel, and N. Xiu, “Vector-valued implicit Lagrangian for symmetric cone complementarity problems,” Asia-Pacific Journal of Operational Research, vol. 26, no. 2, pp. 199-233, 2009. · Zbl 1168.90622
[22] F. Facchinei and J. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, vol. 1-2, Springer, New York, NY, USA, 2003. · Zbl 1062.90002
[23] Y. Liu, L. Zhang, and Y. Wang, “Some properties of a class of merit functions for symmetric cone complementarity problems,” Asia-Pacific Journal of Operational Research, vol. 23, no. 4, pp. 473-495, 2006. · Zbl 1202.90252
[24] R. H. Byrd, P. Lu, J. Nocedal, and C. Zhu, “A limited memory algorithm for bound constrained optimization,” SIAM Journal on Scientific Computing, vol. 16, no. 5, pp. 1190-1208, 1995. · Zbl 0836.65080
[25] J. Nocedal and S. J. Wright, Numerical Optimization, Springer, New York, NY, USA, 1999. · Zbl 0930.65067
[26] L. Grippo, F. Lampariello, and S. Lucidi, “A nonmonotone line search technique for Newton’s method,” SIAM Journal on Numerical Analysis, vol. 23, no. 4, pp. 707-716, 1986. · Zbl 0616.65067
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.