Abstract
Recently Tseng (Math Program 83:159–185, 1998) extended a class of merit functions, proposed by Luo and Tseng (A new class of merit functions for the nonlinear complementarity problem, in Complementarity and Variational Problems: State of the Art, pp. 204–225, 1997), for the nonlinear complementarity problem (NCP) to the semidefinite complementarity problem (SDCP) and showed several related properties. In this paper, we extend this class of merit functions to the second-order cone complementarity problem (SOCCP) and show analogous properties as in NCP and SDCP cases. In addition, we study another class of merit functions which are based on a slight modification of the aforementioned class of merit functions. Both classes of merit functions provide an error bound for the SOCCP and have bounded level sets.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alizadeh F, Schmieta S (2000) Symmetric cones, potential reduction methods, and word-by-word extensions. In: Wolkowicz H, Saigal R, Vandenberghe L (eds) Handbook of semidefinite programming. Kluwer, Boston, pp 195–233
Andersen ED, Roos C, Terlaky T (2003) On implementing a primal-dual interior-point method for conic quadratic optimization. Math Program Ser B 95:249–277
Chen J-S (2006) A new merit function and its related properties for the second-order cone complementarity problem. Pacific J Optim 2:167–179
Chen J-S, Tseng P (2005) An unconstrained smooth minimization reformulation of the second-order cone complementarity problem. Math Program 104:293–327
Chen J-S, Chen X, Tseng P (2004) Analysis of nonsmooth vector-valued functions associated with second-order cone. Math Program 101:95–117
Chen X-D, Sun D, Sun J (2003) Complementarity functions and numerical experiments for second-order cone complementarity problems. Comput Optim Appl 25:39–56
Facchinei F, Pang J-S (2003) Finite-dimensional variational inequalities and complementarity problems Vol I, II. Springer, Berlin Heidelberg New York
Faraut U, Korányi A (1994) Analysis on symmetric cones Oxford Mathematical Monographs. Oxford University Press, New York
Fischer A (1992) A special Newton-type optimization methods. Optimization 24:269–284
Fischer A (1997) Solution of the monotone complementarity problem with locally Lipschitzian functions. Math Program 76:513–532
Fukushima M, Luo Z-Q, Tseng P (2002) Smoothing functions for second-order cone complementarity problems. SIAM J Optim 12:436–460
Goes RMB, Oliveira PR (2002) A new class of merit functions for the semidefinite complementarity problem. Annals of the Brazilian Workshop on Continuous Optimization, pp 1–18
Hayashi S, Yamashita N, Fukushima M (2002) On the coerciveness of merit functions for the second-order cone complementarity problem. Technical Report, Department of Applied Mathematics and Physics, Kyoto University
Hayashi S, Yamashita N, Fukushima M (2005) A combined smoothing and regularization method for monotone second-order cone complementarity problems. SIAM J Optim 15:593–615
Korányi A (1984) Monotone functions on formally real Jordan algebras. Mathematische Annalen 269:73–76
Liu Y-J, Zhang Z-W, Wang Y-H (2005) Some properties of a class of merit functions for symmetric cone complementarity problems. Asia-Pacific J Oper Res (to appear)
Lobo MS, Vandenberghe L, Boyd S, Lebret H (1998) Application of second-order cone programming. Linear Algebra Appl 284:193–228
Luo Z-Q, Tseng P (1997) A new class of merit functions for the nonlinear complementarity problem. In: Ferris MC, Pang J-S (eds) Complementarity and variational problems: state of the art. SIAM, Philadelphia, pp 204–225
Mittelmann HD (2003) An independent benchmarking of SDP and SOCP solvers. Math Program 95:407–430
Monteiro RDC, Tsuchiya T (2000) Polynomial convergence of primal-dual algorithms for the second-order cone programs based on the MZ-family of directions. Math Program 88:61–83
Rockafellar RT (1970) Convex analysis. Princeton Mathematical Series, Princeton
Schmieta S, Alizadeh F (2001) Associative and Jordan algebras, and polynomial time interior-point algorithms for symmetric cones. Math Oper Res 26:543–564
Sim C-K, Zhao G (2005) A note on treating second order cone problem as a special case of semidefinite problem. Math Program 102:609–613
Tao J, Gowda MS (2004) Some P-properties for the nonlinear transformations on Euclidean Jordan Algebra. Technical Report, Department of Mathematics and Statistics, University of Maryland
Tseng P (1998) Merit function for semidefinite complementarity problems. Math Program 83:159–185
Tsuchiya T (1999) A convergence analysis of the scaling-invariant primal-dual path-following algorithms for second-order cone programming. Optim Meth Softw 11:141–182
Author information
Authors and Affiliations
Corresponding author
Additional information
Member of Mathematics Division, National Center for Theoretical Sciences, Taipei Office. The author’s work is partially supported by National Science Council of Taiwan.
Rights and permissions
About this article
Cite this article
Chen, JS. Two classes of merit functions for the second-order cone complementarity problem. Math Meth Oper Res 64, 495–519 (2006). https://doi.org/10.1007/s00186-006-0098-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00186-006-0098-9