Abstract
This paper is to develop first order necessary optimality conditions for a mathematical program with second-order cone complementarity constraints (MPSCC) which includes the mathematical program with (vector) complementarity constraints (MPCC) as a special case. Like the case of MPCC, Robinson’s constraint qualification fails at every feasible point of MPSCC if we treat the MPSCC as an ordinary optimization problem. Using the formulas of regular and limiting coderivatives and generalized Clarke’s Jacobian of the projection operator onto second-order cones from the literature, we present the S-, M-, C- and A-stationary conditions for a MPSCC problem. Moreover, several constraint qualifications including MPSCC-Abadie CQ, MPSCC-LICQ, MPSCC-MFCQ and MPSCC-GMFCQ are proposed, under which a local minimizer of MPSCC is shown to be a S-, M-, C- or A-stationary point.
Similar content being viewed by others
References
Alizadeh, F., Goldfarb, D.: Second order cone programming. Math. Program. 95, 3–51 (2003)
Bonnas, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)
Dempe, S.: Foundation of Bilevel Programming. Kluwer Academic Publishers, Dordrecht (2002)
Ding, C., Sun, D., Ye, J.: First order optimality conditions for mathematical programs with semidefinite cone complementarity constraints. Math. Program. 147, 539–579 (2014)
Ejiri, T.: A smoothing method for mathematical programs with second order cone complementarity constraints. Master thesis, Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Kyoto, Japan (2007)
Flegel, M.L., Kanzow, C.: Abadie-type constraint qualification for mathematical programs with equilibrium constraints. J. Optim. Theory Appl. 124, 595–614 (2005)
Henrion, R., Outrata, J.V.: Calmness of constraint systems with applications. Math. Program. 104, 473–464 (2005)
Jiang, Y.: Optimization Problems with Second-order Cone Equilibrium Constraints. Ph.D thesis, Dalian University of Technology (2011)
Luo, Z.Q., Pang, J.S., Ralph, D.: Math. Prog. Equilib. Constr. Cambridge University Press, Cambridge (1996)
Mordukhovich, B.S.: Metric approximation and necessary optimality conditions for general classes of nonsmooth extremal problems. Soviet Math. Dokl. 22, 526–530 (1980)
Mordukhovich, B.S.: Lipschitzian stability of constraint systems and generalized equations. Nonlinear Anal. 22, 173–206 (1994)
Outrata, J.V., Koc̆vara, M., Zowe, J.: Nonsmooth Approach to Optimization Problems with Equilibrium Constraints. Kluwer Academic Publishers, Boston, MA (1998)
Outrata, J.V., Sun, D.F.: On the coderivative of the projection operator onto the second order cone. Set-Valued Anal 16, 999–1014 (2008)
Robinson, S.M.: First order conditions for general nonlinear optimization. SIAM J. Appl. Math. 30, 597–607 (1976)
Rockafellar, R.T., Wets, R.J.-B.: Var. Anal. Springer, New York (1998)
Scheel, S., Scholtes, S.: Mathematical programs with complementarity constraints: stationarity, optimality, and sensitivity. Math. Oper. Res. 25, 1–22 (2000)
Wu, J., Zhang, L., Zhang, Y.: A smoothing Newton method for mathematical programs governed by second-order cone constrained generalized equations. J. Global Optim. 55, 359–385 (2013)
Yan, T., Fukushima, M.: Smoothing method for mathematical programs with symmetric cone complementarity constraints. Optimization 60, 113–128 (2011)
Ye, J.J.: Necessary and sufficient optimality conditions for mathematical programs with equilibrium constraints. J. Math. Anal. Appl. 307, 350–369 (2005)
Ye, J.J., Zhu, D.L., Zhu, Q.J.: Exact penalization and necessary optimality conditions for generalized bilevel programming problems. SIAM J. Optim. 7, 481–507 (1997)
Zhang, Y., Zhang, L., Wu, J.: Convergence properties of a smoothing approach for mathematical programs with second-order cone complementarity constraints. Set-Valued Var. Anal. 19, 609–646 (2011)
Acknowledgments
The authors are grateful to the two referees for their valuable comments and suggestions on improving the quality of this paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported by the National Natural Science Foundation of China under Projects No. 11401210, No. 11301049, No. 91130007, No. 91330206 and the Fundamental Research Funds for the Central Universities under Project No. 222201314037.
Rights and permissions
About this article
Cite this article
Zhang, Y., Wu, J. & Zhang, L. First order necessary optimality conditions for mathematical programs with second-order cone complementarity constraints. J Glob Optim 63, 253–279 (2015). https://doi.org/10.1007/s10898-015-0295-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-015-0295-2