Abstract
This paper considers the mathematical program with second-order cone complementarity constrains (MPSOCC). As a generalization of the developed mathematical program with complementarity constrains (MPCC), MPSOCC has many applications in practice. Motivated by the MPCC theory, several stationarity concepts, which include the Clarke-type, Mordukhovich-type, and strong stationarities, are presented in this paper. It is further shown that a local minimizer of MPSOCC must be stationary in some sense under suitable conditions. This indicates that these stationarity concepts are reasonable in theory.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alizadeh, F., Goldfarb, D.: Second-order cone programming. Math. Program. 95, 3–51 (2003)
Bonnans, F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)
Chen, J.-S., Chen, X., Tseng, P.: Analysis of nonsmooth vector-valued functions associated with second-order cones. Math. Program. 101, 95–117 (2004)
Chen, J.-S., Pan, S.: A survey on SOC complementarity functions and solution methods for SOCPs and SOCCPs. Pacific Journal of Optimization 8, 33–74 (2012)
Chen, X.D., Sun, D., Sun, J.: Complementarity functions and numerical experiments on some smoothing Newton methods for second-order-cone complementarity problems. Comput. Optim. Appl. 25, 39–56 (2003)
Chen, Y., Florian, M.: The nonlinear bilevel programming problem: Formulations, regularity and optimality conditions. Optimization 32, 193–209 (1995)
Clarke, F.H.: A new approach to Lagrange multipliers. Math. Oper. Res. 1, 165–174 (1976)
Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)
Faraut, U., Koranyi, A.: Analysis on Symmetric Cones. Oxford Mathematical Monographs, Oxford University Press (1994)
Flegel, M.L., Kanzow, C.: On M-stationary points for mathematical programs with equilibrium constraints. J. Math. Anal. Appl. 310, 286–302 (2005)
Fukuda, E.H., Silva, P.J.S., Fukushima, M.: Differentiable exact penalty functions for nonlinear second-order cone programs. SIAM J. Optim. 22, 1607–1633 (2012)
Fukushima, M., Lin, G.-H.: Smoothing methods for mathematical programs with equilibrium constraints. In: Proceedings of the ICKS’04, IEEE Computer Society, pp. 206–213 (2004)
Fukushima, M., Luo, Z.-Q., Tseng, P.: Smoothing functions for second-order cone complementarity problems. SIAM J. Optim. 12, 436–460 (2001)
Hayashi, S., Yamashita, N., Fukushima, M.: A combined smoothing and regularization method for monotone second-order cone complementarity problems. SIAM J. Optim. 15, 593–615 (2005)
Luo, Z.-Q., Pang, J.-S., Ralph, D.: Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge (1996)
Mordukhovich, B.S.: Generalized differential calculus for nonsmooth and set-valued mappings. J. Math. Anal. Appl. 183, 250–288 (1994)
Outrata, J.V., Kocvara, M., Zowe, J.: Nonsmooth approach to Optimization Problems with Equlilibrium Constraints: Theory, Applications, and Numerical Results. Kluwer Academic Publisher, Dordrect (1998)
Outrata, J.V., Ramirez, H.: On the Aubin property of critical points to perturbed second-order cone programs. SIAM J. Optim. 21, 798–823 (2011)
Outrata, J.V., Sun, D.F.: On the coderivative of the projection operator onto the second order cone. Set-Valued Analysis 16, 999–1014 (2008)
Pang, J.-S., Sun, D., Sun, J.: Semismooth homeomorphisms and strong stability of semidefinite and Lorentz cone complementarity problems. Math. Oper. Res. 28, 39–63 (2003)
Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer-Verlag, Berlin (1998)
Treiman, J.S.: Lagrange multipliers for nonconvex generalized gradients with equality, inequality, and set constraints. SIAM J. Control. Optim. 37, 1313–1329 (1999)
Yamamura, H., Okuno, T., Hayashi, S., Fukushima, M.: A smoothing SQP method for mathematical programs with linear second-order cone complementarity constraints. Pac. J. Optim. 9, 345–372 (2013)
Yan, T., Fukushima, M.: Smoothing method for mathematical programs with symmetric cone complementarity. Optimization 60, 113–128 (2011)
Ye, J.J.: Constraint qualifications and necessary optimality conditions for optimization progblems with variational inequality constraints. SIAM J. Optim. 10, 943–962 (2000)
Ye, J.J., Zhu, D.L., Zhu, Q.J.: Exact penalization and neccessary 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 Anal. 19, 609–646 (2011)
Author information
Authors and Affiliations
Corresponding author
Additional information
Yan-Chao Liang, Xi-De Zhu and Gui-Hua Lin are grateful to the anonymous referee for his/her helpful suggestions and comments.
This work was supported in part by NSFC Grant #11071028.
Rights and permissions
About this article
Cite this article
Liang, YC., Zhu, XD. & Lin, GH. Necessary Optimality Conditions for Mathematical Programs with Second-Order Cone Complementarity Constraints. Set-Valued Var. Anal 22, 59–78 (2014). https://doi.org/10.1007/s11228-013-0250-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11228-013-0250-7