Mathematical Programs with Second-Order Cone Complementarity Constraints: Strong Stationarity and Approximation Method

  • Xide Zhu
  • Jin ZhangEmail author
  • Jinchuan Zhou
  • Xinmin Yang


The existence of complementarity constraints causes the difficulties for studying mathematical programs with second-order cone complementarity constraints, since the standard constraint qualification, such as Robinson’s constraint qualification, is invalid. Therefore, various stationary conditions including strong, Mordukhovich and Clarke stationary conditions have been proposed, according to different reformulations of the second-order cone complementarity constraints. In this paper, we present a new reformulation of this problem by taking into consideration the Jordan algebra associated with the second-order cone. It ensures that the classical Karush–Kuhn–Tucker condition coincides with the strong stationary condition of the original problem. Furthermore, we propose a class of approximation methods to solve mathematical programs with second-order cone complementarity constraints. Any accumulation point of the iterative sequences, generated by the approximation method, is Clarke stationary under the corresponding linear independence constraint qualification. This stationarity can be enhanced to strong stationarity with an extra strict complementarity condition. Preliminary numerical experiments indicate that the proposed method is effective.


Mathematical programs with second-order cone complementarity constraints Stationarity conditions Jordan product Calmness conditions Approximation methods 

Mathematics Subject Classification

90C30 90C33 90C46 



This work was supported in part by National Natural Science Foundation of China (11771255, 11601458, 11431004, 11801325), Chongqing Natural Science Foundation (cstc2018jcyj-yszxX0009) and Shandong Province Natural Science Foundation (ZR2016AM07). The authors are grateful to the anonymous reviewer and the editor for their helpful comments and suggestions.


  1. 1.
    Luo, Z.Q., Pang, J.S., Ralph, D.: Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge (1996)CrossRefzbMATHGoogle Scholar
  2. 2.
    Outrata, J., Kocvara, M., Zowe, J.: Nonsmooth Approach to Optimization Problems with Equilibrium Constraints: Theory, Applications and Numerical Results. Springer, Berlin (2013)zbMATHGoogle Scholar
  3. 3.
    Ye, J.J., Zhu, D.L., Zhu, Q.J.: Exact penalization and necessary optimality conditions for generalized bilevel programming problems. SIAM J. Optim. 7(2), 481–507 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Scheel, H., Scholtes, S.: Mathematical programs with complementarity constraints: stationarity, optimality and sensitivity. Math. Oper. Res. 25(1), 1–22 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Ye, J.J.: Necessary and sufficient optimality conditions for mathematical programs with equilibrium constraints. J. Math. Anal. Appl. 307(1), 350–369 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Alizadeh, F., Goldfarb, D.: Second-order cone programming. Math. Program. 95, 3–51 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Fukushima, M., Luo, Z.Q., Tseng, P.: Smoothing functions for second-order-cone complementarity problems. SIAM J. Optim. 12(2), 436–460 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Ye, J.J., Zhou, J.C.: Exact formula for the proximal/regular/limiting normal cone of the second-order cone complementarity set. Math. Program. 162, 33–50 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Ye, J.J., Zhou, J.C.: First-order optimality conditions for mathematical programs with second-order cone complementarity constraints. SIAM J. Optim. 26(4), 2820–2846 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Liang, Y.C., Zhu, X.D., Lin, G.H.: Necessary optimality conditions for mathematical programs with second-order cone complementarity constraints. Set-Valued Var. Anal. 22(1), 59–78 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Outrata, J.V., Sun, D.: On the coderivative of the projection operator onto the second-order cone. Set-Valued Anal. 16(7–8), 999–1014 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Wu, J., Zhang, L., Zhang, Y.: A smoothing Newton method for mathematical programs governed by second-order cone constrained generalized equations. J. Glob. Optim. 55(2), 359–385 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    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)MathSciNetzbMATHGoogle Scholar
  14. 14.
    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(4), 609 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Jiang, Y., Liu, Y.J., Zhang, L.W.: Variational geometry of the complementarity set for second-order cone. Set-Valued Var. Anal. 23, 399–414 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Ye, J.J., Zhou, J.C.: Verifiable sufficient conditions for the error bound property of second-order cone complementarity problems. Math. Program. 171, 361–395 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)CrossRefzbMATHGoogle Scholar
  18. 18.
    Ding, C., Sun, D., Ye, J.J.: First order optimality conditions for mathematical programs with semidefinite cone complementarity constraints. Math. Program. 147(1–2), 539–579 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Gfrerer, H.: First order and second order characterizations of metric subregularity and calmness of constraint set mappings. SIAM J. Optim. 21, 1439–1474 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Gfrerer, H., Klatte, D.: Lipschitz and Hölder stability of optimization problems and generalized equations. Math. Program. 158, 35–75 (2016)MathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.School of ManagementShanghai UniversityShanghaiChina
  2. 2.Faculty of Business AdministrationYokohama National UniversityYokohamaJapan
  3. 3.Department of MathematicsSouthern University of Science and TechnologyShenzhenChina
  4. 4.Department of Statistics, School of Mathematics and StatisticsShandong University of TechnologyZiboChina
  5. 5.College of Mathematics ScienceChongqing Normal UniversityChongqingChina

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