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
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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.
Luo, Z.Q., Pang, J.S., Ralph, D.: Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge (1996)CrossRefzbMATHGoogle Scholar
Outrata, J., Kocvara, M., Zowe, J.: Nonsmooth Approach to Optimization Problems with Equilibrium Constraints: Theory, Applications and Numerical Results. Springer, Berlin (2013)zbMATHGoogle Scholar
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
Scheel, H., Scholtes, S.: Mathematical programs with complementarity constraints: stationarity, optimality and sensitivity. Math. Oper. Res. 25(1), 1–22 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
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
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
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