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July  2018, 14(3): 981-1005. doi: 10.3934/jimo.2017086

## On the convergence properties of a smoothing approach for mathematical programs with symmetric cone complementarity constraints

 1 School of Science, East China University of Science and Technology, Shanghai 200237, China 2 Institute of ORCT, School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China

Received  April 2016 Revised  December 2016 Published  September 2017

Fund Project: This study is supported by the National Natural Science Foundation of China under projects No.11401210, No.11671183, No.11571059, No.91330206 and No.11301049.

This paper focuses on a class of mathematical programs with symmetric cone complementarity constraints (SCMPCC). The explicit expression of C-stationary condition and SCMPCC-linear independence constraint qualification (denoted by SCMPCC-LICQ) for SCMPCC are first presented. We analyze a parametric smoothing approach for solving this program in which SCMPCC is replaced by a smoothing problem $P_{\varepsilon}$ depending on a (small) parameter $\varepsilon$. We are interested in the convergence behavior of the feasible set, stationary points, solution mapping and optimal value function of problem $P_{\varepsilon}$ when $\varepsilon \to 0$ under SCMPCC-LICQ. In particular, it is shown that the convergence rate of Hausdorff distance between feasible sets $\mathcal{F}_{\varepsilon}$ and $\mathcal{F}$ is of order $\mbox{O}(|\varepsilon|)$ and the solution mapping and optimal value of $P_{\varepsilon}$ are outer semicontinuous and locally Lipschitz continuous at $\varepsilon=0$ respectively. Moreover, any accumulation point of stationary points of $P_{\varepsilon}$ is a C-stationary point of SCMPCC under SCMPCC-LICQ.

Citation: Yi Zhang, Liwei Zhang, Jia Wu. On the convergence properties of a smoothing approach for mathematical programs with symmetric cone complementarity constraints. Journal of Industrial & Management Optimization, 2018, 14 (3) : 981-1005. doi: 10.3934/jimo.2017086
##### References:
 [1] A. Ben-Tal and A. Nemirovski, Robust convex optimization methodology and applications, Mathematical Programming, 92 (2002), 453-480.  doi: 10.1007/s101070100286.  Google Scholar [2] G. Bouza and G. Still, Mathematical programs with complementarity constraints: Convergence properties of a smoothing method, Mathematics of Operations Research, 32 (2007), 467-483.  doi: 10.1287/moor.1060.0245.  Google Scholar [3] X. Chen and M. Fukushima, A smoothing method for a mathematical program with P-matrix linear complementarity constraints, Computational Optimization and Applications, 27 (2004), 223-246.  doi: 10.1023/B:COAP.0000013057.54647.6d.  Google Scholar [4] F. Clarke, Optimization and Nonsmooth Analysis, John Wiley and Sons, New York, 1983.   Google Scholar [5] C. Ding, D. Sun and J. Ye, First order optimality conditions for mathematical programs with semidefinite cone complementarity constraints, Mathematical Programming, Ser.A, 147 (2014), 539-579.  doi: 10.1007/s10107-013-0735-z.  Google Scholar [6] F. Facchinei, H. Jiang and L. Qi, A smoothing method for mathematical programs with equilibrium constraints, Mathematical Programming, 85 (1999), 107-134.  doi: 10.1007/s10107990015a.  Google Scholar [7] J. Faraut and A. Korányi, Analysis on Symmetric Cones, Oxford University Press, New York, 1994.   Google Scholar [8] L. Faybusovich, Linear systems in Jordan algebras and primal-dual interior-point algorithm, Journal of Computational and Applied Mathematics, 86 (1997), 149-175.  doi: 10.1016/S0377-0427(97)00153-2.  Google Scholar [9] M. Fukushima and J. Pang, Convergence of a smoothing continuation method for mathematical problems with complementarity constraints, Lecture Notes in Economics and Mathematical Systems, 477 (1999), 99-110.  doi: 10.1007/978-3-642-45780-7_7.  Google Scholar [10] M. Gowda, R. Sznajder and J. Tao, Some P-properties for linear transformations on Euclidean Jordan algebras, Linear algebra and its applications, 393 (2004), 203-232.  doi: 10.1016/j.laa.2004.03.028.  Google Scholar [11] K. Koecher, The Minnesota Notes on Jordan Algebras and Their Applications, edited and annotated by A. Brieg and S. Walcher, Springer, Berlin, 1999. doi: 10.1007/BFb0096285.  Google Scholar [12] G. Lin and M. Fukushima, A modified relaxation scheme for mathematical prgrams with complementarity constraints, Annals of Operations Research, 133 (2005), 63-84.  doi: 10.1007/s10479-004-5024-z.  Google Scholar [13] Z. Luo, J. Pang and D. Ralph, Mathematical Programs with Equilibrium Constraints, Cambridge University Press, Cambridge, United Kingdom, 1996.  doi: 10.1017/CBO9780511983658.  Google Scholar [14] J. Outrata, M. Ko$\breve{c}$vara and J. Zowe, Nonsmooth Approach to Optimization Problems with Equilibrium Constraints, Kluwer Academic Publishers, Boston, MA, 1998. doi: 10.1007/978-1-4757-2825-5.  Google Scholar [15] R. Rockafellar and R. Wets, Variational Analysis, Springer-Verlag, New York, 1998.  doi: 10.1007/978-3-642-02431-3.  Google Scholar [16] S. Scheel and S. Scholtes, Mathematical programs with complementarity constraints: Stationarity, optimality, and sensitivity, Mathematics of Operation Research, 25 (2000), 1-22.  doi: 10.1287/moor.25.1.1.15213.  Google Scholar [17] S. Scholtes, Convergence properties of a regularization scheme for mathematical programs with complementarity constraints, SIAM Journal on Optimization, 11 (2001), 918-936.  doi: 10.1137/S1052623499361233.  Google Scholar [18] D. Sun and J. Sun, Löwner's operator and spectral functions on Euclidean Jordan algebras, Mathematics of Operation Research, 33 (2008), 421-445.  doi: 10.1287/moor.1070.0300.  Google Scholar [19] D. Sun, J. Sun and L. Zhang, The rate of convergence of the augmented Lagrangian method for nonlinear semidefinite programming, Mathematical Programming, 114 (2008), 349-391.  doi: 10.1007/s10107-007-0105-9.  Google Scholar [20] E. Takeshi, A Smoothing Method for Mathematical Programs with Second-Order Cone Complementarity Constraints, Master thesis, Kyoto University in Kyoto, 2007. Google Scholar [21] Y. Wang, Perturbation Analysis of Optimimization Problems over Symmetric Cones, Ph. D. Thesis, Dalian University of Technology, China, 2008. Google Scholar [22] T. Yan and M. Fukushima, Smoothing method for mathematical programs with symmetric cone complementarity constraints, Optimization, 60 (2011), 113-128.  doi: 10.1080/02331934.2010.541458.  Google Scholar [23] Y. Zhang, J. Wu and L. Zhang, First order necessary optimality conditions for mathematical programs with second-order cone complementarity constraint, Journal of Global Optimization, 63 (2015), 253-279.  doi: 10.1007/s10898-015-0295-2.  Google Scholar [24] Y. Zhang, L. Zhang and J. Wu, Convergence properties of a smoothing approach for mathematical programs with second-order cone complementarity constraints, Set-Valued and Variational Analysis, 19 (2011), 609-646.  doi: 10.1007/s11228-011-0190-z.  Google Scholar

show all references

##### References:
 [1] A. Ben-Tal and A. Nemirovski, Robust convex optimization methodology and applications, Mathematical Programming, 92 (2002), 453-480.  doi: 10.1007/s101070100286.  Google Scholar [2] G. Bouza and G. Still, Mathematical programs with complementarity constraints: Convergence properties of a smoothing method, Mathematics of Operations Research, 32 (2007), 467-483.  doi: 10.1287/moor.1060.0245.  Google Scholar [3] X. Chen and M. Fukushima, A smoothing method for a mathematical program with P-matrix linear complementarity constraints, Computational Optimization and Applications, 27 (2004), 223-246.  doi: 10.1023/B:COAP.0000013057.54647.6d.  Google Scholar [4] F. Clarke, Optimization and Nonsmooth Analysis, John Wiley and Sons, New York, 1983.   Google Scholar [5] C. Ding, D. Sun and J. Ye, First order optimality conditions for mathematical programs with semidefinite cone complementarity constraints, Mathematical Programming, Ser.A, 147 (2014), 539-579.  doi: 10.1007/s10107-013-0735-z.  Google Scholar [6] F. Facchinei, H. Jiang and L. Qi, A smoothing method for mathematical programs with equilibrium constraints, Mathematical Programming, 85 (1999), 107-134.  doi: 10.1007/s10107990015a.  Google Scholar [7] J. Faraut and A. Korányi, Analysis on Symmetric Cones, Oxford University Press, New York, 1994.   Google Scholar [8] L. Faybusovich, Linear systems in Jordan algebras and primal-dual interior-point algorithm, Journal of Computational and Applied Mathematics, 86 (1997), 149-175.  doi: 10.1016/S0377-0427(97)00153-2.  Google Scholar [9] M. Fukushima and J. Pang, Convergence of a smoothing continuation method for mathematical problems with complementarity constraints, Lecture Notes in Economics and Mathematical Systems, 477 (1999), 99-110.  doi: 10.1007/978-3-642-45780-7_7.  Google Scholar [10] M. Gowda, R. Sznajder and J. Tao, Some P-properties for linear transformations on Euclidean Jordan algebras, Linear algebra and its applications, 393 (2004), 203-232.  doi: 10.1016/j.laa.2004.03.028.  Google Scholar [11] K. Koecher, The Minnesota Notes on Jordan Algebras and Their Applications, edited and annotated by A. Brieg and S. Walcher, Springer, Berlin, 1999. doi: 10.1007/BFb0096285.  Google Scholar [12] G. Lin and M. Fukushima, A modified relaxation scheme for mathematical prgrams with complementarity constraints, Annals of Operations Research, 133 (2005), 63-84.  doi: 10.1007/s10479-004-5024-z.  Google Scholar [13] Z. Luo, J. Pang and D. Ralph, Mathematical Programs with Equilibrium Constraints, Cambridge University Press, Cambridge, United Kingdom, 1996.  doi: 10.1017/CBO9780511983658.  Google Scholar [14] J. Outrata, M. Ko$\breve{c}$vara and J. Zowe, Nonsmooth Approach to Optimization Problems with Equilibrium Constraints, Kluwer Academic Publishers, Boston, MA, 1998. doi: 10.1007/978-1-4757-2825-5.  Google Scholar [15] R. Rockafellar and R. Wets, Variational Analysis, Springer-Verlag, New York, 1998.  doi: 10.1007/978-3-642-02431-3.  Google Scholar [16] S. Scheel and S. Scholtes, Mathematical programs with complementarity constraints: Stationarity, optimality, and sensitivity, Mathematics of Operation Research, 25 (2000), 1-22.  doi: 10.1287/moor.25.1.1.15213.  Google Scholar [17] S. Scholtes, Convergence properties of a regularization scheme for mathematical programs with complementarity constraints, SIAM Journal on Optimization, 11 (2001), 918-936.  doi: 10.1137/S1052623499361233.  Google Scholar [18] D. Sun and J. Sun, Löwner's operator and spectral functions on Euclidean Jordan algebras, Mathematics of Operation Research, 33 (2008), 421-445.  doi: 10.1287/moor.1070.0300.  Google Scholar [19] D. Sun, J. Sun and L. Zhang, The rate of convergence of the augmented Lagrangian method for nonlinear semidefinite programming, Mathematical Programming, 114 (2008), 349-391.  doi: 10.1007/s10107-007-0105-9.  Google Scholar [20] E. Takeshi, A Smoothing Method for Mathematical Programs with Second-Order Cone Complementarity Constraints, Master thesis, Kyoto University in Kyoto, 2007. Google Scholar [21] Y. Wang, Perturbation Analysis of Optimimization Problems over Symmetric Cones, Ph. D. Thesis, Dalian University of Technology, China, 2008. Google Scholar [22] T. Yan and M. Fukushima, Smoothing method for mathematical programs with symmetric cone complementarity constraints, Optimization, 60 (2011), 113-128.  doi: 10.1080/02331934.2010.541458.  Google Scholar [23] Y. Zhang, J. Wu and L. Zhang, First order necessary optimality conditions for mathematical programs with second-order cone complementarity constraint, Journal of Global Optimization, 63 (2015), 253-279.  doi: 10.1007/s10898-015-0295-2.  Google Scholar [24] Y. Zhang, L. Zhang and J. Wu, Convergence properties of a smoothing approach for mathematical programs with second-order cone complementarity constraints, Set-Valued and Variational Analysis, 19 (2011), 609-646.  doi: 10.1007/s11228-011-0190-z.  Google Scholar
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