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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 |
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.
References:
[1] |
A. Ben-Tal and A. Nemirovski,
Robust convex optimization methodology and applications, Mathematical Programming, 92 (2002), 453-480.
doi: 10.1007/s101070100286. |
[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. |
[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. |
[4] |
F. Clarke, Optimization and Nonsmooth Analysis, John Wiley and Sons, New York, 1983.
![]() |
[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. |
[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. |
[7] |
J. Faraut and A. Korányi, Analysis on Symmetric Cones, Oxford University Press, New York, 1994.
![]() |
[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. |
[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. |
[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. |
[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. |
[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. |
[13] |
Z. Luo, J. Pang and D. Ralph, Mathematical Programs with Equilibrium Constraints, Cambridge University Press, Cambridge, United Kingdom, 1996.
doi: 10.1017/CBO9780511983658.![]() ![]() |
[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. |
[15] |
R. Rockafellar and R. Wets, Variational Analysis, Springer-Verlag, New York, 1998.
doi: 10.1007/978-3-642-02431-3.![]() ![]() |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[4] |
F. Clarke, Optimization and Nonsmooth Analysis, John Wiley and Sons, New York, 1983.
![]() |
[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. |
[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. |
[7] |
J. Faraut and A. Korányi, Analysis on Symmetric Cones, Oxford University Press, New York, 1994.
![]() |
[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. |
[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. |
[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. |
[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. |
[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. |
[13] |
Z. Luo, J. Pang and D. Ralph, Mathematical Programs with Equilibrium Constraints, Cambridge University Press, Cambridge, United Kingdom, 1996.
doi: 10.1017/CBO9780511983658.![]() ![]() |
[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. |
[15] |
R. Rockafellar and R. Wets, Variational Analysis, Springer-Verlag, New York, 1998.
doi: 10.1007/978-3-642-02431-3.![]() ![]() |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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