• Previous Article
    Uniqueness of solutions to fuzzy relational equations regarding Max-av composition and strong regularity of the matrices in Max-av algebra
  • JIMO Home
  • This Issue
  • Next Article
    Single-machine rescheduling problems with learning effect under disruptions
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. DingD. 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. FacchineiH. 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. GowdaR. 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. LuoJ. 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. SunJ. 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. ZhangJ. 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. ZhangL. 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. DingD. 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. FacchineiH. 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. GowdaR. 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. LuoJ. 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. SunJ. 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. ZhangJ. 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. ZhangL. 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

[1]

Li Chu, Bo Wang, Jie Zhang, Hong-Wei Zhang. Convergence analysis of a smoothing SAA method for a stochastic mathematical program with second-order cone complementarity constraints. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1863-1886. doi: 10.3934/jimo.2020050

[2]

Zehui Jia, Xue Gao, Xingju Cai, Deren Han. The convergence rate analysis of the symmetric ADMM for the nonconvex separable optimization problems. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1943-1971. doi: 10.3934/jimo.2020053

[3]

Haibo Cui, Haiyan Yin. Convergence rate of solutions toward stationary solutions to the isentropic micropolar fluid model in a half line. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2899-2920. doi: 10.3934/dcdsb.2020210

[4]

Reza Mazrooei-Sebdani, Zahra Yousefi. The coupled 1:2 resonance in a symmetric case and parametric amplification model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3737-3765. doi: 10.3934/dcdsb.2020255

[5]

Krzysztof A. Krakowski, Luís Machado, Fátima Silva Leite. A unifying approach for rolling symmetric spaces. Journal of Geometric Mechanics, 2021, 13 (1) : 145-166. doi: 10.3934/jgm.2020016

[6]

Xiaofei Liu, Yong Wang. Weakening convergence conditions of a potential reduction method for tensor complementarity problems. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021080

[7]

Yaonan Ma, Li-Zhi Liao. The Glowinski–Le Tallec splitting method revisited: A general convergence and convergence rate analysis. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1681-1711. doi: 10.3934/jimo.2020040

[8]

Xiaoni Chi, Zhongping Wan, Zijun Hao. A full-modified-Newton step $ O(n) $ infeasible interior-point method for the special weighted linear complementarity problem. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021082

[9]

Nikolaz Gourmelon. Generation of homoclinic tangencies by $C^1$-perturbations. Discrete & Continuous Dynamical Systems, 2010, 26 (1) : 1-42. doi: 10.3934/dcds.2010.26.1

[10]

Ugo Bessi. Another point of view on Kusuoka's measure. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3241-3271. doi: 10.3934/dcds.2020404

[11]

David Cantala, Juan Sebastián Pereyra. Endogenous budget constraints in the assignment game. Journal of Dynamics & Games, 2015, 2 (3&4) : 207-225. doi: 10.3934/jdg.2015002

[12]

Antonio Rieser. A topological approach to spectral clustering. Foundations of Data Science, 2021, 3 (1) : 49-66. doi: 10.3934/fods.2021005

[13]

Lorenzo Freddi. Optimal control of the transmission rate in compartmental epidemics. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021007

[14]

Arunima Bhattacharya, Micah Warren. $ C^{2, \alpha} $ estimates for solutions to almost Linear elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021024

[15]

Jie-Wen He, Chi-Chon Lei, Chen-Yang Shi, Seak-Weng Vong. An inexact alternating direction method of multipliers for a kind of nonlinear complementarity problems. Numerical Algebra, Control & Optimization, 2021, 11 (3) : 353-362. doi: 10.3934/naco.2020030

[16]

Amanda E. Diegel. A C0 interior penalty method for the Cahn-Hilliard equation. Electronic Research Archive, , () : -. doi: 10.3934/era.2021030

[17]

Baba Issa Camara, Houda Mokrani, Evans K. Afenya. Mathematical modeling of glioma therapy using oncolytic viruses. Mathematical Biosciences & Engineering, 2013, 10 (3) : 565-578. doi: 10.3934/mbe.2013.10.565

[18]

Alberto Bressan, Ke Han, Franco Rampazzo. On the control of non holonomic systems by active constraints. Discrete & Continuous Dynamical Systems, 2013, 33 (8) : 3329-3353. doi: 10.3934/dcds.2013.33.3329

[19]

Vieri Benci, Sunra Mosconi, Marco Squassina. Preface: Applications of mathematical analysis to problems in theoretical physics. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : i-i. doi: 10.3934/dcdss.2020446

[20]

Luigi Barletti, Giovanni Nastasi, Claudia Negulescu, Vittorio Romano. Mathematical modelling of charge transport in graphene heterojunctions. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021010

2019 Impact Factor: 1.366

Metrics

  • PDF downloads (159)
  • HTML views (1250)
  • Cited by (0)

Other articles
by authors

[Back to Top]