American Institute of Mathematical Sciences

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July  2013, 9(3): 561-577. doi: 10.3934/jimo.2013.9.561

A log-exponential regularization method for a mathematical program with general vertical complementarity constraints

Jie Zhang 1, , Shuang Lin 2, and  Li-Wei Zhang 2,

1. 

School of Mathematics, Liaoning Normal University, Dalian, 116029, China
2. 

School of information Science and Engineering, Dalian Polytechnic University, Dalian, 116029, China, China

Received  March 2012 Revised  March 2013 Published  April 2013

Based on the log-exponential function, a regularization method is proposed for solving a mathematical program with general vertical complementarity constraints (MPVCC) considered by Scheel and Scholtes (Math. Oper. Res. 25: 1-22, 2000). With some known smoothing properties of the log-exponential function, a difficult MPVCC is reformulated as a smooth nonlinear programming problem, which becomes solvable by using available nonlinear optimization software. Detailed convergence analysis of this method is investigated and the results obtained generalize conclusions in Yin and Zhang (Math. Meth. Oper. Res. 64: 255-269, 2006). An example of Stackelberg game is illustrated to show the application of this method.
Keywords: Stackelberg game., mathematical program with general vertical complementarity constraints, Log-exponential regularization.
Mathematics Subject Classification: Primary: 90C30; Secondary: 90C46, 90C4.
Citation: Jie Zhang, Shuang Lin, Li-Wei Zhang. A log-exponential regularization method for a mathematical program with general vertical complementarity constraints. Journal of Industrial & Management Optimization, 2013, 9 (3) : 561-577. doi: 10.3934/jimo.2013.9.561
References:
[1]

S. I. Birbil, G. Gürkan and O. Listes, Solving stochastic mathematical programs with complementarity constraints using simulation,, Math. Oper. Res., 31 (2006), 739.  doi: 10.1287/moor.1060.0215.  Google Scholar

[2]

R. W. Cottle and G. B. Dantzig, A generalization of the linear complementarity problem,, J. Combinatorial Theory, 8 (1970), 79.  doi: 10.1016/S0021-9800(70)80010-2.  Google Scholar

[3]

F. H. Clarke, "Optimization and Nonsmooth Analysis,", John Wiley and Sons, (1983).   Google Scholar

[4]

M. Fukushima and J. S. Pang, Convergence of a smoothing continuation method for mathematical programs with complementarity constraints,, in, (1999), 99.  doi: 10.1007/978-3-642-45780-7_7.  Google Scholar

[5]

A. Fischer, A special Newton-type optimization method,, Optimization, 24 (1992), 269.  doi: 10.1080/02331939208843795.  Google Scholar

[6]

M. S. Gowda and R. Sznajder, A generalization of the Nash equilibrium theorem on bimatrix games,, International Journal of Game Theory, 25 (1996), 1.  doi: 10.1007/BF01254380.  Google Scholar

[7]

Z. Huang and J. Sun, A smoothing Newton algorithm for mathematical programs with complementarity constraints,, Journal of Industrial and Management Optimization, 1 (2005), 153.  doi: 10.3934/jimo.2005.1.153.  Google Scholar

[8]

X. Liu and J. Sun, Generalized stationary points and an interior point method for mathematical programs with equilibrium constraints,, Mathematical Programming, 101 (2004), 231.  doi: 10.1007/s10107-004-0543-6.  Google Scholar

[9]

X. Liu, G. Perakis and J. Sun, A robust SQP method for mathematical programs with linear complementarity constraints,, Computational Optimization and Applications, 34 (2006), 5.  doi: 10.1007/s10589-005-3075-y.  Google Scholar

[10]

Z. Q. Luo, J. S. Pang and D. Ralph, "Mathematical Programs with Equilibrium Constraints,", Cambridge University Press, (1996).  doi: 10.1017/CBO9780511983658.  Google Scholar

[11]

M. Kočvara and J. V. Outrata, Optimization problems with equilibrium constraints and their numerical solution,, Math. Program., 101 (2004), 119.  doi: 10.1007/s10107-004-0539-2.  Google Scholar

[12]

J. Peng and Z. Lin, A non-interior continuation method for generalized linear complementarity problems,, Mathematical Programming, 86 (1999), 533.  doi: 10.1007/s101070050104.  Google Scholar

[13]

H. Qi and L. Liao, A smoothing Newton method for extended vertical linear complementarity problems,, SIAM Joural of Matrix Analysis and Applications, 21 (1999), 45.  doi: 10.1137/S0895479897329837.  Google Scholar

[14]

R. T. Rockafellar and R. J. B. Wets, "Variational Analysis,", Berlin Heidelberg, (1998).  doi: 10.1007/978-3-642-02431-3.  Google Scholar

[15]

R. T. Rockafellar, "Convex Analysis,", Princeton University Press, (1970).   Google Scholar

[16]

H. Scheel and S. Scholtes, Mathematical programs with complementarity constraints: Stability, optimality, and sensitivity,, Math. Oper. Res., 25 (2000), 1.  doi: 10.1287/moor.25.1.1.15213.  Google Scholar

[17]

H. V. Stackelberg, "The Theory of Market Economy,", Oxford University Press, (1952).   Google Scholar

[18]

H. Yin and J. Zhang, Global convergence of a smooth approximation method for mathematical programs with complementarity constraints,, Math. Meth. Oper. Res., 64 (2006), 255.  doi: 10.1007/s00186-006-0076-2.  Google Scholar

[19]

N. D. Yen, Stability of the solution set of perturbed nonsmooth inequality systems and application,, Journal of Optimization Theory and Applications, 93 (1997), 199.  doi: 10.1023/A:1022662120550.  Google Scholar

[20]

J. Zhang, L. Zhang and S. Lin, A class of smoothing SAA methods for a stochastic mathematical program with complementarity constraints,, J. Math. Anal. Appl., 387 (2012), 201.  doi: 10.1016/j.jmaa.2011.08.073.  Google Scholar

[21]

J. Zhang, L. Zhang and W. Wang, On constraint qualifications in terms of approximate Jacobians for nonsmooth continuous optimization problems,, Nonlinear Analysis, 75 (2012), 2566.  doi: 10.1016/j.na.2011.11.003.  Google Scholar

show all references

References:
[1]

S. I. Birbil, G. Gürkan and O. Listes, Solving stochastic mathematical programs with complementarity constraints using simulation,, Math. Oper. Res., 31 (2006), 739.  doi: 10.1287/moor.1060.0215.  Google Scholar

[2]

R. W. Cottle and G. B. Dantzig, A generalization of the linear complementarity problem,, J. Combinatorial Theory, 8 (1970), 79.  doi: 10.1016/S0021-9800(70)80010-2.  Google Scholar

[3]

F. H. Clarke, "Optimization and Nonsmooth Analysis,", John Wiley and Sons, (1983).   Google Scholar

[4]

M. Fukushima and J. S. Pang, Convergence of a smoothing continuation method for mathematical programs with complementarity constraints,, in, (1999), 99.  doi: 10.1007/978-3-642-45780-7_7.  Google Scholar

[5]

A. Fischer, A special Newton-type optimization method,, Optimization, 24 (1992), 269.  doi: 10.1080/02331939208843795.  Google Scholar

[6]

M. S. Gowda and R. Sznajder, A generalization of the Nash equilibrium theorem on bimatrix games,, International Journal of Game Theory, 25 (1996), 1.  doi: 10.1007/BF01254380.  Google Scholar

[7]

Z. Huang and J. Sun, A smoothing Newton algorithm for mathematical programs with complementarity constraints,, Journal of Industrial and Management Optimization, 1 (2005), 153.  doi: 10.3934/jimo.2005.1.153.  Google Scholar

[8]

X. Liu and J. Sun, Generalized stationary points and an interior point method for mathematical programs with equilibrium constraints,, Mathematical Programming, 101 (2004), 231.  doi: 10.1007/s10107-004-0543-6.  Google Scholar

[9]

X. Liu, G. Perakis and J. Sun, A robust SQP method for mathematical programs with linear complementarity constraints,, Computational Optimization and Applications, 34 (2006), 5.  doi: 10.1007/s10589-005-3075-y.  Google Scholar

[10]

Z. Q. Luo, J. S. Pang and D. Ralph, "Mathematical Programs with Equilibrium Constraints,", Cambridge University Press, (1996).  doi: 10.1017/CBO9780511983658.  Google Scholar

[11]

M. Kočvara and J. V. Outrata, Optimization problems with equilibrium constraints and their numerical solution,, Math. Program., 101 (2004), 119.  doi: 10.1007/s10107-004-0539-2.  Google Scholar

[12]

J. Peng and Z. Lin, A non-interior continuation method for generalized linear complementarity problems,, Mathematical Programming, 86 (1999), 533.  doi: 10.1007/s101070050104.  Google Scholar

[13]

H. Qi and L. Liao, A smoothing Newton method for extended vertical linear complementarity problems,, SIAM Joural of Matrix Analysis and Applications, 21 (1999), 45.  doi: 10.1137/S0895479897329837.  Google Scholar

[14]

R. T. Rockafellar and R. J. B. Wets, "Variational Analysis,", Berlin Heidelberg, (1998).  doi: 10.1007/978-3-642-02431-3.  Google Scholar

[15]

R. T. Rockafellar, "Convex Analysis,", Princeton University Press, (1970).   Google Scholar

[16]

H. Scheel and S. Scholtes, Mathematical programs with complementarity constraints: Stability, optimality, and sensitivity,, Math. Oper. Res., 25 (2000), 1.  doi: 10.1287/moor.25.1.1.15213.  Google Scholar

[17]

H. V. Stackelberg, "The Theory of Market Economy,", Oxford University Press, (1952).   Google Scholar

[18]

H. Yin and J. Zhang, Global convergence of a smooth approximation method for mathematical programs with complementarity constraints,, Math. Meth. Oper. Res., 64 (2006), 255.  doi: 10.1007/s00186-006-0076-2.  Google Scholar

[19]

N. D. Yen, Stability of the solution set of perturbed nonsmooth inequality systems and application,, Journal of Optimization Theory and Applications, 93 (1997), 199.  doi: 10.1023/A:1022662120550.  Google Scholar

[20]

J. Zhang, L. Zhang and S. Lin, A class of smoothing SAA methods for a stochastic mathematical program with complementarity constraints,, J. Math. Anal. Appl., 387 (2012), 201.  doi: 10.1016/j.jmaa.2011.08.073.  Google Scholar

[21]

J. Zhang, L. Zhang and W. Wang, On constraint qualifications in terms of approximate Jacobians for nonsmooth continuous optimization problems,, Nonlinear Analysis, 75 (2012), 2566.  doi: 10.1016/j.na.2011.11.003.  Google Scholar

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