A New Exact Penalty Function for Solving Constrained Finite Min-Max Problems

MA Cheng; LI Xun; YIU Ka-Fai Cedric; ZHANG Lian-sheng

doi:10.3879/j.issn.1000-0887.2012.02.010

Volume 33 Issue 2

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MA Cheng, LI Xun, YIU Ka-Fai Cedric, ZHANG Lian-sheng. A New Exact Penalty Function for Solving Constrained Finite Min-Max Problems[J]. Applied Mathematics and Mechanics, 2012, 33(2): 250-264. doi: 10.3879/j.issn.1000-0887.2012.02.010

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A New Exact Penalty Function for Solving Constrained Finite Min-Max Problems

doi: 10.3879/j.issn.1000-0887.2012.02.010

¹Department of Applied Mathematics, Hong Kong Polytechnic University, Kowloon, Hong Kong, P.R.China;²Department of Mathematics, Shanghai University, Shanghai 200444, P.R.Chian

Received Date: 2011-03-31
Rev Recd Date: 2011-11-23
Publish Date: 2012-02-15

Abstract

Abstract

A new exact yet smooth penalty function to tackle constrained min-max problems was introduced. Using this new penalty function and adding just one extra variable, a constrained min-max problem was transformed into an unconstrained optimization one. It was proved that, under certain reasonable assumptions and when the penalty parameter was sufficiently large, the minimizer of this unconstrained optimization problem was equivalent to the minimizer of the original constrained one. Moreover, the local exactness property was also studied. The numerical results demonstrate that this penalty function method is an effective and promising approach for solving constrained finite min-max problems.
- min-max problem,
- constrained optimization,
- penalty function

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References

[1]	Rustem B, Howe M A.Algorithms for Worst-Case Design With Applications to Risk Management[M]. Princeton: Princeton University Press, 2001.
[2]	Zhu S S, Fukushima M. Worst-case conditional value-at-risk with application to robust portfolio management[J]. Operations Research, 2009, 57(5): 1155-1168.
[3]	Lemarechal C. Nondifferentiable optimization. Nemhauser G L, Rinnooy Kan A H G, Todd M J. Handbooks in Operations Research and Management Science. Vol 1: Optimization[M]. Amsterdam, Netherlands: North-Holland, 1989.
[4]	Rockafellar R T. Linear-quadric programming and optimal control[J]. SIAM Journal on Control and Optimization, 1987, 25(3): 781-814.
[5]	Rockafellar R T. Computational schemes for large-scale problems in extended linear-quadratic programming[J]. Mathematical Programming, 1990, 48(1/3): 447-474.
[6]	Gaudioso M, Monaco M F. A bundle type approach to the unconstrained minimization of convex nonsmooth functions[J]. Mathematical Programming, 1982, 23(1): 216-226.
[7]	Polak E, Mayne D H, Higgins J E. Superlinearly convergent algorithm for min-max problems[J]. Journal of Optimization Theory and Applications, 1991, 69(3): 407-439.
[8]	Zowe J. Nondifferentiable optimization: a motivation and a short introduction into the subgradient and the bundle concept[C]Schittkowski K. Computational Mathematical Programming, NATO SAI Series. New York:Springer, 1985.
[9]	DiPillo G, Grippo L, Lucidi S. A smooth method for the finite minimax problem[J]. Mathematical Programming, 1993, 60(1/3): 187-214.
[10]	Gigola C, Gomez S. A regularization method for solving the finite convex min-max problems[J]. SIAM Journal on Numerical Analysis, 1990, 27(6): 1621-1634.
[11]	Li X S, Pan S. Solving the finite min-max problem via an exponential penalty method[J]. Comput Technol, 2003, 8(2): 3-15.
[12]	Ye F, Liu H, Zhou S, Liu S. A smoothing trust-region Newton-CG method for minimax problem[J]. Applied Mathematics and Computation, 2008, 199(2): 581-589.
[13]	Zang I. A smoothing technique for min-max optimization[J]. Mathematical Programming, 1980, 19(1): 61-77.
[14]	Zhou J L, Tits A L. An SQP algorithm for finely discretized continuous minimax problems and other minimax problems with many objective functions[J]. SIAM J Optimization, 1996, 6(2): 461-487.
[15]	Zhu Z, Cai X, Jian J. An improved SQP algorithm for solving minimax problems[J]. Applied Mathematics Letters, 2009, 22(4): 464-469.
[16]	Obasanjo E, Tzallas-Regas G, Rustem B. An interior-point algorithm for nonlinear minimax problems[J]. Journal of Optimization Theory and Applications, 2010, 144(2): 291-318.
[17]	Huyer W, Neumaier A. A new exact penalty function[J]. SIAM Journal on Optimization, 2003, 13(4): 1141-1158.
[18]	Burke J V. An exact penalization viewpoint of constrained optimization[J]. SIAM J Control and Optimization, 1991, 29(4): 968-998.
[19]	Pang J S. Error bound in mathematical programming[J]. Mathematical Programming, 1997, 79(1/3): 299-332.