一种新的求解带约束的有限极大极小问题的精确罚函数

马骋; 李迅; 姚家晖; 张连生

doi:10.3879/j.issn.1000-0887.2012.02.010

一种新的求解带约束的有限极大极小问题的精确罚函数

doi: 10.3879/j.issn.1000-0887.2012.02.010

香港理工大学应用数学系, 九龙,香港;

基金项目: AMSS-PolyU联合研究所资助项目

详细信息

通讯作者:
马骋(1983—),男,山东青岛人,博士生(联系人.E-mail: mc-0812@163.com).

中图分类号: O221.2
计量
- 文章访问数: 1532
- HTML全文浏览量: 108
- PDF下载量: 854
- 被引次数: 0
出版历程
- 收稿日期: 2011-03-31
- 修回日期: 2011-11-23
- 刊出日期: 2012-02-15

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

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

摘要

摘要: 提出了一种新的精确光滑罚函数求解带约束的极大极小问题．仅仅添加一个额外的变量，利用这个精确光滑罚函数，将带约束的极大极小问题转化为无约束优化问题．证明了在合理的假设条件下，当罚参数充分大，罚问题的极小值点就是原问题的极小值点．进一步，研究了局部精确性质．数值结果表明这种罚函数算法是求解带约束有限极大极小问题的一种有效算法．
- 带约束的极大极小问题 /
- 约束优化问题 /
- 罚函数
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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参考文献(19)

[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.