October  2010, 6(4): 895-910. doi: 10.3934/jimo.2010.6.895

A new exact penalty function method for continuous inequality constrained optimization problems

1. 

Department of Mathematics and Statistics, Curtin University of Technology, Kent Street, Bentley 6102, WA, Australia, Australia

2. 

Department of Mathematics, Shanghai University, 99, Shangda Road, 200444, Shanghai, China, China

Received  March 2010 Revised  July 2010 Published  September 2010

In this paper, a computational approach based on a new exact penalty function method is devised for solving a class of continuous inequality constrained optimization problems. The continuous inequality constraints are first approximated by smooth function in integral form. Then, we construct a new exact penalty function, where the summation of all these approximate smooth functions in integral form, called the constraint violation, is appended to the objective function. In this way, we obtain a sequence of approximate unconstrained optimization problems. It is shown that if the value of the penalty parameter is sufficiently large, then any local minimizer of the corresponding unconstrained optimization problem is a local minimizer of the original problem. For illustration, three examples are solved using the proposed method. From the solutions obtained, we observe that the values of their objective functions are amongst the smallest when compared with those obtained by other existing methods available in the literature. More importantly, our method finds solution which satisfies the continuous inequality constraints.
Citation: Changjun Yu, Kok Lay Teo, Liansheng Zhang, Yanqin Bai. A new exact penalty function method for continuous inequality constrained optimization problems. Journal of Industrial & Management Optimization, 2010, 6 (4) : 895-910. doi: 10.3934/jimo.2010.6.895
References:
[1]

Proc. IEEE, 69 (1981), 1334-1362. doi: 10.1109/PROC.1981.12170.  Google Scholar

[2]

in The State of the Art in Numerical Analysis, I. S. Duff and G. A. Watson, eds., Clarendon Press, Oxford, (1997), 363-390.  Google Scholar

[3]

Journal of Industrial and Management Optimization, 5 (2009), 141-151. doi: 10.3934/jimo.2009.5.141.  Google Scholar

[4]

IEEE Transactions on Automatic Control, AC-25 (1980), 49-54. doi: 10.1109/TAC.1980.1102227.  Google Scholar

[5]

SIAM Review, 35 (1993), 380-429. doi: 10.1137/1035089.  Google Scholar

[6]

SIAM J.Optim, 13 (2003), 1141-1158. doi: 10.1137/S1052623401390537.  Google Scholar

[7]

Annals of Operations Research, 98 (2000), 189-213. doi: 10.1023/A:1019208524259.  Google Scholar

[8]

Automatica, 26 (1990), 371-375. doi: 10.1016/0005-1098(90)90131-Z.  Google Scholar

[9]

Journal of Global Optimization, 30 (2004), 169-194. doi: 10.1007/s10898-004-8266-z.  Google Scholar

[10]

Journal of Optimization Theory and Applications, 108 (2001), 617-632. doi: 10.1023/A:1017539525721.  Google Scholar

[11]

Journal of Global Optimization, 24 (2002), 205-217. doi: 10.1023/A:1020234019886.  Google Scholar

[12]

Computing, Supplement, 15 (2001), 119-132.  Google Scholar

[13]

Journal of Global Optimization, 29 (2004), 401-413. doi: 10.1023/B:JOGO.0000047910.80739.95.  Google Scholar

[14]

Communications in Information and Systems, 2 (2002), 399-410.  Google Scholar

[15]

SIAM Journal of Optimization, 16 (2006), 1137-1154. doi: 10.1137/040619867.  Google Scholar

[16]

IEEE Transactions on Automatic Control, AC-21 (1976), 184-193. doi: 10.1109/TAC.1976.1101196.  Google Scholar

[17]

Proc. IEEE, 72 (1984), 1777-1794. doi: 10.1109/PROC.1984.13086.  Google Scholar

[18]

Automatica, 18 (1982), 267-283. doi: 10.1016/0005-1098(82)90087-5.  Google Scholar

[19]

I. S. Duff and G. A. Watson, eds., Clarendon Press, Oxford, 1997, 339-362.  Google Scholar

[20]

Automatica, 29 (1993), 789-792. doi: 10.1016/0005-1098(93)90076-6.  Google Scholar

[21]

Annals of Operations Research, 98 (2000), 215-234. doi: 10.1023/A:1019260508329.  Google Scholar

[22]

Journal of Industrial and Management Optimization, 2 (2005), 235-250.  Google Scholar

[23]

Optimization Methods and Software, 20 (2005), 629-643. doi: 10.1080/10556780500094739.  Google Scholar

[24]

Annals of Operations Research, 103 (2001), 235-250. doi: 10.1023/A:1012911307208.  Google Scholar

[25]

Report in the 4th Australia-China workshop on optimization: Theory, Methods and Applications, 2009. Google Scholar

show all references

References:
[1]

Proc. IEEE, 69 (1981), 1334-1362. doi: 10.1109/PROC.1981.12170.  Google Scholar

[2]

in The State of the Art in Numerical Analysis, I. S. Duff and G. A. Watson, eds., Clarendon Press, Oxford, (1997), 363-390.  Google Scholar

[3]

Journal of Industrial and Management Optimization, 5 (2009), 141-151. doi: 10.3934/jimo.2009.5.141.  Google Scholar

[4]

IEEE Transactions on Automatic Control, AC-25 (1980), 49-54. doi: 10.1109/TAC.1980.1102227.  Google Scholar

[5]

SIAM Review, 35 (1993), 380-429. doi: 10.1137/1035089.  Google Scholar

[6]

SIAM J.Optim, 13 (2003), 1141-1158. doi: 10.1137/S1052623401390537.  Google Scholar

[7]

Annals of Operations Research, 98 (2000), 189-213. doi: 10.1023/A:1019208524259.  Google Scholar

[8]

Automatica, 26 (1990), 371-375. doi: 10.1016/0005-1098(90)90131-Z.  Google Scholar

[9]

Journal of Global Optimization, 30 (2004), 169-194. doi: 10.1007/s10898-004-8266-z.  Google Scholar

[10]

Journal of Optimization Theory and Applications, 108 (2001), 617-632. doi: 10.1023/A:1017539525721.  Google Scholar

[11]

Journal of Global Optimization, 24 (2002), 205-217. doi: 10.1023/A:1020234019886.  Google Scholar

[12]

Computing, Supplement, 15 (2001), 119-132.  Google Scholar

[13]

Journal of Global Optimization, 29 (2004), 401-413. doi: 10.1023/B:JOGO.0000047910.80739.95.  Google Scholar

[14]

Communications in Information and Systems, 2 (2002), 399-410.  Google Scholar

[15]

SIAM Journal of Optimization, 16 (2006), 1137-1154. doi: 10.1137/040619867.  Google Scholar

[16]

IEEE Transactions on Automatic Control, AC-21 (1976), 184-193. doi: 10.1109/TAC.1976.1101196.  Google Scholar

[17]

Proc. IEEE, 72 (1984), 1777-1794. doi: 10.1109/PROC.1984.13086.  Google Scholar

[18]

Automatica, 18 (1982), 267-283. doi: 10.1016/0005-1098(82)90087-5.  Google Scholar

[19]

I. S. Duff and G. A. Watson, eds., Clarendon Press, Oxford, 1997, 339-362.  Google Scholar

[20]

Automatica, 29 (1993), 789-792. doi: 10.1016/0005-1098(93)90076-6.  Google Scholar

[21]

Annals of Operations Research, 98 (2000), 215-234. doi: 10.1023/A:1019260508329.  Google Scholar

[22]

Journal of Industrial and Management Optimization, 2 (2005), 235-250.  Google Scholar

[23]

Optimization Methods and Software, 20 (2005), 629-643. doi: 10.1080/10556780500094739.  Google Scholar

[24]

Annals of Operations Research, 103 (2001), 235-250. doi: 10.1023/A:1012911307208.  Google Scholar

[25]

Report in the 4th Australia-China workshop on optimization: Theory, Methods and Applications, 2009. Google Scholar

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