# American Institute of Mathematical Sciences

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] R. K. Brayton, G. D. Hachtel and A. L. Sangiovanni-Vincentlli, A survey of optimization techniques for integrated circuit design,, Proc. IEEE, 69 (1981), 1334. doi: 10.1109/PROC.1981.12170. [2] A. R. Conn, N. I. M. Gould and Ph. L. Toint, Methods for nonlinear constraints in optimization calculations,, in The State of the Art in Numerical Analysis, (1997), 363. [3] Z. G. Feng, K. L. Teo and V. Rehbock, A smoothing approach for semi-infinite programming with projected Newton-type algorithm,, Journal of Industrial and Management Optimization, 5 (2009), 141. doi: 10.3934/jimo.2009.5.141. [4] G. Gonzaga, E. Polak and R. Trahan, An improved algorithm for optimization problems with functional inequality constraints,, IEEE Transactions on Automatic Control, AC-25 (1980), 49. doi: 10.1109/TAC.1980.1102227. [5] R. Hettich and K. O. Kortanek, Semi-infinite programming: Theory, methods, and applications,, SIAM Review, 35 (1993), 380. doi: 10.1137/1035089. [6] W. Huyer and A. Neumaier, A new exact penalty function,, SIAM J.Optim, 13 (2003), 1141. doi: 10.1137/S1052623401390537. [7] S. Ito, Y. Liu and K. L. Teo, A dual parametrization method for convex semi-infinite programming,, Annals of Operations Research, 98 (2000), 189. doi: 10.1023/A:1019208524259. [8] L. S. Jennings and K. L. Teo, A computational algorithm for functional inequality constrained optimization problems,, Automatica, 26 (1990), 371. doi: 10.1016/0005-1098(90)90131-Z. [9] D. H. Li, L. Q. Qi, J. Tam and S. Y. Wu, A smoothing newton method for semi-infinite programming,, Journal of Global Optimization, 30 (2004), 169. doi: 10.1007/s10898-004-8266-z. [10] Y. Liu, S. Ito, H. W. J. Lee and K. L. Teo, Semi-infinite programming approach to continuously-constrained linear-quadratic optimal control problems,, Journal of Optimization Theory and Applications, 108 (2001), 617. doi: 10.1023/A:1017539525721. [11] Y. Liu and K. L. Teo, An adaptive dual parametrization algorithm for quadratic semi-infinite programming problems,, Journal of Global Optimization, 24 (2002), 205. doi: 10.1023/A:1020234019886. [12] Y. Liu, K. L. Teo and S. Ito, Global optimization in linear-quadratic semi-infinite programming,, Computing, 15 (2001), 119. [13] Y. Liu, K. L. Teo and S. Y. Wu, A new quadratic semi-infinite programming algorithm based on dual parametri,, Journal of Global Optimization, 29 (2004), 401. doi: 10.1023/B:JOGO.0000047910.80739.95. [14] Y. Liu, C. H. Tseng and K. L. Teo, A unified quadratic semi-infinite programming approach to time and frequence domain constrained digital filter design,, Communications in Information and Systems, 2 (2002), 399. [15] Q. Ni, C. Ling, L. Q. Qi and K. L. Teo, A truncated projected Newton-type algorithm for large-scale semi-infinite programming,, SIAM Journal of Optimization, 16 (2006), 1137. doi: 10.1137/040619867. [16] E. Polak and D. Q. Mayne, An algorithm for optimization problems with functional inequality constraints,, IEEE Transactions on Automatic Control, AC-21 (1976), 184. doi: 10.1109/TAC.1976.1101196. [17] E. Polak, D. Q. Mayne and D. M. Stimler, Control system design via semi-infinite optimization: A review,, Proc. IEEE, 72 (1984), 1777. doi: 10.1109/PROC.1984.13086. [18] E. Polak and Y. Wardi, Nondifferential optimization algorithm for designing control systems having singular value inequalities,, Automatica, 18 (1982), 267. doi: 10.1016/0005-1098(82)90087-5. [19] D. F. Shanno and E. M. Simantiraki, Interior point methods for linear and nonlinear programming, in The State of the Art in Numerical Analysis,, I. S. Duff and G. A. Watson, (1997), 339. [20] K. L. Teo, V. Rehbock and L. S. Jennings, A new computational algorithm for functional inequality constrained optimization problems,, Automatica, 29 (1993), 789. doi: 10.1016/0005-1098(93)90076-6. [21] K. L. Teo, X. Q. Yang and L. S. Jennings, Computational discretization algorithms for functional inequality constrained optimization,, Annals of Operations Research, 98 (2000), 215. doi: 10.1023/A:1019260508329. [22] X. J. Tong and S. Z. Zhou, A smoothing projected Newton-type method for semismooth equations with bound constraints,, Journal of Industrial and Management Optimization, 2 (2005), 235. [23] S. Y. Wu, D. H. Li, L. Q. Qi and G. L. Zhou, An iterative method for solving KKT system of the semi-infinite programming,, Optimization Methods and Software, 20 (2005), 629. doi: 10.1080/10556780500094739. [24] X. Q. Yang and K. L. Teo, Nonlinear Lagrangian functions and applications to semi-infinite programs,, Annals of Operations Research, 103 (2001), 235. doi: 10.1023/A:1012911307208. [25] L. S. Zhang, "New Simple Exact Penalty Function for Constrained Minimization on $\mathbbR^n$,", Report in the 4th Australia-China workshop on optimization: Theory, (2009).

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##### References:
 [1] R. K. Brayton, G. D. Hachtel and A. L. Sangiovanni-Vincentlli, A survey of optimization techniques for integrated circuit design,, Proc. IEEE, 69 (1981), 1334. doi: 10.1109/PROC.1981.12170. [2] A. R. Conn, N. I. M. Gould and Ph. L. Toint, Methods for nonlinear constraints in optimization calculations,, in The State of the Art in Numerical Analysis, (1997), 363. [3] Z. G. Feng, K. L. Teo and V. Rehbock, A smoothing approach for semi-infinite programming with projected Newton-type algorithm,, Journal of Industrial and Management Optimization, 5 (2009), 141. doi: 10.3934/jimo.2009.5.141. [4] G. Gonzaga, E. Polak and R. Trahan, An improved algorithm for optimization problems with functional inequality constraints,, IEEE Transactions on Automatic Control, AC-25 (1980), 49. doi: 10.1109/TAC.1980.1102227. [5] R. Hettich and K. O. Kortanek, Semi-infinite programming: Theory, methods, and applications,, SIAM Review, 35 (1993), 380. doi: 10.1137/1035089. [6] W. Huyer and A. Neumaier, A new exact penalty function,, SIAM J.Optim, 13 (2003), 1141. doi: 10.1137/S1052623401390537. [7] S. Ito, Y. Liu and K. L. Teo, A dual parametrization method for convex semi-infinite programming,, Annals of Operations Research, 98 (2000), 189. doi: 10.1023/A:1019208524259. [8] L. S. Jennings and K. L. Teo, A computational algorithm for functional inequality constrained optimization problems,, Automatica, 26 (1990), 371. doi: 10.1016/0005-1098(90)90131-Z. [9] D. H. Li, L. Q. Qi, J. Tam and S. Y. Wu, A smoothing newton method for semi-infinite programming,, Journal of Global Optimization, 30 (2004), 169. doi: 10.1007/s10898-004-8266-z. [10] Y. Liu, S. Ito, H. W. J. Lee and K. L. Teo, Semi-infinite programming approach to continuously-constrained linear-quadratic optimal control problems,, Journal of Optimization Theory and Applications, 108 (2001), 617. doi: 10.1023/A:1017539525721. [11] Y. Liu and K. L. Teo, An adaptive dual parametrization algorithm for quadratic semi-infinite programming problems,, Journal of Global Optimization, 24 (2002), 205. doi: 10.1023/A:1020234019886. [12] Y. Liu, K. L. Teo and S. Ito, Global optimization in linear-quadratic semi-infinite programming,, Computing, 15 (2001), 119. [13] Y. Liu, K. L. Teo and S. Y. Wu, A new quadratic semi-infinite programming algorithm based on dual parametri,, Journal of Global Optimization, 29 (2004), 401. doi: 10.1023/B:JOGO.0000047910.80739.95. [14] Y. Liu, C. H. Tseng and K. L. Teo, A unified quadratic semi-infinite programming approach to time and frequence domain constrained digital filter design,, Communications in Information and Systems, 2 (2002), 399. [15] Q. Ni, C. Ling, L. Q. Qi and K. L. Teo, A truncated projected Newton-type algorithm for large-scale semi-infinite programming,, SIAM Journal of Optimization, 16 (2006), 1137. doi: 10.1137/040619867. [16] E. Polak and D. Q. Mayne, An algorithm for optimization problems with functional inequality constraints,, IEEE Transactions on Automatic Control, AC-21 (1976), 184. doi: 10.1109/TAC.1976.1101196. [17] E. Polak, D. Q. Mayne and D. M. Stimler, Control system design via semi-infinite optimization: A review,, Proc. IEEE, 72 (1984), 1777. doi: 10.1109/PROC.1984.13086. [18] E. Polak and Y. Wardi, Nondifferential optimization algorithm for designing control systems having singular value inequalities,, Automatica, 18 (1982), 267. doi: 10.1016/0005-1098(82)90087-5. [19] D. F. Shanno and E. M. Simantiraki, Interior point methods for linear and nonlinear programming, in The State of the Art in Numerical Analysis,, I. S. Duff and G. A. Watson, (1997), 339. [20] K. L. Teo, V. Rehbock and L. S. Jennings, A new computational algorithm for functional inequality constrained optimization problems,, Automatica, 29 (1993), 789. doi: 10.1016/0005-1098(93)90076-6. [21] K. L. Teo, X. Q. Yang and L. S. Jennings, Computational discretization algorithms for functional inequality constrained optimization,, Annals of Operations Research, 98 (2000), 215. doi: 10.1023/A:1019260508329. [22] X. J. Tong and S. Z. Zhou, A smoothing projected Newton-type method for semismooth equations with bound constraints,, Journal of Industrial and Management Optimization, 2 (2005), 235. [23] S. Y. Wu, D. H. Li, L. Q. Qi and G. L. Zhou, An iterative method for solving KKT system of the semi-infinite programming,, Optimization Methods and Software, 20 (2005), 629. doi: 10.1080/10556780500094739. [24] X. Q. Yang and K. L. Teo, Nonlinear Lagrangian functions and applications to semi-infinite programs,, Annals of Operations Research, 103 (2001), 235. doi: 10.1023/A:1012911307208. [25] L. S. Zhang, "New Simple Exact Penalty Function for Constrained Minimization on $\mathbbR^n$,", Report in the 4th Australia-China workshop on optimization: Theory, (2009).
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