An Exact Penalty Approach for Mixed Integer Nonlinear Programming Problems

ABSTRACT

We propose an exact penalty approach for solving mixed integer nonlinear programming (MINLP) problems by converting a general MINLP problem to a finite sequence of nonlinear programming (NLP) problems with only continuous variables. We express conditions of exactness for MINLP problems and show how the exact penalty approach can be extended to constrained problems.

We propose an exact penalty approach for solving mixed integer nonlinear programming (MINLP) problems by converting a general MINLP problem to a finite sequence of nonlinear programming (NLP) problems with only continuous variables. We express conditions of exactness for MINLP problems and show how the exact penalty approach can be extended to constrained problems.

KEYWORDS

Mixed Integer Nonlinear Programming, Continuous Programming, Exact Penalty Method, Exact Penalty Functions

Mixed Integer Nonlinear Programming, Continuous Programming, Exact Penalty Method, Exact Penalty Functions

Cite this paper

nullR. Shandiz and N. Mahdavi-Amiri, "An Exact Penalty Approach for Mixed Integer Nonlinear Programming Problems,"*American Journal of Operations Research*, Vol. 1 No. 3, 2011, pp. 185-189. doi: 10.4236/ajor.2011.13021.

nullR. Shandiz and N. Mahdavi-Amiri, "An Exact Penalty Approach for Mixed Integer Nonlinear Programming Problems,"

References

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[2] J. Cai and G. Thierauf, “Discrete Optimization of Structures Using an Improved Penalty Function Method,” Engineering Optimization, Vol. 21, No. 4, 1993, pp. 293-306. doi:10.1080/03052159308940981

[3] J. F. Fu, R. G. Fenton and W. L. Cleghorn, “A Mixed Integer-Discrete-Continuous Programming Method and Its Application to Engineering Design Optimization,” Engineering Optimization, Vol. 17, No. 4, 1991, pp. 263- 280. doi:10.1080/03052159108941075

[4] S. S. Rao, “Engineering Optimization: Theory and Practice,” 4th Edition, Wiley, Hoboken, 2009.

[5] D. K. Shin, Z. Gürdal and O. H. Griffin Jr., “A Penalty Approach for Nonlinear Optimization with Discrete Design Variables,” Engineering Optimization, Vol. 16, No. 1, 1990, pp. 29-42. doi:10.1080/03052159008941163

[6] W. Murray and K. M. Ng, “An Algorithm for Nonlinear Optimization Problems with Binary Variables,” Computational Optimization and Applications, Vol. 47, No. 2, 2008, pp. 257-288. doi:10.1007/s10589-008-9218-1

[7] F. Giannessi and F. Niccolucci, “Connections between Nonlinear and Integer Programming Problems,” Symposia Mathematica, Vol. 19, 1976, pp. 161-176.

[8] F. Rinaldi, “New Results on the Equivalence between Zero-One Programming and Continuous Concave Programming,” Optimation Letters, Vol. 3, No. 3, 2009, pp. 377-386. doi:10.1007/s11590-009-0117-x

[9] S. Lucidi and F. Rinaldi, “Exact Penalty Functions for Nonlinear Integer Programming Problems,” Journal of Optimization Theory and Applications, Vol. 145, No. 3, 2010, pp. 479-488. doi:10.1007/s10957-010-9700-7

[10] D. Li and X. Sun, “Nonlinear Integer Programming,” Springer, New York, 2006.

[11] D. G. Luenberger and Y. Ye, “Linear and Nonlinear Programming,” 3rd Edition, Springer, New York, 2008.

[1] M. Ragavachari, “On Connections between Zero-One Integer Programming and Concave Programming under Linear Constraints,” Operations Research, Vol. 17, No. 4, 1969, pp. 680-684. doi:10.1287/opre.17.4.680

[2] J. Cai and G. Thierauf, “Discrete Optimization of Structures Using an Improved Penalty Function Method,” Engineering Optimization, Vol. 21, No. 4, 1993, pp. 293-306. doi:10.1080/03052159308940981

[3] J. F. Fu, R. G. Fenton and W. L. Cleghorn, “A Mixed Integer-Discrete-Continuous Programming Method and Its Application to Engineering Design Optimization,” Engineering Optimization, Vol. 17, No. 4, 1991, pp. 263- 280. doi:10.1080/03052159108941075

[4] S. S. Rao, “Engineering Optimization: Theory and Practice,” 4th Edition, Wiley, Hoboken, 2009.

[5] D. K. Shin, Z. Gürdal and O. H. Griffin Jr., “A Penalty Approach for Nonlinear Optimization with Discrete Design Variables,” Engineering Optimization, Vol. 16, No. 1, 1990, pp. 29-42. doi:10.1080/03052159008941163

[6] W. Murray and K. M. Ng, “An Algorithm for Nonlinear Optimization Problems with Binary Variables,” Computational Optimization and Applications, Vol. 47, No. 2, 2008, pp. 257-288. doi:10.1007/s10589-008-9218-1

[7] F. Giannessi and F. Niccolucci, “Connections between Nonlinear and Integer Programming Problems,” Symposia Mathematica, Vol. 19, 1976, pp. 161-176.

[8] F. Rinaldi, “New Results on the Equivalence between Zero-One Programming and Continuous Concave Programming,” Optimation Letters, Vol. 3, No. 3, 2009, pp. 377-386. doi:10.1007/s11590-009-0117-x

[9] S. Lucidi and F. Rinaldi, “Exact Penalty Functions for Nonlinear Integer Programming Problems,” Journal of Optimization Theory and Applications, Vol. 145, No. 3, 2010, pp. 479-488. doi:10.1007/s10957-010-9700-7

[10] D. Li and X. Sun, “Nonlinear Integer Programming,” Springer, New York, 2006.

[11] D. G. Luenberger and Y. Ye, “Linear and Nonlinear Programming,” 3rd Edition, Springer, New York, 2008.