Decrease of the Penalty Parameter in Differentiable Penalty Function Methods

Affiliation(s)

Faculty of Mathematical Sciences, Sharif University of Technology, Tehran, Iran.

Ferdowsi University of Mashhad, Mashhad, Iran.

Faculty of Mathematical Sciences, Sharif University of Technology, Tehran, Iran.

Ferdowsi University of Mashhad, Mashhad, Iran.

ABSTRACT

We propose a simple modification to the differentiable penalty methods for solving nonlinear programming problems. This modification decreases the penalty parameter and the ill-conditioning of the penalty method and leads to a faster convergence to the optimal solution. We extend the modification to the augmented Lagrangian method and report some numerical results on several nonlinear programming test problems, showing the effectiveness of the proposed approach.

We propose a simple modification to the differentiable penalty methods for solving nonlinear programming problems. This modification decreases the penalty parameter and the ill-conditioning of the penalty method and leads to a faster convergence to the optimal solution. We extend the modification to the augmented Lagrangian method and report some numerical results on several nonlinear programming test problems, showing the effectiveness of the proposed approach.

Cite this paper

nullR. Shandiz and E. Tohidi, "Decrease of the Penalty Parameter in Differentiable Penalty Function Methods,"*Theoretical Economics Letters*, Vol. 1 No. 1, 2011, pp. 8-14. doi: 10.4236/tel.2011.11003.

nullR. Shandiz and E. Tohidi, "Decrease of the Penalty Parameter in Differentiable Penalty Function Methods,"

References

[1] R. Courant, “Variational Methods for the Solution of Problems of Equilibrium and Vibrations,” Bulletin of the American Mathematical Sociaty, Vol. 49, No. 1, 1943, pp. 1-23. doi:10.1090/S0002-9904-1943-07818-4

[2] A. V. Fiacco and G. P. McCormick, “Nonlinear Programming: Sequential Unconstrained Minimization Techniques,” Society for Industrial and Applied Mathematics, McLean, Virginia, 1990.

[3] D. M. Murray and S. J. Yakowitz, “Constrained Differential Dynamic Programming and Its Application to Multireservior Control,” Water Resources Research, Vol. 15, No. 5, 1979, pp. 1017-1027. doi:10.1029/WR015i005p01017

[4] W. I. Zangwill, “Nonlinear Programming via Penalty Functions,” Man-agement Science, Vol. 13, No. 5, 1967, pp. 344-358. doi:10.1287/mnsc.13.5.344

[5] R. Fletcher, “A Class of Methods for Nonlinear Programming with Termination and Convergence Properties,” Integer and Nonlinear Pro-gramming, Amsterdam, 1970, pp. 157-173.

[6] M. S. Bazaraa, H. D. Sherali and C. M. Shetty, “Nonlinear Pro-gramming: Theory and Algorithms,” 3rd Edition, Wiley, New York, 2006.

[7] C. Charalambous, “A Lower Bound for the Controlling Parameters of the Exact Penalty Functions,” Mathematical Programming, Vol. 15, No. 1, 1978, pp. 278-290. doi:10.1007/BF01609033

[8] A. R. Conn, “Constrained Optimization Using a Nondifferentiable Penalty Function,” SIAM Journal of Numerical Analysis, Vol. 10, No. 4, 1973, pp. 760-784. doi:10.1137/0710063

[9] G. D. Pillo and L. Grippo, “A Continuously Differentiable Exact Penalty Function for Nonlinear Programming Problems with Inequality Constraints,” SIAM Journal of Control and Optimization, Vol. 23, No. 1, 1985, pp. 72-84. doi:10.1137/0323007

[10] G. D. Pillo and L. Grippo, “Exact Penalty Functions in Constrained Optimization,” SIAM Journal of Control and Optimization, Vol. 27, No. 6, 1989, pp. 1333-1360. doi:10.1137/0327068

[11] J.-P. Dussault, “Improved Convergence Order for Augmented Penalty Algorithms,” Computational Optimization and Applications, Vol. 44, No. 3, 2009, pp. 373-383. doi:10.1007/s10589-007-9159-0

[12] A. L. Peressini, F. E. Sullivan and J. J. Uhl, “The Mathematics of Nonlinear Programming,” Springer-Verlag, New York, 1988.

[13] T. Pietrzykowski, “An Exact Potential Me-thod for Constrained Maxima,” SIAM Journal of Numer-ical Analysis, Vol. 6, No. 2, 1969, pp. 294-304. doi:10.1137/0706028

[14] M. Mongeau and A. Sartenaer, “Automatic Decrease of the Penalty Parameterin Exact Penalty Function Methods,” European Journal of Operational Research, Vol. 83, No. 3, 1995, pp. 686-699. doi:10.1016/0377-2217(93)E0339-Y

[15] D. G. Luen-berger and Y. Ye, “Linear and Nonlinear Programming,” 3rd Edition, Springer, New York, 2008.

[16] M. J. D. Powell, “A Fast Algorithm for Nonlinearly Constrained Optimization Calculations,” Lecture Notes in Mathematics, Vol. 630, 1978, pp. 144-157.

[17] W. Hock and K. Schittkowski, “Test Examples for Nonlinear Programming Codes,” Journal of Optimization Theory and Applications, Vol. 30, No. 1, 1980, pp. 127-129.

[18] K. Schittkowski, “More Test Examples for Nonlinear Programming Codes (Lecture Notes in Economics and Mathematical Systems),” Springer, Berlin, 1987.

[19] Princeton Library of Nonlinear Programming Models, 2011. http://www.gamsworld.org/performance/princetonlib/princetonlib.htm

[1] R. Courant, “Variational Methods for the Solution of Problems of Equilibrium and Vibrations,” Bulletin of the American Mathematical Sociaty, Vol. 49, No. 1, 1943, pp. 1-23. doi:10.1090/S0002-9904-1943-07818-4

[2] A. V. Fiacco and G. P. McCormick, “Nonlinear Programming: Sequential Unconstrained Minimization Techniques,” Society for Industrial and Applied Mathematics, McLean, Virginia, 1990.

[3] D. M. Murray and S. J. Yakowitz, “Constrained Differential Dynamic Programming and Its Application to Multireservior Control,” Water Resources Research, Vol. 15, No. 5, 1979, pp. 1017-1027. doi:10.1029/WR015i005p01017

[4] W. I. Zangwill, “Nonlinear Programming via Penalty Functions,” Man-agement Science, Vol. 13, No. 5, 1967, pp. 344-358. doi:10.1287/mnsc.13.5.344

[5] R. Fletcher, “A Class of Methods for Nonlinear Programming with Termination and Convergence Properties,” Integer and Nonlinear Pro-gramming, Amsterdam, 1970, pp. 157-173.

[6] M. S. Bazaraa, H. D. Sherali and C. M. Shetty, “Nonlinear Pro-gramming: Theory and Algorithms,” 3rd Edition, Wiley, New York, 2006.

[7] C. Charalambous, “A Lower Bound for the Controlling Parameters of the Exact Penalty Functions,” Mathematical Programming, Vol. 15, No. 1, 1978, pp. 278-290. doi:10.1007/BF01609033

[8] A. R. Conn, “Constrained Optimization Using a Nondifferentiable Penalty Function,” SIAM Journal of Numerical Analysis, Vol. 10, No. 4, 1973, pp. 760-784. doi:10.1137/0710063

[9] G. D. Pillo and L. Grippo, “A Continuously Differentiable Exact Penalty Function for Nonlinear Programming Problems with Inequality Constraints,” SIAM Journal of Control and Optimization, Vol. 23, No. 1, 1985, pp. 72-84. doi:10.1137/0323007

[10] G. D. Pillo and L. Grippo, “Exact Penalty Functions in Constrained Optimization,” SIAM Journal of Control and Optimization, Vol. 27, No. 6, 1989, pp. 1333-1360. doi:10.1137/0327068

[11] J.-P. Dussault, “Improved Convergence Order for Augmented Penalty Algorithms,” Computational Optimization and Applications, Vol. 44, No. 3, 2009, pp. 373-383. doi:10.1007/s10589-007-9159-0

[12] A. L. Peressini, F. E. Sullivan and J. J. Uhl, “The Mathematics of Nonlinear Programming,” Springer-Verlag, New York, 1988.

[13] T. Pietrzykowski, “An Exact Potential Me-thod for Constrained Maxima,” SIAM Journal of Numer-ical Analysis, Vol. 6, No. 2, 1969, pp. 294-304. doi:10.1137/0706028

[14] M. Mongeau and A. Sartenaer, “Automatic Decrease of the Penalty Parameterin Exact Penalty Function Methods,” European Journal of Operational Research, Vol. 83, No. 3, 1995, pp. 686-699. doi:10.1016/0377-2217(93)E0339-Y

[15] D. G. Luen-berger and Y. Ye, “Linear and Nonlinear Programming,” 3rd Edition, Springer, New York, 2008.

[16] M. J. D. Powell, “A Fast Algorithm for Nonlinearly Constrained Optimization Calculations,” Lecture Notes in Mathematics, Vol. 630, 1978, pp. 144-157.

[17] W. Hock and K. Schittkowski, “Test Examples for Nonlinear Programming Codes,” Journal of Optimization Theory and Applications, Vol. 30, No. 1, 1980, pp. 127-129.

[18] K. Schittkowski, “More Test Examples for Nonlinear Programming Codes (Lecture Notes in Economics and Mathematical Systems),” Springer, Berlin, 1987.

[19] Princeton Library of Nonlinear Programming Models, 2011. http://www.gamsworld.org/performance/princetonlib/princetonlib.htm