A General Class of Convexification Transformation for the Noninferior Frontier of a Multiobjective Program

Tao Li, Yanjun Wang, Zhian Liang

Department of Mathematics Shanghai University of Finance and Economics, Shanghai, China.
Department of Statistics, Missouri University of Science & Technology, Rolla, USA.

DOI: 10.4236/ajor.2013.33036                

Abstract

A general class of convexification transformations is proposed to convexify the noninferior frontier of a multiobjective program. We prove that under certain assumptions the noninferior frontier could be convexified completely or partly after transformation and then weighting method can be applied to identify the noninferior solutions. Numerical experiments are given to vindicate our results.

Keywords

Noninferior Frontier; Convexification; Weighting Method; Multiobjective Optimization

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	D. Li, X. L. Sun, M. P. Biswal and F. Gao, “Convexification, Concavification and Monotonization in Global Optimization,” Annals of Operations Research, Vol. 105, No. 1-4, 2001, pp. 213-226. doi:10.1023/A:1013313901854
[2]	X. L. Sun, K. McKinnon and D. Li, “A Convexification Method for a Class of Global Optimization Problem with Application to Reliability Optimization,” Journal of Global Optimization, Vol. 21, No. 2, 2001, pp. 185-199. doi:10.1023/A:1011962605464
[3]	Z. Y. Wu, F. S. Bai and L. S. Zhang, “Convexification and Concavification for a General Class of Global Optimization Problems,” Journal of Global Optimization, Vol. 31, No. 1, 2005, pp. 45-60. doi:10.1007/s10898-004-0569-6
[4]	D. Li, “Zero Duality Gap for a Class of Nonconvex Optimization Problems,” Journal of Optimization Theory and Applications, Vol. 85, No. 2, 1995, pp. 309-324. doi:10.1007/BF02192229
[5]	D. Li and X .L. Sun, “Local Convexification of Lagrangian Function in Nonconvex Optimization,” Journal of Optimization Theory and Applications, Vol. 104, No. 1, 2000, pp. 109-120. doi:10.1023/A:1004628822745
[6]	Z. Y. Wu, F. S. Bai and L. S. Zhang, “Convexification and Concavification for a General Class of Global Optimization Problems,” Journal of Global Optimization, Vol. 31, No. 1, 2005, pp. 45-60. doi:10.1007/s10898-004-0569-6
[7]	Z. Y. Wu, H. W. J. Lee and X. M. Yang, “A Class of Convexification and Concavification Methods for NonMonotone Optimization Problems,” Optimization, Vol. 54, No. 6, 2005, pp. 605-625. doi:10.1080/02331930500342807
[8]	T. Li, Y. J. Wang, Z. Liang and P. M. Pardalos, “Local Saddle Point and a Class of Convexification Methods for Nonconvex Optimization Problems,” Journal of Global Optimization, Vol. 38, No. 3, 2007, pp. 405-419. doi:10.1007/s10898-006-9090-4
[9]	H. L. Li, J. F. Tsai and C. A. Floudas, “Convex Underestimation for Posynomial Functions of Positive Variables,” Optimization Letter, Vol. 2, No. 3, 2008, pp. 333-340. doi:10.1007/s11590-007-0061-6
[10]	D. Li, “Convexification of Noninferior Frontier,” Journal of Optimization Theory and Applications, Vol. 88, No. 1, 1996, pp. 177-196. doi:10.1007/BF02192028
[11]	C. J. Goh and X. Q. Yang, “Convxification of a Noninferior Frontier,” Journal of Optimization Theory and Applications, Vol. 97, No. 3, 1998, pp. 759-768. doi:10.1023/A:1022654528902
[12]	D. Li and M. P. Biswal, “Exponential Transformation in Convexifying a Noninferior Frontier and Exponential Generating Method,” Journal of Optimization Theory and Applications, Vol. 99, No. 1, 1998, pp. 183-199. doi:10.1023/A:1021708412776
[13]	M. S. Bazaraa, H. D. Sherali and C. M. Shetty, “Nonlinear Programming, Theory and Algorithms,” 2nd Edition, John Wiley & Sons, Inc., New York, 1993.