×
Peng, Z. Y.; Lin, Z.; Li, X. B.

Semistrict \(G\)-preinvexity and optimality in nonlinear programming. (English) Zbl 1278.90388

Abstr. Appl. Anal. 2013, Article ID 103864, 7 p. (2013).
Summary: A class of semistrictly \(G\)-preinvex functions and optimality in nonlinear programming are further discussed. Firstly, the relationships between semistrictly \(G\)-preinvex functions and \(G\)-preinvex functions are further discussed. Then, two interesting properties of semistrictly \(G\)-preinvexity are given. Finally, two optimality results for nonlinear programming problems are obtained under the assumption of semistrict \(G\)-preinvexity. The obtained results are new and different from the corresponding ones in the literature. Some examples are given to illustrate our results.

MSC:

90C30 Nonlinear programming

Keywords:

optimality results
Cite Review PDF
Full Text: DOI
Creative Commons CC license icon Creative Commons BY license icon

References:

[1] Mangasarian, O. L., Nonlinear Programming, xiii+220 (1969), New York, NY, USA: McGraw-Hill, New York, NY, USA · Zbl 0194.20201
[2] Bazaraa, M. S.; Sherali, H. D.; Shetty, C. M., Nonlinear Programming: Theory and Algorithms, xiv+560 (1979), New York, NY, USA: John Wiley & Sons, New York, NY, USA · Zbl 0476.90035
[3] Schaible, S.; Ziemba, W. T., Generalized Concavity in Optimization and Economics (1981), London, UK: Academic Press Inc., London, UK
[4] Mishra, S. K.; Giorgi, G., Invexity and Optimization. Invexity and Optimization, Nonconvex Optimization and Its Applications, 88, x+266 (2008), Berlin, Germany: Springer, Berlin, Germany · Zbl 1155.90016 · doi:10.1007/978-3-540-78562-0
[5] Hanson, M. A., On sufficiency of the Kuhn-Tucker conditions, Journal of Mathematical Analysis and Applications, 80, 2, 545-550 (1981) · Zbl 0463.90080 · doi:10.1016/0022-247X(81)90123-2
[6] Weir, T.; Mond, B., Pre-invex functions in multiple objective optimization, Journal of Mathematical Analysis and Applications, 136, 1, 29-38 (1988) · Zbl 0663.90087 · doi:10.1016/0022-247X(88)90113-8
[7] Weir, T.; Jeyakumar, V., A class of nonconvex functions and mathematical programming, Bulletin of the Australian Mathematical Society, 38, 2, 177-189 (1988) · Zbl 0639.90082 · doi:10.1017/S0004972700027441
[8] Yang, X. M.; Li, D., On properties of preinvex functions, Journal of Mathematical Analysis and Applications, 256, 1, 229-241 (2001) · Zbl 1016.90056 · doi:10.1006/jmaa.2000.7310
[9] Yang, X. M.; Li, D., Semistrictly preinvex functions, Journal of Mathematical Analysis and Applications, 258, 1, 287-308 (2001) · Zbl 0985.26007 · doi:10.1006/jmaa.2000.7382
[10] Yang, X. M.; Yang, X. Q.; Teo, K. L., Characterizations and applications of prequasi-invex functions, Journal of Optimization Theory and Applications, 110, 3, 645-668 (2001) · Zbl 1064.90038 · doi:10.1023/A:1017544513305
[11] Peng, J. W.; Yang, X. M., Two properties of strictly preinvex functions, Operations Research Transactions, 9, 1, 37-42 (2005)
[12] Avriel, M.; Diewert, W. E.; Schaible, S.; Zang, I., Generalized Concavity (1975), New York, NY, USA: Plenum Press, New York, NY, USA
[13] Antczak, T., On \(G\)-invex multiobjective programming. I. Optimality, Journal of Global Optimization, 43, 1, 97-109 (2009) · Zbl 1191.90052 · doi:10.1007/s10898-008-9299-5
[14] Antczak, T., \(G\)-pre-invex functions in mathematical programming, Journal of Computational and Applied Mathematics, 217, 1, 212-226 (2008) · Zbl 1219.90126 · doi:10.1016/j.cam.2007.06.026
[15] Luo, H. Z.; Wu, H. X., On the relationships between \(G\)-preinvex functions and semistrictly \(G\)-preinvex functions, Journal of Computational and Applied Mathematics, 222, 2, 372-380 (2008) · Zbl 1154.90010 · doi:10.1016/j.cam.2007.11.006
[16] Peng, Z. Y., Semistrict \(G\)-preinvexity and its application, Journal of Inequalities and Applications, 2012 (2012) · Zbl 1305.26035 · doi:10.1186/1029-242X-2012-198
[17] Long, X. J., Optimality conditions and duality for nondifferentiable multiobjective fractional programming problems with \((C, \alpha, \rho, d)\)-convexity, Journal of Optimization Theory and Applications, 148, 1, 197-208 (2011) · Zbl 1229.90258 · doi:10.1007/s10957-010-9740-z
[18] Long, X.-J.; Huang, N.-J., Lipschitz \(B\)-preinvex functions and nonsmooth multiobjective programming, Pacific Journal of Optimization, 7, 1, 97-107 (2011)
[19] Mishra, S. K., Topics in Nonconvex Optimization: Theory and Applications. Topics in Nonconvex Optimization: Theory and Applications, Springer Optimization and Its Applications, 50, xviii+268 (2011), New York, NY, USA: Springer, New York, NY, USA · Zbl 1216.90003 · doi:10.1007/978-1-4419-9640-4
[20] Mishra, S. K.; Rueda, N. G., Generalized invexity-type conditions in constrained optimization, Journal of Systems Science & Complexity, 24, 2, 394-400 (2011) · Zbl 1254.90234 · doi:10.1007/s11424-011-8234-x
[21] Liu, X.; Yuan, D. H., On semi-\((B, G)\)-preinvex functions, Abstract and Applied Analysis, 2012 (2012) · Zbl 1237.26010 · doi:10.1155/2012/530468
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.