×

Second and higher order duality in Banach space under \(\rho-(\eta,\theta)\)-invexity. (English) Zbl 1242.49073

Summary: We introduce the concept of second and higher order duality in Banach space using \(\rho - (\eta ,\theta )\)-invexity type conditions. Duality theorems for Mangasarian-type and Mond–Wei- type are established under weaker conditions. We also give counter examples to justify our work.

MSC:

49N15 Duality theory (optimization)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Jeyakumar, V., \(p\)-convexity and second order duality, Utilitas Math., 29, 71-85 (1986) · Zbl 0557.90080
[2] Mangasarian, O. L., Second and higher order duality in nonlinear programming, J. Math. Anal. Appl., 51, 607-620 (1975) · Zbl 0313.90052
[3] Meo, M.; Zumpano, G., Damage assessment on plate-like structures using a global-local optimization approach, Optim. Eng., 9, 161-177 (2008) · Zbl 1175.74066
[4] Mond, B., Second order duality for nonlinear programs, Opsearch, 11, 90-99 (1974)
[5] Bagivov, A. M.; Rubinov, A. M.; Zhang, J., Local optimization method with global multidimensional search, J. Global Optim., 3, 161-179 (2005) · Zbl 1123.90079
[6] Dorn, W. S., A duality theorem for convex programs, IBM. J. Res. Devlop, 4, 407-413 (1960) · Zbl 0095.14503
[7] Schechter, M., More on subgradient duality, J. Math. Anal. Appl., 71, 251-262 (1979) · Zbl 0421.90062
[8] Hanson, M. A., On sufficiency of the Kuhn-Tucker conditions, J. Math. Anal. Appl., 80, 545-550 (1981) · Zbl 0463.90080
[9] Ben-Israel, A.; Mond, B., What is invexity?, J. Aust. Math. Soc. Ser. B, 28, 1-9 (1986) · Zbl 0603.90119
[10] M.A. Hanson, B. Mond, Self-duality and invexity, FSU Statistics Report M 716, Department of Statistics, Florida State University, Florida, (1986).; M.A. Hanson, B. Mond, Self-duality and invexity, FSU Statistics Report M 716, Department of Statistics, Florida State University, Florida, (1986).
[11] Nahak, C.; Nanda, S., Multiobjective duality with \(\rho -(\eta, \theta)\)-invexity, J. Appl. Math. Stoch. Anal., 2, 175-180 (2005) · Zbl 1274.90360
[12] Zalmai, G. J., Generalized sufficiency criteria in continuous-time programming with application to a class of variational-type inequalities, J. Math. Anal. Appl., 153, 331-355 (1990) · Zbl 0718.49018
[13] Behera, N.; Nahak, C.; Nanda, S., Generalized \((\rho, \theta) - \eta \)-invexity and generalized \((\rho, \theta) - \eta \)-invariant-monotonicity, Nonlinear Anal., 68, 2495-2506 (2008) · Zbl 1211.90178
[14] Varaiya, P. P., Nonlinear programming in Banach space, SIAM J. Appl. Math., 15, 284-293 (1967) · Zbl 0171.18004
[15] Ritter, K., Duality for nonlinear programming in a Banach space, SIAM J. Appl. Math., 15, 294-302 (1967) · Zbl 0152.18404
[16] Mishra, S. K.; Rueda, N. G., Higher order generalized invexity and duality in mathematical programming, J. Math. Anal. Appl., 247, 173-182 (2000) · Zbl 1056.90136
[17] Mond, B.; Zhang, J., Higher order invexity and duality in mathematical programming, (Crouzeix, J. P., Generalized Convexity, Generalized Monotonicity: Recent Results, 372 (1998), Kluwer Academic: Kluwer Academic DordrechtrNorwell, MA), 357-372 · Zbl 0932.90039
[18] Lokenath, D.; Mikusinski, P., Introduction to Hilbert Spaces with Applications (1990), Academic Press, Inc. · Zbl 0715.46009
[19] Luenberger, D. G., Optimization by Vector Space Methods (1968), John Wiley and sons, Inc.: John Wiley and sons, Inc. New York · Zbl 0184.44502
[20] Mangasarian, O. L., Nonlinear Programming (1969), McGraw-Hill: McGraw-Hill New York · Zbl 0194.20201
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.