[KSCI] Korea Science Citation Index Service

http://dx.doi.org/10.4134/JKMS.2013.50.4.727

SECOND-ORDER UNIVEX FUNCTIONS AND GENERALIZED DUALITY MODELS FOR MULTIOBJECTIVE PROGRAMMING PROBLEMS CONTAINING ARBITRARY NORMS

Zalmai, G.J. (Department of Mathematics and Computer Science, Northern Michigan University)

Publication Information

Journal of the Korean Mathematical Society / v.50, no.4, 2013 , pp. 727-753 More about this Journal

Abstract

In this paper, we introduce three new broad classes of second-order generalized convex functions, namely, ( $\mathcal{F}$ , $b$ , ${\phi}$ , ${\rho}$ , ${\theta}$ )-sounivex functions, ( $\mathcal{F}$ , $b$ , ${\phi}$ , ${\rho}$ , ${\theta}$ )-pseudosounivex functions, and ( $\mathcal{F}$ , $b$ , ${\phi}$ , ${\rho}$ , ${\theta}$ )-quasisounivex functions; formulate eight general second-order duality models; and prove appropriate duality theorems under various generalized ( $\mathcal{F}$ , $b$ , ${\phi}$ , ${\rho}$ , ${\theta}$ )-sounivexity assumptions for a multiobjective programming problem containing arbitrary norms.

Keywords

multiobjective programming; generalized ( $\mathcal{F}$ , b, ${\phi}$ , ${\rho}$ , ${\theta}$ )-sounivex functions; arbitrary norms; dual problems; duality theorems;

Citations & Related Records

Reference

1	I. Ahmad and Z. Husain, Nondifferentiable second-order symmetric duality, Asia-Pac. J. Oper. Res. 22 (2005), no. 1, 19-31. DOI ScienceOn
2	I. Ahmad and Z. Husain, Second order (F, ${\alpha}$ , ${\rho}$ , d)-convexity and duality in multiobjective programming, Inform. Sci. 176 (2006), no. 20, 3094-3103. DOI ScienceOn
3	I. Ahmad and Z. Husain, Second order symmetric duality in multiobjective programming involving cones, Optim. Lett. (2012) (forthcoming).
4	I. Ahmad and Z. Husain, On multiobjective second order symmetric duality with cone constraints, European J. Oper. Res. 204 (2010), no. 3, 402-409. DOI ScienceOn
5	C. R. Bector and B. K. Bector, (Generalized)-binvex functions and second order duality for a nonlinear programming problem, Congr. Numer. 52 (1986), 37-52.
6	C. R. Bector and B. K. Bector, On various duality theorems for second-order duality in nonlinear programming, Cahiers Centre Etudes Rech. Oper. 28 (1986), no. 4, 283-292.
7	C. R. Bector and S. Chandra, Second-order duality for generalized fractional programming, Contributions to mathematical economics and optimization theory, 11-28, Methods Oper. Res., 56, Athenaum/Hain/Hanstein, Konigstein, 1986.
8	C. R. Bector and S. Chandra, Second order symmetric and self dual programs, Opsearch 23 (1986), no. 2, 89-95.
9	C. R. Bector and S. Chandra, First and second order duality for a class of nondifferentiable fractional programming problems, J. Inform. Optim. Sci. 7 (1986), no. 3, 335-348.
10	C. R. Bector and S. Chandra, (Generalized)-bonvexity and higher order duality for fractional programming, Opsearch 24 (1987), no. 3, 143-154.
11	C. R. Bector, S. Chandra, S. Gupta, and S. K. Suneja, Univex sets, functions, and univex nonlinear programming, in Generalized convexity (Pecs, 1992), 3-18, Lecture Notes in Econom. and Math. Systems, 405, Springer, Berlin, 1994.
12	C. R. Bector, S. Chandra, and I. Husain, Second-order duality for a minimax programming problem, Opsearch 28 (1991), 249-263.
13	R. R. Egudo and M. A. Hanson, Second order duality in multiobjective programming, Opsearch 30 (1993), 223-230.
14	T. R. Gulati and I. Ahmad, Second order symmetric duality for nonlinear minimax mixed integer programming problems, European J. Oper. Res. 101 (1997), no. 1, 122-129. DOI ScienceOn
15	T. R. Gulati, I. Ahmad, and I. Husain, Second order symmetric duality with generalized convexity, Opsearch 38 (2001), no. 2, 210-222. DOI
16	M. Hachimi and B. Aghezzaf, Second order duality in multiobjective programming in-volving generalized type-I functions, Numer. Funct. Anal. Optim. 25 (2004), no. 7-8. 725-736.
17	M. A. Hanson, Second order invexity and duality in mathematical programming, Opsearch 30 (1993), 313-320.
18	R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, New York, 1985.
19	I. Husain and Z. Jabeen, Second order duality for fractional programming with support functions, Opsearch 41 (2004), no. 2, 121-134. DOI
20	V. Jeyakumar, p-convexity and second order duality, Util. Math. 29 (1986), 71-85.
21	V. Jeyakumar, First and second order fractional programming duality, Opsearch 22 (1985), no. 1, 24-41.
22	J. C. Liu, Second order duality for minimax programming, Util. Math. 56 (1999), 53-63.
23	O. L. Mangasarian, Second- and higher-order duality in nonlinear programming, J. Math. Anal. Appl. 51 (1975), no. 3, 607-620. DOI ScienceOn
24	S. K. Mishra, Second order generalized invexity and duality in mathematical programming, Optimization 42 (1997), no. 1, 51-69. DOI ScienceOn
25	S. K. Mishra, Second order symmetric duality in mathematical programming with F-convexity, European J. Oper. Res. 127 (2000), no. 3, 507-518. DOI ScienceOn
26	S. K. Mishra and N. G. Rueda, Higher-order generalized invexity and duality in mathematical programming, J. Math. Anal. Appl. 247 (2000), no. 1, 173-182. DOI ScienceOn
27	B. Mond, Second order duality for nonlinear programs, Opsearch 11 (1974), no. 2-3, 90-99.
28	B. Mond and T. Weir, Generalized convexity and higher-order duality, J. Math. Sci. 16-18 (1981-1983), 74-94.
29	B. Mond and T. Weir, Generalized concavity and duality, in Generalized Concavity in Optimization and Economics, S. Schaible and W. T. Ziemba, editors, pp. 263-279, Academic Press, New York, 1981.
30	B. Mond and J. Zhang, Duality for multiobjective programming involving second-order V-invex functions, in Proceedings of the Optimization Miniconference II, B. M. Glover and V. Jeyakumar, editors, pp. 89-100, University of New South Wales, Sydney, Australia, 1995.
31	B. Mond and J. Zhang, Higher order invexity and duality in mathemaical programming, in Generalized Convexity, Generalized Monotonicity: Recent Results, J. P. Crouzeix, et al., editors, pp. 357-372, Kluwer Academic Publishers, printed in the Netherlands, 1998.
32	R. B. Patel, Second order duality in multiobjective fractional programming, Indian J. Math. 38 (1996), no. 1, 39-46.
33	M. N. Vartak and I. Gupta, Duality theory for fractional programming problems under-convexity, Opsearch 24 (1987), 163-174.
34	X. M. Yang, X. Q. Yang, and K. L. Teo, Huard type second-order converse duality for nonlinear programming, Appl. Math. Lett. 18 (2005), no. 2, 205-208. DOI ScienceOn
35	X. Yang, Generalized convex duality for multiobjective fractional programs, Opsearch 31 (1994), no. 2, 155-163.
36	X. Q. Yang, Second-order global optimality conditions for convex composite optimization, Math. Programming 81 (1998), no. 3, Ser. A, 327-347.
37	X. M. Yang, X. Q. Yang, and K. L. Teo, Nondifferentiable second order symmetric duality in mathematical programming with F-convexity, European J. Oper. Res. 144 (2003), 554-559. DOI ScienceOn
38	X. M. Yang, X. Q. Yang, K. L. Teo, and S. H. Hou, Second order duality for nonlinear programming, Indian J. Pure Appl. Math. 35 (2004), no. 5, 699-708.
39	G. J. Zalmai, Optimality conditions and duality models for a class of nonsmooth constrained fractional variational problems, Optimization 30 (1994), no. 1, 15-51. DOI ScienceOn
40	G. J. Zalmai, Generalized ( ${\eta},{\rho}$ )-invex functions and global semiparametric sufficient efficiency conditions for multiobjective fractional programming problems containing arbitrary norms, J. Global Optim. 36 (2006), no. 1, 51-85. DOI
41	G. J. Zalmai, Generalized second-order ( $F,{\beta},{\phi},{\rho},{\theta}$ )-univex functions and parametric duality models in semiinfinite discrete minmax fractional programming, Adv. Nonlinear Var. Inequal. 15 (2012), no. 2, 63-91.
42	J. Zhang and B. Mond, Second order b-invexity and duality in mathematical programming, Util. Math. 50 (1996), 19-31.
43	B. Aghezzaf, Second order mixed type duality in multiobjective programming problems, J. Math. Anal. Appl. 285 (2003), no. 1, 97-106. DOI ScienceOn
44	J. Zhang and B. Mond, Second order duality for multiobjective nonlinear programming involving generalized convexity, in Proceedings of the Optimization Miniconference III, B. M. Glover, B. D. Craven, and D. Ralph, editors, pp. 79-95, University of Ballarat, 1997.