**AUTHORS:**Najeeb Abdulaleem

**Download as PDF**

In this paper, the class of E-differentiable vector optimization problems with both inequality and equality constraints is considered. For such (not necessarily) differentiable vector optimization problems, The so-called scalar and vector Wolfe E-dual problems are defined for the considered E-differentiable multiobjective programming problem with both inequality and equality constraints and several E-dual theorems are established also under (generalized) E-invexity hypotheses.

**KEYWORDS:**
E-invex set, E-invex function, E-differentiable function, Wolfe E-duality

**REFERENCES:**

[1] N. Abdulaleem: E-invexity and generalized Einvexity in E-differentiable multiobjective programming, to be published.

[2] T. Antczak, N. Abdulaleem: Optimality conditions for E-differentiable vector optimization problems with the multiple interval-valued objective function, to be published.

[3] T. Antczak: r-preinvexity and r-invexity in mathematical programming, J. Comput. Math. Appl. 50(3-4), (2005), 551-566.

[4] T. Antczak: Optimality and duality for nonsmooth multiobjective programming problems with V-r-invexity, J. Global Optim. 45 (2009) 319334.

[5] A. Ben-Israel, B. Mond: What is invexity?, J. Austral. Math. Soc. Ser. B 28 (1986) 1-9.

[6] M. S. Bazaraa, H. D. Sherali, and C. M. Shetty: Nonlinear Programming: Theory and Algorithms. John Wiley and Sons, New York 1991.

[7] G. R. Bitran: Duality for nonlinear multiplecriteria optimization problems, J. Optim. Theory Appl. 35 (1981), 367-401.

[8] B. D. Craven: Invex functions and constrained local minima, Bull. Aust. Math. Soc. 25 (1981), 37-46.

[9] B. D. Craven and B.M. Glover: Invex functions and duality, J. Aust. Math. Soc. (Series A) 39 (1985), 1-20.

[10] B. D. Craven: A modified Wolfe dual for weak vector minimization, Numer. Fund. Anal. Optim. 10 (1989), 899-907.

[11] W. S. Dorn: A duality theorem for convex programs, IBM J. Res. Dev. (4) 1960, 407-413.

[12] R. R. Egudo and M. A. Hanson: Multiobjective duality with invexity, J. Math. Anal. Appl. 126 (1987), 469-477.

[13] M. A. Hanson: On sufficiency of the KuhnTucker conditions, J. Math. Anal. Appl. (1981), 545-550.

[14] V. Jeyakumar, B. Mond: On generalized convex mathematical programming, J. Aust. Math. Soc. Ser. B 34 (1992), 43-53.

[15] V. Jeyakumar: Equivalence of a saddle-points and optima, and duality for a class of nonsmooth nonconvex problems, J. Math. Anal. Appl. 130 (1988), 334-343.

[16] D. T. Luc, C. Malivert: Invex optimisation problems, Bull. Aust. Math. Soc. 46 (1992), 47-66.

[17] O. L. Mangasarian: Nonlinear programming, Society for Industrial and Applied Mathematics, (1994).

[18] A. A. Megahed, H. G. Gomaa, E. A. Youness, A. Z. El-Banna, Optimality conditions of E-convex programming for an E-differentiable function, J. Inequal. Appl., 2013 (2013), 246.

[19] P. Wolfe, A duality theorem for non-linear programming, Quart. appl. math. 19 (1961), 239- 244.

[20] X. M. Yang: On E-convex sets, E-convex functions, and E-convex programming, J. Optim. Theory Appl. 109 (2001), 699-704.

[21] E. A. Youness: E-convex sets, E-convex functions, and E-convex programming, J. Optim. Theory Appl. 102 (1999), 439-450