\(d-\rho -(\eta ,\theta )\)-invexity in multiobjective optimization. (English) Zbl 1155.90451

Summary: A generalization of convexity is considered in the case of multiobjective optimization problems, where the functions involved are non-differentiable. Under \(d-\rho -(\eta ,\theta )\)-invexity assumptions on the functions involved, weak, strong and converse duality results are proved to relate weak Pareto (efficient) solutions of the multiobjective programming problems (PVP),(DVP) and (MWD). We have also established the Karush-Kuhn-Tucker sufficient optimality condition.


90C29 Multi-objective and goal programming
26B25 Convexity of real functions of several variables, generalizations
49N15 Duality theory (optimization)
Full Text: DOI


[1] Antczak, T., Multiobjective programming under \(d\)-invexity, Eur. J. oper. res., 137, 28-36, (2002) · Zbl 1027.90076
[2] Ben-Isreal, A.; Mond, B., What is invexity?, J. aust. math. soc. ser. B, 28, 1-9, (1986) · Zbl 0603.90119
[3] Craven, B.D., Generalized concavity and duality, (), 473-489 · Zbl 0488.49022
[4] Das, L.N.; Nanda, S., Proper efficiency conditions and duality for multiobjective programming problems involving semilocally invex functions, Optimization, 34, 43-51, (1995) · Zbl 0857.90111
[5] Egudo, R.R.; Hanson, M.A., Multiobjective duality with invexity, J. math. anal. appl., 126, 469-477, (1987) · Zbl 0635.90086
[6] Hanson, M.A., On sufficiency of kunh – tucker conditions, J. math. anal. appl., 80, 545-550, (1981) · Zbl 0463.90080
[7] M.A. Hanson, B. Mond, Self-Duality and Invexity FSU Statistics Report M 716, Florida State University, Department of Statistics, Tahhahassee, Florida 32306-3033, 1986
[8] Mangasarian, O.L., Nonlinear programming, (1969), McGraw-Hill New York · Zbl 0194.20201
[9] Mishra, S.K.; Wang, S.Y.; Lai, K.K., Nondifferentiable multiobjective programming under generalized \(d\)-univexity, Eur. J. oper. res., 160, 218-226, (2005) · Zbl 1067.90150
[10] Mond, B.; Weir, T., Generalized concavity and duality, (), 263-279 · Zbl 0538.90081
[11] Nahak, C.; Nanda, S., Multiobjective duality with \(\rho -(\eta, \theta)\)-invexity, J. appl. math. stoch. anal., 2005, 2, 175-180, (2005) · Zbl 1274.90360
[12] Reiland, T.W., Nonsmooth invexity, Bulletin of the austral. math. soc., 42, 437-446, (1990) · Zbl 0711.90072
[13] Xu, Z., Mixed type duality in multiobjective programming problems, J. math. anal. appl., 198, 621-635, (1996) · Zbl 0847.90131
[14] Ye, Y.L., D-invexity and optimality conditions, J. math. anal. appl., 153, 242-249, (1991) · Zbl 0755.90074
[15] 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
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.