Abstract
In this paper, a new type of generalized cone convexity is introduced in the case of set-valued optimization problem where the maps involved are contingent epiderivable. It extends the notion of \(\rho -(\eta , \theta )\)-invexity from vector optimization to set-valued optimization. The sufficient Karush–Kuhn–Tucker (KKT) optimality conditions are investigated under the stated assumptions. We also study the duality results of Mond–Weir type (MWD), Wolfe type (WD) and mixed type (Mix D) for weak solutions of a pair of set-valued optimization problems.
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References
Aubin, J.P., Frankowsa, H.: Set-Valued Analysis. Birkhauser, Boston (1990)
Bao-huai, S., San-yang, L.: Kuhn–Tucker condition and Wolfe duality of preinvex set-valued optimization. Appl. Math. Mech. Engl. 27(12), 1655–1664 (2006)
Ben-Israel, A., Mond, B.: What is invexity? J. Aust. Math. Soc. (Ser. B) 28, 1–9 (1986)
Bhatia, D., Mehra, A.: Lagrangian duality for preinvex set-valued functions. J. Math. Anal. Appl. 214, 599–612 (1997)
Corley, H.W.: Existence and Lagrangian duality for maximizations of set-valued functions. J. Optim. Theory Appl. 54, 489–501 (1987)
Craven, B.D.: Generalized concavity and duality. In: Schaible, S., Ziemba, W.T. (eds.) Generalized Concavity in Optimization and Economics, pp. 473–489. Academic Press, New York (1981)
Craven, B.D.: Invex functions and constrained local minima. Bull. Aust. Math. Soc. 24, 357–366 (1981)
Hanson, M.A.: On sufficiency of Kuhn–Tucker conditions. J. Math. Anal. Appl. 80, 545–550 (1981)
Hanson, M.A., Mond, B.: Self-duality and invexity. FSU Technical Report No. M 716, Department of Statistics, Florida State University, Tahhahassee, Florida 32306–3033 (1986)
Jahn, J.: Vector Optimization: Theory. Applications and Extensions. Springer, Berlin (2003)
Jahn, J., Rauh, R.: Contingent epiderivative and set-valued optimization. Math. Methods Oper. Res. 46, 193–211 (1997)
Lin, L.: Optimization of set-valued functions. J. Math. Anal. Appl. 186, 30–51 (1994)
Luc, D.T., Malivert, C.: Invex optimization problems. Bull. Aust. Math. Soc. 49, 47–66 (1992)
Nahak, C., Mohapatra, R.N.: \(d-\rho -(\eta, \theta )-\)invexity in multiobjective optimization. Nonlinear Anal. 70, 2288–2296 (2009)
Rodriguez-Marin, L., Sama, M.: About contingent epiderivative. J. Math. Anal. Appl. 327, 745–762 (2007)
Rueda, N.G., Hanson, M.A.: Optimality criteria in mathematical programming involving generalized invexity. J. Math. Anal. Appl. 130(2), 375–385 (1988)
Sach, P.H., Craven, B.D.: Invex multifunction and duality. Numer. Funct. Anal. Opt. 12, 575–591 (1991)
Weir, T., Jeyakumar, V.: A class of nonconvex functions and mathematical programming. Bull. Aust. Math. Soc. 38(1), 177–189 (1988)
Weir, T., Mond, B.: Preinvex functions in multiple-objective optimization. J. Math. Anal. Appl. 136(1), 29–38 (1988)
Ye, Y.L.: \(d\)-invexity and optimality conditions. J. Math. Anal. Appl. 153, 242–249 (1991)
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)
Acknowledgments
The authors are grateful to the reviewers for their valuable comments which improved the presentation of the paper. The first author is thankful to Council of Scientific and Industrial Research (CSIR), India, for his financial support in executing this study.
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Das, K., Nahak, C. Sufficient optimality conditions and duality theorems for set-valued optimization problem under generalized cone convexity. Rend. Circ. Mat. Palermo 63, 329–345 (2014). https://doi.org/10.1007/s12215-014-0163-9
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DOI: https://doi.org/10.1007/s12215-014-0163-9
Keywords
- Convex cone
- Contingent epiderivative
- Set-valued optimization problem
- \(\rho -(\eta , \theta )\)-invexity
- Duality