Abstract
Trade-off information is important in multiobjective optimization. It describes the relationships of changes in objective function values. For example, in interactive methods we need information about the local behavior of solutions when looking for improved search directions.
Henig and Buchanan have generalized in Mathematical Programming 78(3), 1997 the concept of trade-offs in convex multiobjective optimization problems. With the help of tangent cones they define a cone of trade-off directions.
In this paper, we examine the possibility of extending the results of Henig and Buchanan for nonconvex multiobjective optimization problems. We carry out the generalization in the sense of Clarke’s nonconvex analysis.
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References
Amahroq, T., Taa, A. (1997), On Lagrang e-Kuhn-Tucker Multipliers for Multiobjective Optimization Problems, Optimization 41, No. 2, 159–172.
Chankong, V., Haimes, Y.Y. (1983), “Multiobjective Decision Making Theory and Methodology,” North-Holland, New York.
Clarke, F.H. (1983), “Optimization and Nonsmooth Analysis,” John Wiley & Sons, Inc., New York.
Coladas, L., Li, Z., Wang, S. (1994), Optimality Conditions for Multiobjective and Nonsmooth Minimization in Abstract Spaces, Bulletin of the Australian Mathematical Society 50, No. 2, 203–218.
Craven, B.D. (1989), Nonsmooth Multiobjective Programming, Numerical Functional Analysis and Optimization 10, No. 1–2, 49–64.
Doležal, J. (1985), Necessary Conditions for Pareto Optimality in Non-differentiable Problems, Problems of Control and Information Theory 14, No. 2, 131–141.
El Abdouni, B., Thibault, L. (1992), Lagrange Multipliers for Pareto Nonsmooth Programming Problems in Banach Spaces, Optimization 26, No. 3-4, 277–285.
Haimes, Y.Y., Chankong, V. (1979), Kuhn-Tucker Multipliers as TradeOffs in Multiobjective Decision-Making Analysis, Automática 15, No. 1, 59–72.
Halme, M., Korhonen, P. (1989), Nondominated Tradeoffs and Termination in Interactive Multiple Objective Linear Programming, in “Improving Decision Making in Organisations,” Lecture Notes in Economics and Mathematical Systems 335, Edited by Lockett, A.G, Islei, G., Springer-Verlag, Berlin,Heidelberg, 410–423.
Henig, M.I. (1982), Proper Efficiency with Respect to Cones, Journal of Optimization Theory and Applications 36, No. 3, 387–407.
Henig, M.I., Buchanan, J.T. (1997), Tradeoff Directions in Multiobjective Optimization Problems, Mathematical Programming 78, No. 3, 357–374.
Kaliszewski, I., Michalowski, W. (1995), Generation of Outcomes with Selectively Bounded Trade-Offs, Foundations of Computing and Decision Sciences 20, No. 2, 113–122.
Kaliszewski, L, Michalowski, W. (1997), Efficient Solutions and Bounds on Tradeoffs, Journal of Optimization Theory and Applications 94, No. 2, 381–394.
Mäkelä, M.M., Neittaanmäki, P. (1992), “Nonsmooth Optimization: Analysis and Algorithms with Applications to Optimal Control,” World Scientific, Singapore.
Miettinen, K. (1999), “Nonlinear Multiobjective Optimization,” Kluwer Academic Publishers, Boston.
Minami, M. (1983), Weak Pareto-Optimal Necessary Conditions in a Nondifferentiable Multiobjective Program on a Banach Space, Journal of Optimization Theory and Applications 41, No. 3, 451–461.
Rockafellar, R.T. (1970), “Convex Analysis,” Princeton University Press, Princeton, New Jersey.
Sakawa, M., Yano, H. (1990), Trade-Off Rates in the Hyperplane Method for Multiobjective Optimization Problems, European Journal of Operational Research 44, No. 1, 105–118.
Wang, S. (1984), Lagrange Conditions in Nonsmooth and Multiobjective Mathematical Programming, Mathematics in Economics 1, 183–193.
Yano, H., Sakawa, M. (1987), Trade-Off Rates in the Weighted Tchebycheff Norm Method, Large Scale Systems 13, No. 2, 167–177.
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Miettinen, K., Mäkelä, M.M. (2000). Tangent and Normal Cones in Nonconvex Multiobjective Optimization. In: Haimes, Y.Y., Steuer, R.E. (eds) Research and Practice in Multiple Criteria Decision Making. Lecture Notes in Economics and Mathematical Systems, vol 487. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-57311-8_9
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DOI: https://doi.org/10.1007/978-3-642-57311-8_9
Publisher Name: Springer, Berlin, Heidelberg
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