Abstract
A class of scalarizations of vector optimization problems is studied in order to characterize weakly efficient, efficient, and properly efficient points of a nonconvex vector problem. A parallelism is established between the different solutions of the scalarized problem and the various efficient frontiers. In particular, properly efficient points correspond to stable solutions with respect to suitable perturbations of the feasible set.
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KUHN, H. W., and TUCKER, A.W., Nonlinear Programming, 2nd Berkeley Symposium on Mathematical Statistics and Probability, Berkeley, California, pp. 481–492, 1951.
GUERRAGGIO, A., MOLHO, E., and ZAFFARONI, A., On the Notion of Proper Efficiency in Vector Optimization, Journal of Optimization Theory and Applications, Vol. 82, pp. 1–21, 1994.
SAWARAGI, Y., NAKAYAMA, H., and TANINO, T., Theory of Multiobjective Optimization, Academic Press, New York, NY, 1985.
WIERZBICKI, A. P., The Use of Reference Objectives in Multiobjective Optimization, Multiple-Criteria Decision Making: Theory and Applications, Edited by G. Fandel and T. Gal, Lecture Notes in Economics and Mathematical Systems, Springer, Berlin, Germany, pp. 468–486, 1980.
LUC, D. T., Theory of Vector Optimization, Lecture Notes in Economics and Mathematical Systems, Springer, Berlin, Germany, Vol. 319, 1989.
JAHN, J., Mathematical Vector Optimization in Partially-Ordered Linear Spaces, Peter Lang Verlag, Frankfurt am Main, Germany, 1986.
AMAHROQ, T., and TAA, A., On Lagrange-Kuhn-Tucker Multipliers for Multiobjective Optimization Problems, Optimization, Vol. 41, pp. 159–172, 1997.
CILIGOT-TRAVAIN, M., On Lagrange-Kuhn-Tucker Multipliers for Pareto Optimization Problems, Numerical Functional Analysis and Optimization, Vol. 15, pp. 689–693, 1994.
MIGLIERINA, E., Characterization of Solutions of Multiobjective Optimization Problems, Rendiconti del Circolo Matematico di Palermo, Vol. 50, 2001.
BORWEIN, J. M., The Geometry of Pareto Efficiency over Cones, Mathematische Operationforschung und Statistik, Serie Optimization, Vol. 11, pp. 235–248, 1980.
SONNTAG, Y., and ZALINESCU, C., Set Convergences: A Survey and a Classification, Set-Valued Analysis, Vol. 2, pp. 339–356, 1994.
DOLECKI, S., and MALIVERT, C., Stability of Efficient Sets: Continuity of Mobile Polarities, Nonlinear Analysis: Theory, Methods and Applications, Vol. 12, pp. 1461–1486, 1988.
PENOT, J. P., and STERNA-KARWAT, A., Parametrized Multicriteria Optimization: Continuity and Closedness of Optimal Multifunctions, Journal of Mathematical Analysis and Applications, Vol. 120, pp. 150–168, 1986.
LUC, D. T., LUCCHETTI, R., and MALIVERT, C., Convergence of the Efficient Sets, Set-Valued Analysis, Vol. 2, pp. 207–218, 1994.
LUC, D. T., Recession Cones and the Domination Property in Vector Optimization, Mathematical Programming, Vol. 49, pp. 113–122, 1990.
BAZARAA, M. S., and SHETTY, C. M., Foundations of Optimization, Lecture Notes in Economics and Mathematical Systems, Springer, Berlin, Germany, Vol. 122, 1976.
HENIG, M. I., Proper Efficiency with Respect to Cones, Journal of Optimization Theory and Applications, Vol. 36, pp. 387–407, 1982.
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Miglierina, E., Molho, E. Scalarization and Stability in Vector Optimization. Journal of Optimization Theory and Applications 114, 657–670 (2002). https://doi.org/10.1023/A:1016031214488
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DOI: https://doi.org/10.1023/A:1016031214488