Abstract
Many applications require the optimization of multiple conflicting goals at the same time. Such a problem can be modeled as a vector optimization problem. Vector optimization deals with the problem of finding efficient elements of a vector-valued function. In that sense, vector optimization generalizes the concept of scalar optimization. In scalar optimization, there is only one concept for efficiency which characterizes efficient elements, namely the solution which generates the smallest function value. But, due to the lack of a total order in general spaces, order relations that are defined within the optimality concept need to be chosen. In this chapter, we discuss several solution concepts for a vector optimization problem. In particular, solution concepts for vector optimization problem equipped with a variable domination structure are studied. Moreover, we present some existence results for solutions of vector optimization problems.
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References
H.P. Benson, An improved definition of proper efficiency for vector maximization with respect to cones. J. Optim. Theory Appl. 71, 232–241 (1979)
J. Borwein, Proper efficient points for maximization with respect to cones. SIAM J. Control Optim. 15(1), 57–63 (1977)
V. Chankong, Y.Y. Haimes, Multiobjective Decision Making: Theory and Methodology (Elsevier Science Publishing Co., Inc., New York, 1983)
F.Y. Edgeworth, Mathematical Psychics (Kegan Paul, London, 1881)
M. Ehrgott, Multicriteria Optimization, Lecture Notes in Economics and Mathematical Sciences, vol. 491 (Springer, Berlin, 2000)
G. Eichfelder, Variable Ordering Structures in Vector Optimization, Habilitation Thesis, University Erlangen-Nürnberg, 2011
G. Eichfelder, Variable ordering structures in vector optimization, in Recent Developments in Vector Optimization, ed. by Q.H. Ansari, J.-C. Yao (Springer, Heidelberg, 2012), pp. 95–126
G. Eichfelder, Variable Ordering Structures in Vector Optimization (Springer, Berlin, 2014)
A. Engau, Definition and characterization of Geoffrion proper efficiency for real vector optimization with infinitely many criteria. J. Optim. Theory Appl. 165, 439–457 (2015)
A.M. Geoffrion, Proper efficiency and the theory of vector maximization. J. Math. Anal. Appl. 22, 618–630 (1968)
J. Jahn, Vector Optimization: Theory, Applications, and Extensions (Springer, Berlin, Heidelberg, 2004)
H.W. Kuhn, A.W. Tucker, Nonlinear programming, in Proceedings of the second Berkeley Symposium on Mathematical Statistics and Probability, 1950 (University of California Press, Berkeley, 1951), pp. 481–492
D.T. Luc, Structure of the efficient point set. Proc. Amer. Math. Soc. 95(3), 433–440 (1985)
D.T. Luc, Theory of Vector Optimization, Lecture Notes in Economics and Mathematical Systems, vol. 319 (Springer, Berlin, 1989)
D.T. Luc, Multiobjective Linear Programming: An Introduction (Springer, Berlin, 2016)
V. Pareto, Manuale di economia politica (Societa Editrice Libraria, Milano, 1906). English translation: V. Pareto, Manual of Bibliography 467 political economy, translated by A.S. Schwier, Augustus M. Kelley Publishers, New York (1971)
Y. Sawaragi, H. Nakayama, T. Tanino, Theory of Multiobjective Optimization (Academic Press Inc., Orlando, 1985)
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Ansari, Q.H., Köbis, E., Yao, JC. (2018). Solution Concepts in Vector Optimization. In: Vector Variational Inequalities and Vector Optimization. Vector Optimization. Springer, Cham. https://doi.org/10.1007/978-3-319-63049-6_3
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DOI: https://doi.org/10.1007/978-3-319-63049-6_3
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