Abstract
We study necessary and sufficient conditions to attain solutions of set optimization problems in terms of variational inequalities of Stampacchia and Minty type. The notion of solution we deal with has been introduced by Heyde and Löhne in 2011. To define the set-valued variational inequality, we introduce a set-valued directional derivative that we relate to Dini derivatives of a family of scalar problems. Optimality conditions are given by Stampacchia and Minty type variational inequalities, defined both by set-valued directional derivatives and by Dini derivatives of the scalarizations. The main results allow to obtain known variational characterizations for vector optimization problems as special cases.
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References
Aliprantis, C.D., Border, K.C.: Infinite Dimensional Analysis: A Hitchhiker’s Guide. Studies in Economic Theory, vol. 4. Springer, Berlin (2006)
Aubin, J.-P., Frankowska, H.: Set-Valued Analysis. Systems & Control: Foundations & Applications, vol. 2. Birkhäuser Boston Inc., Boston, MA (1990)
Crespi, G.P., Ginchev, I., Rocca, M.: First-order optimality conditions in set-valued optimization. Math. Methods Oper. Res. 63(1), 87–106 (2006)
Crespi, G.P., Ginchev, I., Rocca, M.: Some remarks on the minty vector variational principle. J. Math. Anal. Appl. 345(1), 165–175 (2008)
Dedekind, R.: Stetigkeit und irrationale Zahlen. In: Fricke, R., Noether, E., Ore, Ö. (eds.) Richard Dedekind Gesammelte Mathematische Werke, 5th edn 1927 (1872). Druck und Verlag von Friedr. Vieweg & Sohn Akt.-Ges, Braunschweig (1932)
Fuchs, L.: Teilweise Geordnete Algebraische Strukturen. Vandenhoeck u. Ruprecht, Göttingen (1966)
Galatos, N.: Residuated Lattices: An Algebraic Glimpse at Substructural Logics. Elsevier Science Ltd, Boston (2007)
Getan, J., Martinez-Legaz, J.E., Singer, I.: (*, s)-Dualities. J. Math. Sci. 115(4), 2506–2541 (2003)
Giannessi, F.:Theorems of alternative, quadratic programs and complementarity problems, in Variational inequalities and complementarity problems. In: Proc. Internat. School, Erice, pp. 151–186. Wiley, Chichester, 1978 (1980)
Giannessi, F.: On Minty variational principle, in New Trends in Mathematical Programming. Series on Applied Optimization, 13:93–99. Kluwer Academic Publishers, Boston (1998)
Göpfert, A., Riahi, H., Tammer, C., Zălinescu, C.: Variational Methods in Partially Ordered Spaces. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 17. Springer, New York (2003)
Hamel, A.H.: Variational Principles on Metric and Uniform Spaces. Habilitation thesis, MLU Halle-Wittenberg, Halle (2005)
Hamel, A.H.: A duality theory for set-valued functions i: fenchel conjugation theory. Set-Valued Var. Anal. 17(2), 153–182 (2009)
Hamel, A.H., Schrage, C.: Notes on extended real- and set-valued functions. J. Convex Anal. 19(2), 355–384 (2012)
Hamel, A.H., Schrage, C.: Directional derivative and subdifferentials of set–valued convex functions. Pac. J. Optim. 10(4), 667–689 (2014)
Hernández, E., Rodríguez-Marín, L.: Duality in set optimization with set-valued maps. Pac. J. Optim. 3(2), 245–255 (2007)
Hernández, E., Rodríguez-Marín, L.: Existence theorems for set optimization problems. Nonlinear Anal. 67(6), 1726–1736 (2007)
Heyde, F., Löhne, A.: Solution concepts in vector optimization. a fresh look at an old story. Optim. 60(12), 1421–1440 (2011)
Heyde, F., Schrage, C.: Continuity of set-valued maps and a fundamental duality formula for set-valued optimization. J. Math. Anal. Appl. 397(2), 772–784 (2013)
Jahn, J.: Vector Optimization: Theory, Applications, and Extensions. Springer, Berlin (2004)
Löhne, A.: Vector Optimization with Infimum and Supremum. Springer, Berlin (2011)
Martınez-Legaz, J.E., Singer, I.: Dualities associated to binary operations on \(\bar{R}\). J. Convex Anal. 2(1–2), 185–209 (1995)
Rockafellar, R.T., Wets, R.J.B.: Variational Analysis. Springer, Berlin (2004)
Schrage, C.: Set-valued Convex Analyis. Ph.D. thesis, MLU Halle-Wittenberg, Halle (2009)
Schrage, C.: Scalar representation and conjugation of set-valued functions. Optim. 64(2), 197–223 (2015)
Yang, X.M., Yang, X.Q., Teo, K.L.: Some remarks on the minty vector variational inequality. J. Optim. Theory Appl. 121, 193–201 (2004)
Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific Publishing Co., Inc., River Edge (2002)
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We thank the two anonymous referees for their valuable comments that helped to improve the manuscript.
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Crespi, G.P., Schrage, C. (2015). Set Optimization Meets Variational Inequalities. In: Hamel, A., Heyde, F., Löhne, A., Rudloff, B., Schrage, C. (eds) Set Optimization and Applications - The State of the Art. Springer Proceedings in Mathematics & Statistics, vol 151. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48670-2_7
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DOI: https://doi.org/10.1007/978-3-662-48670-2_7
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