Abstract
It is proven that a proper closed convex function with values in the power set of a preordered, separated locally convex space is the pointwise supremum of its set-valued affine minorants. A new concept of Legendre–Fenchel conjugates for set-valued functions is introduced and a Moreau–Fenchel theorem is proven. Examples and applications are given, among them a dual representation theorem for set-valued convex risk measures.
Similar content being viewed by others
References
Azimov, A.Y.: Duality of multiobjective problems. Math. USSR Sbornik 59(2), 515–531 (1988)
Azimov, A.Y.: Duality for set-valued multiobjective optimization problems, part 1: mathematical programming. J. Optim. Theory Appl. 137, 61–74 (2008)
Borwein, J.M.: A Lagrange multiplier theorem and a sandwich theorem for convex relations. Math. Scand. 48, 189–204 (1981)
Borwein, J.M.: Continuity and differentiability properties of convex operators. Proc. London Math. Soc., 44(3), 420–444 (1982)
Borwein, J.M.: Subgradients of convex operators. Math. Oper.forsch. Stat. Ser. Optim. 15(2), 179–191 (1984)
Borwein, J., Penot, J.P., Thera, M.: Conjugate convex operators. J. Math. Anal. Appl. 102, 399–414 (1984)
Breckner, W.W., Kolumbán, I.: Konvexe Optimierungsaufgaben in topologischen Vektorräumen. Math. Scand. 25, 227–247 (1969)
Brink, C.: Power structures. Algebra Univers. 30, 177–216 (1993)
Brumelle, S.L.: Convex operators and supports. Math. Oper. Res. 3(2), 171–175 (1978)
Brumelle, S.L.: Duality for multiple objective convex programs. Math. Oper. Res. 6(2), 159–172 (1981)
Cascos, I., Molchanov, I.: Multivariate risks and depth-trimmed regions. Finance Stoch. 11, 373–397 (2007)
Chen, G., Huang, X., Yang, Y.: Vector optimization. In: Lecture Notes in Economics and Mathematical Systems, no. 541. Springer, Berlin Heidelberg New York (2005)
Dolecki, S., Malivert, C.: General duality in vector optimization. Optimization 27(1–2), 97–119 (1993)
Ekeland, I., Temam, R.: Convex analysis and variational problems. In: Studies in Mathematics and its Applications, vol. 1. North Holland, Elsevier (1976)
Elster, K.-H., Nehse, R.: Konjugierte operatoren und subdifferentiale. Math. Oper.forsch. Stat. 6(4), 641–657 (1975)
Fel’dman, M.M.: On sufficient conditions for the existence of supports to sublinear operators. Sib. Math. J. 16(1), 106–111 (1975), [translation from Sib. Mat. Zh. 16, 132–138 (1975)]
Föllmer, H., Schied, A.: Stochastic finance. In: de Gruyter Studies in Mathematics, Extended Edition, vol. 27. Walter de Gruyter, Berlin (2004)
Fuchssteinert, B., Lusky, W.: Convex cones. In: Mathematics Study, vol. 58. North Holland, Amsterdam (1981)
Godini, G.: A framework for best simultaneous approximation: normed almost linear spaces. J. Approx. Theory 43, 338–358 (1985)
Göpfert, A., Riahi, H., Tammer, C., Zălinescu, C.: Variational methods in partially ordered spaces. In: CMS Books in Mathematics, vol. 17. Springer, New York (2003)
Hamel, A.H.: Variational principles on metric and uniform spaces. Habilitationsschrift. Martin-Luther-Universität Halle-Wittenberg. http://sundoc.bibliothek.uni-halle.de/habil-online/05/05H167/habil.pdf (2005)
Hamel, A.H., Heyde, F., Höhne, M.: Set-valued measures of risk. In: Report on Optimization and Stochastics 15. Martin-Luther-Universität Halle-Wittenberg. http://www.mathematik.uni-halle.de/reports/sources/2007/07-15report.pdf (2007)
Heyde, F., Löhne, A.: Geometric duality in multiple objective linear programming. Report on optimization and stochastics 15. Martin-Luther-Universität Halle-Wittenberg. http://ito.mathematik.uni-halle.de/~loehne/pdf/ (2006)
Heyde, F., Löhne, A., Tammer, C.: Set-valued duality theory for multiple objective linear programs and application to mathematical finance. Report on optimization and stochastics 4. Martin-Luther-Universität Halle-Wittenberg. http://ito.mathematik.uni-halle.de/~loehne/pdf/ (2007)
Ioffe, A.D., Levin, V.L.: Subdifferentials of convex functions. Tr. Mosk. Mat. Obs. 26, 3–73 (1972)
Jahn, J.: Scalarization in vector optimization. Math. Program. 29(2), 203–218 (1984)
Jahn, J.: Vector optimization. In: Theory, Applications, and Extensions. Springer, Berlin (2004)
Jouini, E., Meddeb, M., Touzi, N.: Vector-valued coherent risk measures. Finance Stoch. 8(4), 531–552 (2004)
Kannai, Y., Peleg, B.: A note on the extension of an order on a set to the power set. J. Econom. Theory 32(1), 172–175 (1984)
Kantorovitch, L.: The method of successive approximations for functional equations. Acta Math. 71, 63–97 (1939)
Kasahara, S.: On formulations of topological linear spaces by topological semifields. Math. Japon. 19, 121–134 (1974)
Kuroiwa, D., Tanaka, T., Truong, X.D.H.: On cone convexity of set-valued maps. Nonlinear Anal. 30(3), 1487–1496 (1997)
Linke, Yu.È.: Sublinear operators without subdifferentials. Sibirsk. Mat. Zh. 32(3), 219–221 (1991)
Löhne, A.: Optimization with set relations. Ph.D. thesis, Martin-Luther-Universität Halle-Wittenberg (2005)
Löhne, A.: Optimization with set relations: conjugate duality. Optimization 54(3), 265–282 (2005)
Löhne, A.: On convex functions with values in conlinear spaces. J. Nonlinear Convex Anal. 7(1), 115–122 (2006)
Löhne, A., Tammer, C.: A new approach to duality in vector optimization. Optimization 56(1) (2007)
Luc, D.T.: On duality theory in multiobjective programming. J. Optim. Theory Appl. 43(4), 557–582 (1984)
Luc, D.T.: Theory of Vector Optimization. In: Lecture Notes in Economics and Mathematical Systems, vol. 319. Springer, Berlin (1989)
Malivert, C.: Fenchel duality in vector optimization. In: Advances in Optimization (Lambrecht, 1991). Lecture Notes in Econom. and Math. Systems, vol. 382, pp. 420–438. Springer, Berlin (1992)
Postolică, V.: A generalization of Fenchel’s duality theorem. Ann. Sci. Math. Québec 10(2), 199–206 (1986)
Postolică, V.: Vectorial optimization programs with multifunctions and duality. Ann. Sci. Math. Québec 10(1), 85–102 (1986)
Pshenichnyi, B.N.: Convex multivalued mappings and their conjugates. Kibernetika 3, 94–102 (1972)
Raffin, C.: Sur les programmes convexes définis dans des espaces vectoriels topologiques. Ann. Inst. Fourier (Grenoble) 20, 457–491 (1970)
Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin Heidelberg New York (1998)
Rubinov, A.M.: Sublinear operators and their applications. Russian Math. Surveys 32(4), 115–175 (1977) (translation from Usp. Mat. Nauk)
Sawaragi, Y., Nakayama, H., Tanino, T.: Theory of multiobjective optimization. In: Mathematics in Science and Engineering, vol. 176. Academic, Orlando (1985)
Singer, I.: Abstract convex analysis. In: Canadian Mathematical Society Series of Monographs and Advanced Texts. Wiley, New York (1997)
Song, W.: Conjugate duality in set-valued vector optimization. J. Math. Anal. Appl. 216(1), 265–283 (1997)
Song, W.: A generalization of fenchel duality in set-valued vector optimization. Math. Methods Oper. Res. 48(2), 259–272 (1998)
Song, W.: Duality in set-valued optimization. Dissertationes Math. (Rozprawy Mat.) 375, 69 (1998)
Tanino T.: Conjugate duality in vector optimization. J. Math. Anal. Appl. 167, 84–97 (1992)
Tanino T., Sawaragi, Y.: Conjugate maps and duality in multiobjective optimization. J. Optim. Theory Appl. 31(4), 473–499 (1980)
Valadier, M.: Sous-différentiabilité de fonctions convexes à valeurs dans un espace vectoriel ordonné. Math. Scand. 30, 65–74 (1972)
Zowe, J.: Subdifferentiability of convex functions with values in an ordered vector space. Math. Scand. 34, 69–83 (1974)
Zowe, J.: A duality theorem for a convex programming problem in order complete vector lattices. J. Math. Anal. Appl. 50, 273–287 (1975)
Zowe, J.: Sandwich theorems for convex operators with values in an ordered vector space. J. Math. Anal. Appl. 66(2), 282–296 (1978)
Zălinescu, C.: Duality for vectorial nonconvex optimization by convexification and applications. An. Stiint. Univ. “Al. I. Cuza” Iasi XXIX, 16–34 (1983)
Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific, Singapore (2002)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hamel, A.H. A Duality Theory for Set-Valued Functions I: Fenchel Conjugation Theory. Set-Valued Anal 17, 153–182 (2009). https://doi.org/10.1007/s11228-009-0109-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11228-009-0109-0
Keywords
- Set order relations
- Legendre–Fenchel conjugate
- Moreau–Fenchel theorem
- Set-valued function
- Conlinear space
- Set-valued risk measure