Abstract
In this paper, we study nonnegative, increasing, co-radiant and quasi-concave functions over real locally convex topological vector spaces. Such functions have frequently been employed in microeconomic analysis. We next characterize the abstract concavity, the upper support set and the superdifferential of this class of functions by applying a type of duality, which is used in microeconomic theory.
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References
Rubinov, A.M.: Abstract Convexity and Global Optimization. Kluwer Academic Publishers, Dordrecht-Boston-London (2000)
Martínez-Legaz, J.E.: Quasiconvex duality theory by generalized conjugation methods. Optimization 19, 603–652 (1988)
Martínez-Legaz, J.E., Rubinov, A.M., Schaible, S.: Increasing quasi-concave co-radiant functions with applications in mathematical economics. Math. Meth. Oper. Res. 61, 261–280 (2005)
Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)
Singer, I.: Abstract Convex Analysis. Wiley-Interscience, Hoboken (1997)
Dutta, J., Martínez-Legaz, J.E., Rubinov, A.M.: Monotonic analysis over cones: I. Optimization 53, 165–177 (2004)
Dutta, J., Martínez-Legaz, J.E., Rubinov, A.M.: Monotonic analysis over cones: II. Optimization 53, 529–547 (2004)
Mohebi, H., Sadeghi, H.: Monotonic analysis over ordered topological vector spaces: I. Optimization 56, 305–321 (2007)
Mohebi, H., Sadeghi, H.: Monotonic analysis over ordered topological vector spaces: II. Optimization 58, 241–249 (2009)
Rubinov, A.M., Glover, B.M.: Increasing convex-along-rays functions with application to global optimization. J. Optim. Theory Appl. 102, 615–642 (1999)
Daryaei, M.H., Mohebi, H.: Abstract convexity of extended real valued ICR functions. Optimization 62, 835–855 (2013)
Doagooei, A.R., Mohebi, H.: Monotonic analysis over ordered topological vector spaces: IV. J. Glob. Optim. 45, 355–369 (2009)
Dutta, J., Martínez-Legaz, J.E., Rubinov, A.M.: Monotonic analysis over cones: III. J. Convex Anal. 15, 581–592 (2008)
Avriel, M., Diewert, W.E., Schaible, S., Zang, I.: Generalized Concavity. Plenum Publishing Corporation, New York (1988)
Diewert, W.E.: Duality approaches to microeconomic theory. In: Arrow, K.J., Intriligator, M.D. (eds.) Handbook of Mathematical Economics, vol. 2, pp. 535–599. North-Holland Publishing Company, Amsterdam (1982)
Martínez-Legaz, J.E.: Duality between direct and indirect utility functions under minimal hypotheses. J. Math. Econ. 20, 199–209 (1991)
Passy, V., Prisman, E.Z.: Conjugacy in quasi-convex programming. Math. Progr. 30, 121–146 (1984)
Penot, J.-P., Volle, M.: On quasi-convex duality. Math. Oper. Res. 15, 597–625 (1990)
Rubinov, A.M., Glover, B.M.: On generalized quasi-convex conjugation. Contemp. Math. 204, 199–216 (1997)
Thach, P.T.: Quasi-conjugates of functions: duality relationship between quasi-convex minimization under a reverse convex constraint and applications. J. Math. Anal. Appl. 159, 299–322 (1991)
Penot, J.-P.: Duality for radiant and shady programs. Acta Math. Vietnam. 22, 541–566 (1997)
Penot, J.-P.: Radiant and coradiant dualities. Pac. J. Optim. 6, 263–279 (2010)
Martínez-Legaz, J.E., Santos, M.S.: On expenditure functions. J. Math. Econom. 25, 143–163 (1996)
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The authors are very grateful to the anonymous referees for their useful suggestions on an earlier version of this paper. The comments of the referees were very fruitful, and these comments have enabled the authors to improve the paper significantly.
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Communicated by Constantin Zalinescu.
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Mirzadeh, S., Mohebi, H. Abstract Concavity of Increasing Co-radiant and Quasi-Concave Functions with Applications in Mathematical Economics. J Optim Theory Appl 169, 443–472 (2016). https://doi.org/10.1007/s10957-016-0901-6
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DOI: https://doi.org/10.1007/s10957-016-0901-6
Keywords
- Abstract concavity
- Abstract convexity
- Duality
- Co-radiant function
- Quasi-concave function
- Increasing function
- Upper semi-continuous function
- Upper support set
- Superdifferential
- Production function