Abstract
Functions which are increasing, co-radiant and quasi-concave have found many applications in microeconomic analysis. In production theory it is commonly assumed that the production function is increasing and quasi-concave. Likewise in consumer theory one often assumes that the utility function has these properties. In this paper, we first examine characterizations of the dual problem for the difference of two increasing, co-radiant and quasi-concave functions. Next, we give various characterizations of the minimal elements of the upper support set of co-radiant functions, by applying a type of duality, which is used in microeconomic theory. As an application, we obtain necessary and sufficient conditions for the global minimum of the difference of two increasing, co-radiant and quasi-concave functions defined on a real locally convex topological vector space X.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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)
Doagooei, A.R., Mohebi, H.: Optimization of the difference of ICR functions. Nonlinear Anal. 71, 4493–4499 (2009)
Doagooei, A.R., Mohebi, H.: Optimization of the difference of topical functions. J. Glob. Optim. 57(4), 1349–1358 (2013)
Glover, B.M., Jeyakumar, V.: Nonlinear extensions of Farkas’ lemma with applications to global optimization and least squares. Math. Oper. Res. 20, 818–837 (1995)
Glover, B.M., Ishizuka, Y., Jeyakumar, V., Rubinov, A.M.: Inequality systems and global optimization. J. Math. Anal. Appl. 202, 900–919 (1996)
Hiriart-Urruty, J.B.: From Convex to Nonconvex Minimization: Necessary and Sufficient Conditions for Global Optimality, in Nonsmooth Optimization and Related Topics. Plenum, New York (1989)
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. Methods Oper. Res. 61, 261–280 (2005)
Mirzadeh, S., Mohebi, H.: Increasing co-radiant and quasi-concave functions with applications in mathematical economics. J. Optim. Theory Appl. 169(2), 443–472 (2016)
Renner, P., Schmedders, K.: A polynomial optimization approach to principal-agent problems. Econometrica 83(2), 729–769 (2015)
Rubinov, A.M.: Abstract Convexity and Global Optimization. Kluwer Academic Publishers, Dordrecht (2000)
Singer, I.: Abstract Convex Analysis. Wiley-Interscience, New York (1997)
Toland, J.F.: Duality in nonconvex optimization. J. Math. Anal. Appl. 66, 399–415 (1978)
Acknowledgements
The authors are very grateful to two anonymous referees for their useful suggestions regarding an earlier version of this paper. The comments of the referees were very useful and they helped us to improve the paper significantly. Funding was provided by Shahid Bahonar University of Kerman (Grant No. 93500).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Mirzadeh, S., Mohebi, H. Global minimization of the difference of increasing co-radiant and quasi-concave functions. Optim Lett 12, 885–902 (2018). https://doi.org/10.1007/s11590-017-1155-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11590-017-1155-4