Abstract
We introduce a new concept of asymptotic functions which allows us to simultaneously study convex and concave functions as well as quasiconvex and quasiconcave functions. We provide some calculus rules and most relevant properties of the new asymptotic functions for application purpose. We also compare them with the classical asymptotic functions of convex analysis. By using the new concept of asymptotic functions we establish sufficient conditions for the nonemptiness and for the boundedness of the solution set of quasiconvex minimization problems and quasiconcave maximization problems. Applications are given for quadratic and fractional quadratic problems.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Adly, S., Goeleven, D., Théra, M.: Recession mappings and noncoercive variational inequalities. Nonlinear Anal. 26, 1573–1603 (1996)
Ait Mansour, M., Chbani, Z., Riahi, J.: Recession bifunction and solvability of noncoercive equilibrium problems. Commun. Appl. Anal. 7, 369–377 (2003)
Amara, Ch.: Directions de majoration d’une fonction quasiconvexe et applications. Serdica Math. J. 24, 289–306 (1998)
Auslender, A., Teboulle, M.: Asymptotic Cones and Functions in Optimization and Variational Inequalities. Springer, New York (2003)
Baiocchi, C., Buttazzo, G., Gastaldi, F., Tomarelli, F.: General existence theorems for unilateral problems in continuum mechanics. Arch. J. Ration. Mech. Anal. 100, 149–189 (1988)
Cambini, A., Martein, L.: Generalized Convexity and Optimization. Springer, Berlin (2009)
Debreu, G.: Theory of Value. Yale University Press, London (1975)
Fenchel, W.: Convex Cones, Sets and Functions. Mimeographed Lecture Notes. Princeton University, Princeton (1951)
Flores-Bazán, F., Flores-Bazán, F., Vera, C.: Maximizing and minimizing quasiconvex functions: related properties, existence and optimality conditions via radial epiderivates. J. Glob. Optim. 63, 99–123 (2015)
Flores-Bazán, F., Hadjisavvas, N., Lara, F.: Second order asymptotic analysis: basic theory. J. Convex Anal. 22, 1173–1196 (2015)
Flores-Bazán, F., Hadjisavvas, N., Lara, F., Montenegro, I.: First- and second-order asymptotic analysis with applications in quasiconvex optimization. J. Optim. Theory Appl. 170, 372–393 (2016)
Goeleven, D.: Complementarity and Variational Inequalities in Electronics. Academic Press, London (2017)
Hadjisavvas, N., Lara, F., Martínez-Legaz, J.E.: A quasiconvex asymptotic function with applications in optimization. J. Optim. Theory Appl. 180, 170–186 (2019)
Iusem, A., Lara, F.: Second order asymptotic functions and applications to quadratic programming. J. Convex Anal. 25, 271–291 (2018)
Iusem, A., Lara, F.: Optimality conditions for vector equilibrium problems with applications. J. Optim. Theory Appl. 180, 187–206 (2019)
Lara, F.: Quadratic fractional programming under asymptotic analysis. J. Convex Anal. 26, 15–32 (2019)
Lara, F., López, R., Svaiter, B.F.: A further study on asymptotic functions via variational analysis. J. Optim. Theory Appl. 182, 366–382 (2019)
Luc, D.T.: Theory of Vector Optimization. Lecture Notes in Economics and Mathematical Systems, vol. 319. Springer, New York (1989)
Luc, D.T.: Recession maps and applications. Optimization 27, 1–15 (1989)
Luc, D.T.: Recession cones and the domination property in vector optimization. Math. Programm. 49, 113–122 (1990)
Luc, D.T., Penot, J.-P.: Convergence of asymptotic directions. Trans. Am. Math. Soc. 353, 4095–4121 (2001)
Luc, D.T., Théra, M.: Derivatives with support and applications. Math. Oper. Res. 19, 659–675 (1994)
Penot, J.-P.: What is quasiconvex analysis? Optimization 47, 35–110 (2000)
Penot, J.-P.: Noncoercive problems and asymptotic conditions. Asymptot. Anal. 49, 205–215 (2006)
Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)
Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998)
Schaible, S.: Fractional programming. In: Horst, R., Pardalos, P. (eds.) Handbook of Global Optimization, pp. 495–608. Kluwer Academic Publishers, Dordrecht (1995)
Schaible, S., Ziemba, W.T.: Generalized Concavity in Optimization and Economics. Academic Press, New York (1981)
Stancu-Minasian, I.M.: Fractional Programming: Theory, Methods and Applications. Kluwer Academic Publishers, Dordrecht (1997)
Steinitz, E.: Bedingt konvergente Reihen und konvexe Systeme I. J. Reine Angew. Math. 143, 128–175 (1913)
Steinitz, E.: Bedingt konvergente Reihen und konvexe Systeme II. J. Reine Angew. Math. 144, 1–40 (1914)
Steinitz, E.: Bedingt konvergente Reihen und konvexe Systeme III. J. Reine Angew. Math. 146, 1–52 (1916)
Acknowledgements
Part of this work was carried out F. Lara was visiting the Département de Mathématiques, Université d’Avignon, Avignon, France, during June 2018, and the Department of Product and Systems Design Engineering, University of the Aegean, Hermoupolis, Syros, Greece, during July 2018. This author wishes to thank both departments for their hospitality. This research was partially supported by Conicyt–Chile under project Fondecyt Iniciación 11180320 (F. Lara).
Finally, we would like to thank the referee for corrections and pertinent remarks that contributed to the improvement of our paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Hadjisavvas, N., Lara, F. & Luc, D.T. A general asymptotic function with applications in nonconvex optimization. J Glob Optim 78, 49–68 (2020). https://doi.org/10.1007/s10898-020-00891-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-020-00891-2