Abstract
In this paper, we define the concepts of monotonicity and generalized monotonicity for semidifferentiable maps. Further, we present the characterizations of convexity and generalized convexity in case of semidifferentiable functions. These results rely on general mean-value theorem for semidifferentiable functions (J Glob Optim 40:503–508, 2010).
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Karamardian, S.: Complementarity problems over cones with monotone and pseudomonotone maps. J. Optim. Theory Appl. 18, 445–454 (1976)
Rockafellar, R.T.: Characterization of the subdifferentials of convex functions. Pacific J. Math. 17, 497–510 (1966)
Karamardian, S., Schaible, S.: Seven kinds of monotone maps. J. Optim. Theory Appl. 66(1), 37–46 (1990)
Minty, G.J.: On the monotonicity of the gradient of a convex function. Pacific J. Math. 14, 243–247 (1964)
Ye, M., He, Y.: A double projection method for solving variational inequalities without monotonicity. Comput. Optim. Appl. 60(1), 141–150 (2015)
Kaul, R.N., Kaur, S.: Generalizations of convex and related functions. European J. Oper. Res. 9(4), 369–377 (1982)
Delfour, M.C.: Introduction to optimization and semidifferential calculus. Society for Industrial and Applied Mathematics (SIAM). Philadelphia (2012)
Penot, J.-P., Quang, P.H.: Generalized convexity of functions and generalized monotonicity of set-valued maps. J. Optim. Theory Appl. 92(2), 343–356 (1997)
Komlósi, S.: Generalized monotonicity and generalized convexity. J. Optim. Theory Appl. 84(2), 361–376 (1995)
Mangasarian, O.L.: Nonlinear Programming. McGraw-Hill Book Co., New York-London-Sydney (1969)
Castellani, M., Pappalardo, M.: On the mean value theorem for semidifferentiable functions. J. Global Optim. 46(4), 503–508 (2010)
Durdil, J.: On Hadamard differentiability. Comment. Math. Univ. Carolinae 14, 457–470 (1973)
Penot, J.-P.: Calcul sous-différentiel et optimisation. J. Funct. Anal. 27(2), 248–276 (1978)
Delfour, M.C., Zolésio, J.-P.: Shapes and geometries, Society for Industrial and Applied Mathematics (SIAM). Philadelphia (2001)
Giannessi, F., Maugeri, A. (eds.): Variational Inequalities and Network Equilibrium Problems. Plenum Press, New York (1995)
Acknowledgements
The first author is financially supported by Department of Science and Technology, SERB, New Delhi, India, through grant no.: MTR/2018/000121. The second author is financially supported by CSIR-UGC JRF, New Delhi, India, through Reference no.: 1272/(CSIR-UGC NET DEC.2016). The third author is financially supported by UGC-BHU Research Fellowship, through sanction letter no: Ref.No./Math/Res/ Sept.2015/2015-16/918.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this paper
Cite this paper
Mishra, S.K., Singh, S.K., Shahi, A. (2020). On Monotone Maps: Semidifferentiable Case. In: Le Thi, H., Le, H., Pham Dinh, T. (eds) Optimization of Complex Systems: Theory, Models, Algorithms and Applications. WCGO 2019. Advances in Intelligent Systems and Computing, vol 991. Springer, Cham. https://doi.org/10.1007/978-3-030-21803-4_19
Download citation
DOI: https://doi.org/10.1007/978-3-030-21803-4_19
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-21802-7
Online ISBN: 978-3-030-21803-4
eBook Packages: Intelligent Technologies and RoboticsIntelligent Technologies and Robotics (R0)