Abstract
Recently, several kinds of generalized monotone maps were introduced by Karamardian and the author. They play a role in complementarity problems and variational inequality problems and are related to generalized convex functions. Following a presentation of seven kinds of (generalized) monotone maps, various characterizations of differentiable and affine generalized monotone maps are reported which can simplify the identification of such properties. Finally, pseudomonotone maps are related to sufficient matrices studied in complementarity theory.
The author gratefully acknowledges the research support he received as Visiting Professor of the Dipartimento di Statistica e Matematica Applicata All ‘Economica, Universita’ di Pisa, Spring 1992.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Arrow, K.J. and A.C. Enthoven, Quasi-concave programming, Econometrica 29, 1961, 779–800.
Avriel, M., r-convex functions, Mathematical Programming 2, 1972, 309–323.
Avriel, M., Diewert, W.E., Schaible, S. and I. Zang, Generalized concavity, Plenum Publishing Corporation, New York 1988.
Avriel, M. and S. Schaible, Second-order characterizations of pseudoconvex functions, Mathematical Programming 14, 1978, 170–185.
Cambini, A., Castagnoli, E., Martein, L., Mazzoleni, P. and S. Schaible (eds.), Generalized convexity and fractional programming with economic applications, Springer-Verlag, Berlin—Heidelberg—New York 1990.
Castagnoli, E., On order-preserving functions, in: Fedrizzi, M. and J. Kacprzyk (eds.), Proceedings of the 8th Italian—Polish Symposium on Systems Analysis and Decision Support in Economics and Technology, Levico Terme, September 1989, Omnitech Press, Warszawa, 1990, 151–165.
Castagnoli, E. and P. Mazzoleni, Order-preserving functions and generalized convexity, Rivista di Matematica per le Scienze Economiche e Sociali, 14, 1991, 33–45.
Castagnoli, E. and P. Mazzoleni, Generalized monotonicity and poor vector order relations, presented at the International Workshop on Generalized Convexity and Fractional Programming, University of California, Riverside, October 1989.
Cottle, R. W., The principal pivoting method revisited, Mathematical Programming B, 48, 1990, 369–385.
Cottle, R. W. and Y.-Y. Chang, Least-index resolution of degeneracy in linear complementarity problems with sufficient matrices, Technical Report Sol 909, Department of Operations Research, Stanford University, June 1990.
Cottle, R. W., Pang, J.-S. and V. Ventkateswaran, Sufficient matrices and the linear complementarity problem, Linear Algebra and its Applications 114 /115, 1989, 231–249.
Cottle, R. W. and J. C. Yao, Pseudomonotone complementarity problems in Hilbert space, J. of Optimization Theory and Applications 75, 1992, 281–295.
Crouzeix, J. P. and J. A. Ferland, Criteria for quasiconvexity and pseudoconvexity: relationships and comparisons, Mathematical Programming 23, 1982, 193–205.
Gowda, M. S. Pseudomonotone and copositive star matrices, Linear Algebra and Its Applications 113, 1989, 107–118.
Gowda, M. S. Affine pseudomonotone mappings and the linear complementarity problem, SIAM J. of Matrix Analysis and Applications 11, 1990, 373–380.
Hadjisavvas, N. and S. Schaible, On strong pseudomonotonicity and (semi) strict quasimonotonicity, J. of Optimization Theory and Applications 79, No. 1, 1993.
Harker, P. T. and J. S. Pang, Finite dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications in: Cottle, R. W., Kyparisis, J. and J. S. Pang (eds.), Variational inequality problems, Mathematical Programming 48, Series B, 1990, 161–220.
Hassouni, A., Sous-differentiels des fonctions quasi-convexes, Thèse de 3 ème cycle, Mathématique Appliquées, Toulouse 1983.
Hassouni, A. and R. Ellaia, Characterizations of nonsmooth functions through their generalized gradients, Optimization 22, 1991.
Karamardian, S., Complementarity over cones with monotone and pseudomonotone maps, J. of Optimization Theory and Applications 18, 1976, 445–454.
Karamardian, S. and S. Schaible, Seven kinds of monotone maps, J. of Optimization Theory and Applications 66, 1990, 37–46.
Karamardian, S. and S. Schaible, First-order characterizations of generalized monotone maps, Working Paper 90–5, Graduate School of Management, University of California, Riverside, December 1989.
Karamardian, S. Schaible, S. and J. P. Crouzeix, Characterizations of generalized monotone maps, J. of Optimization Theory and Applications 76, 1993, 399–413.
Komlósi, S., On generalized upper quasidifferentiability, in: Giannessi, F. (ed.), Nonsmooth optimization methods and applications. Gordon and Breach, Amsterdam 1992, 189–200.
Mazzoleni, P., Monotonicity properties and generalized concavity, presented at the International Workshop on Generalized Convexity and Fractional Programming, University of California, Riverside, October 1989.
Pini, R. and S. Schaible, Some invariance properties of generalized mono-tonicity, this volume.
Schaible, S., Second-order characterization of pseudoconvex quadratic functions, J. of Optimization Theory and Applications 21, 1977, 15–26.
Schaible, S. Quasiconvex, pseudoconvex and strictly pseudoconvex quadratic functions, J. of Optimization Theory and Applications 35, 1981, 303–338.
Schaible, S. and W. T. Ziemba (eds), Generalized concavity in optimization and economics, Academic Press, New York 1981.
Singh, C. and B. K. Dass (eds), Continuous-time, fractional and multiobjective programming, Analytic Publishing Company, New Delhi 1989.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1994 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Schaible, S. (1994). Generalized monotonicity — a survey. In: Komlósi, S., Rapcsák, T., Schaible, S. (eds) Generalized Convexity. Lecture Notes in Economics and Mathematical Systems, vol 405. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-46802-5_18
Download citation
DOI: https://doi.org/10.1007/978-3-642-46802-5_18
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-57624-2
Online ISBN: 978-3-642-46802-5
eBook Packages: Springer Book Archive