Abstract
Characterizations of different kinds of generalized monotonicity are surveyed for the following subclasses of maps: affine maps, differentiable maps, locally Lipschitz maps.
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References
Aussel, D., Corvallec, J. N. and M. Lassonde, “Subdifferential characterization of quasiconvexity and convexity”, Journal of Convex Analysis, 1 (1994) 195201.
Avriel, M., Diewert, W. E., Schaible, S. and I. Zang, “Generalized concavity”, Plenum Publishing Corporation, New York, 1988.
Castagnoli, E. and P. Mazzoleni, “Order-preserving functions and generalized convexity”, Rivista di Matematica per le Scienze Economiche e Sociali, 14 (1991) 33–46.
Crouzeix, J.-P. and J. A. Ferland, “Criteria for differentiable generalized monotone maps”, Mathematical Programming, 75 (1996) 399–406.
Crouzeix, J.-P. and S. Schaible, “Generalized monotone affine maps ”, SIAM Journal on Matrix Analysis and Applications, 17 (1996) 992–997.
Ellaia, R. and A. Hassouni, “Characterization of nonsmooth functions through their generalized gradients”, Optimization, 22 (1991) 401–416.
Ferland, J. A., “Quasi-convex and pseudo-convex functions on solid convex sets”, Technical Report 71–4, Department of Operations Research, Stanford University, 1971.
Hadjisavvas, N. and S. Schaible, “On strong pseudomonotonicity and (semi)strict quasimonotonicity”, Journal of Optimization Theory and Applications, 79 (1993) 139–155.
Hadjisavvas, N. and S. Schaible, “Errata corrige. On strong pseudomonotonicity and (semi)strict quasimonotonicity”, Journal of Optimization Theory and Applications, 85 (1995) 741–742.
Hadjisavvas, N. and S. Schaible, “Quasimonotonicity and pseudomonotonicity in variational inequalities and equilibrium problems”, in: Crouzeix, J.-P., Martinez-Legaz, J.-E. and M. Volle (eds.) “Generalized convexity and mono-tonicity: recent developments”, Kluwer Academic Publishers, Dordrecht, London, Boston, 1997, forthcoming.
Hassouni, A., “Sous-differentiels des fonctions quasi-convexes”, Thèse de 3ème cycle, Universite Paul Sabatier, Toulouse I II, 1983.
Hassouni, A., “Operateurs quasimonotones, applications a certains problemes variationnels”, Thèse de Ph.D., Universite Paul Sabatier, Toulouse I II, 1993.
Jeyakumar, V., Luc, D. T. and S. Schaible “Characterizations of generalized monotone nonsmooth continuous maps using approximate Jacobians”, Working Paper 96–22, A. G. Anderson Graduate School of Management, University of California, Riverside, October 1996.
Karamardian, S., “Complementarity over cones with monotone and pseudomonotone maps”, Journal of Optimization Theory and Applications, 18 (1976) 445–454.
Karamardian, S. and S. Schaible, “Seven kinds of monotone maps”, Journal 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, 1989.
Karamardian, S., Schaible, S. and J.-P. Crouzeix, “Characterizations of generalized monotone maps”, Journal of Optimization Theory and Applications, 76 (1993) 399–413.
Komlôsi, S., “On generalized upper quasidifferentiability”, in: Giannessi, F., (ed.), “Nonsmooth optimization methods and applications”, Gordon and Breach, Amsterdam, 1992. 189–200.
Komlôsi, S., “Generalized monotonicity in nonsmooth analysis”, in: Komlôsi, S., Rapcsâk, T. and S. Schaible, “Generalized Convexity”, Springer Verlag, Berlin-Heidelberg-New York, 1994, 263–275.
Komlsi, S., “Generalized monotonicity and generalized convexity”, Journal of Optimization Theory and Applications, 84 (1995) 361–376.
Komlsi, S., “Monotonicity and quasimonotonicity in nonlinear analysis”, in: Du, D. Z., Qi, L. and R. S. Womersley (eds.), “Recent advances in nonsmooth optimization”, World Scientific Publishing Co, Singapore, 1995, 193–214.
Luc, D. T., “Characterizations of quasiconvex functions”, Bulletin of the Australian Mathematical Society, 48 (1993) 393–405.
Luc, D. T., “On generalized convex nonsmooth functions”, Bulletin of the Australian Mathematical Society, 49 (1994) 139–149.
Luc, D. T. and S. Schaible, “Generalized monotone nonsmooth maps”, Journal of Convex Analysis, 3, No. 2, (1996) forthcoming.
Mangasarian, O. L., “Nonlinear Programming ”, McGraw Hill, New York, 1969.
Martos, B., “Subdefinite matrices and quadratic forms”, SIAM Journal of Applied Mathematics, 17 (1969) 1215–1223.
Martos, B., “Quadratic programming with a quasiconvex objective function”, Operations Research, 19 (1971) 82–97.
Penot, J.-P., “Generalized convexity in the light of nonsmooth analysis”, in: Duriez, R. and C. Michelot (eds.), Lecture Notes in Mathematical Systems and Economics 429, Springer Verlag, Berlin-Heidelberg-New York, 1995, 269–290.
Penot, J.-P. and P. H. Quang, “Generalized convexity of functions and generalized monotonicity of set-valued maps”, Journal of Optimization Theory and Applications, 92 (1997) 343–356.
Penot, J.-P. and P. H. Sach, “Generalized monotonicity of subdifferentials and generalized convexity”, Journal of Optimization Theory and Applications, forthcoming.
Pini, R. and S. Schaible, “Invariance properties of generalized monotonicity”, Optimization, 28 (1994) 211–222.
Schaible, S., “Beiträge zur quasikonvexen Programmierung”, Doctoral Dissertation, Universität Köln, 1971.
Schaible, S., “Quasiconvex, pseudoconvex and strictly pseudoconvex quadratic functions”, Journal of Optimization Theory and Applications, 35 (1981) 303338.
Schaible, S., “Generalized monotonicity–concepts and uses”, in: Giannessi, F. and A. Maugeri (eds.), “Variational inequalities and network equilibrium problems ”, Plenum Publishing Corporation, New York, 1995, 289–299.
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Schaible, S. (1998). Criteria for Generalized Monotonicity. In: Giannessi, F., Komlósi, S., Rapcsák, T. (eds) New Trends in Mathematical Programming. Applied Optimization, vol 13. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-2878-1_20
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DOI: https://doi.org/10.1007/978-1-4757-2878-1_20
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