Continuity and maximal quasimonotonicity of normal cone operators
DOI:
https://doi.org/10.24193/subbmath.2022.1.03Keywords:
Quasimonotone operator, maximal quasimonotone operator, cone upper semicontinuity, upper sign continuity, quasiconvex function.Abstract
In this paper, we study some properties of the adjusted normal cone operator of quasiconvex functions. In particular, we introduce a new notion of maximal quasimotonicity for set-valued maps different from similar ones recently appeared in literature, and we show that it is enjoyed by this operator. Moreover, we prove the s x w* cone upper semicontinuity of the normal cone operator in the domain of f in case the set of global minima has non empty interior.
Mathematics Subject Classification (2010): 47H05, 47H04, 49J53, 90C33.
Received 16 November 2021; Accepted 20 December 2021.
References
Al-Homidan, S., Hadjisavvas, N., Shaalan, L., Tranformation of quasiconvex functions to eliminate local minima, J. Optim. Theory Appl., 177(2018), 93-105.
Aliprantis, C.D., Border, K., Infinite Dimensional Analysis, Springer-Verlag Berlin Heidelberg, 2006.
Aubin, J.P., Frankowska, H., Set-Valued Analysis, Birkha¨user, 1990.
Aussel, D., Subdifferential properties of quasiconvex and pseudoconvex functions: Unified approach, J. Optim. Theory Appl., 97(1998), 29-45.
Aussel, D., Corvellec, J.-N., Lassonde, M., Subdifferential characterization of quasiconvexity and convexity, J. Convex Anal., 1(1994), 195-201.
Aussel, D., Eberhard, A., Maximal quasimonotonicity and dense single-directional properties of quasimonotone operators, Math. Program. Ser. B, 139(2013), 27-54.
Aussel, D., Hadjisavvas, N., Adjusted sublevel sets, normal operator, and quasiconvex programming, SIAM J. Optim., 16(2005), 358-367.
Borde, J., Crouzeix, J.-P., Continuity properties of the normal cone to the level sets of a quasiconvex function, J. Optim. Theory Appl., 66(1990), 415-429.
Borwein, J.M, Fifty years of maximal monotonicity, Optim. Lett., 4(2010), 473-490.
Borwein, J.M, Lewis, A.S., Partially finite convex programming, Part I: Quasi relative interiors and duality theory, Math. Program., 57(1992), 15-48.
Bueno, O., Cotrina, J., On maximality of quasimonotone operators, Set-Valued Var. Anal., 27(2019), 87-101.
Clarke, F.H., Optimization and Nonsmooth Analysis, Wiley Interscience, New York, 1983.
Csetnek, E.R., Overcoming the failure of the classical generalized interior-point regularity conditions in convex optimization. Applications of the duality theory to enlargements of maximal monotone operators, PhD Thesis, Logos Verlag, Berlin, 2010.
Hadjisavvas, N., Continuity and maximality properties of pseudo-monotone operators, J. Convex Anal., 178(2003), 459-469.
Hu, S., Papageorgiou, S., Handbook of Multivalued Analysis, Vol. I: Theory, Kluwer Academic Publisher, Norwell, MA, 1997.
Karamardian, S., Schaible, S., Seven kinds of monotone maps, J. Optim. Theory Appl., 66(1990), 37-46.
Martınez-Legaz, J.-E., Svaiter, B.F., Monotone operators representable by l.s.c. convex functions, Set-Valued Anal., 13(2005), 21-46.
Schirotzek, W., Nonsmooth Analysis, Springer Verlag, Berlin Heidelberg, 2007.
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