Abstract
We consider the question of integration of a multivalued operator T, that is the question of finding a function f such that T⊑∂f. If ∂ is the Fenchel–Moreau subdifferential, the above problem has been completely solved by Rockafellar, who introduced cyclic monotonicity as a necessary and sufficient condition. In this article we consider the case where f is quasiconvex and ∂ is the lower subdifferential ∂<. This leads to the introduction of a property that is reminiscent to cyclic monotonicity. We also consider the question of the density of the domains of subdifferential operators.
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Aussel, D., Corvellec, J.-N. and Lassonde, M.: Subdifferential characterization of quasiconvexity and convexity, J. Convex Anal. 1 (1994), 195-201.
Benoist, J. and Hiriart-Urruty, J.-B.: What is the subdifferential of the closed convex hull of a function? SIAM J. Math. Anal. 27 (1996), 1661-1679.
Borwein, J., Moors, W. and Shao, Y.: Subgradient representation of multifunctions, J. Austral. Math. Soc. Ser. B40 (1998), 1-13.
Clarke, F. H.: Optimization and Nonsmooth Analysis, Wiley, New York, 1983.
Daniilidis, A. and Hadjisavvas, N.: On the subdifferentials of generalized convex functions and cyclicity, J. Math. Anal. Appl. 237 (1999), 30-42.
Hassouni, A.: Opérateurs quasimonotones; applications à certains problè mes variationnels, Thè se, Université Paul Sabatier, Toulouse, 1993.
Janin, R.: Sur des multiapplications qui sont des gradients généralisés, C.R. Acad. Sci. Paris Ser. I 294 (1982), 115-117.
Martinez-Legaz, J.-E.: On lower subdifferentiable functions, In: K.-H. Hoffmann et al. (eds), Trends in Mathematical Optimization, Internat. Ser. Numer. Math. 84, Birkhäuser, Basel, 1988, pp. 197-232.
Martinez-Legaz, J.-E. and Romano-Rodriguez, S.: α-lower subdifferentiable functions, SIAM J. Optim. 3 (1993), 800-825.
Martinez-Legaz, J.-E. and Sach, P.: A new subdifferential in quasiconvex analysis, J. Convex Anal. 6 (1999), 1-12.
Penot, J.-P.: Generalized convexity in the light of nonsmooth analysis, In: R. Durier and C. Michelot (eds), Recent Developments in Optimization, Lecture Notes in Econom. and Math. Systems 429, Springer-Verlag, Berlin, 1995, pp. 269-290.
Penot, J.-P.: What is quasiconvex analysis? Optimization 47 (2000), 35-110.
Phelps, R.: Convex Functions, Monotone Operators and Differentiability, 2nd edn, Lecture Notes in Math. 1364, Springer-Verlag, Berlin, 1993.
Plastria, F.: Lower subdifferentiable functions and their minimization by cutting plane, J. Optim. Theory Appl. 46 (1985), 37-54.
Poliquin, R.: Integration of subdifferentials of nonconvex functions, Nonlinear Anal. 17 (1991), 385-398.
Rockafellar, R. T.: On the maximal monotonicity of subdifferential mappings, Pacific J. Math. 33 (1970), 209-216.
Rockafellar, R. T.: Generalized directional derivatives and subgradients of nonconvex functions, Canad. J. Math. 32 (1980), 257-280.
Thibault, L. and Zagrodny, D.: Integration of subdifferentials of lower semi-continuous functions on Banach spaces, J. Math. Anal. Appl. 189 (1995), 33-58.
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Bachir, M., Daniilidis, A. & Penot, JP. Lower Subdifferentiability and Integration. Set-Valued Analysis 10, 89–108 (2002). https://doi.org/10.1023/A:1014460029093
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DOI: https://doi.org/10.1023/A:1014460029093