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Journal of Convex Analysis 07 (2000), No. 2, 243--270
Copyright Heldermann Verlag 2000

Elements of Quasiconvex Subdifferential Calculus

Jean-Paul Penot
Dép. de Mathématiques, Faculté des Sciences, Université de Pau, 64000 Pau, France

Constantin Zalinescu
Faculté de Mathématiques, Université "Al. I. Cuza", 6600 Iasi, Romania

A number of rules for the calculus of subdifferentials of generalized convex functions are displayed. The subdifferentials we use are among the most significant for this class of functions, in particular for quasiconvex functions: we treat the Greenberg-Pierskalla's subdiffential and its relatives and the Plastria's lower subdifferential. We also deal with a recently introduced subdifferential constructed with the help of a generalized derivative. We emphasize the case of the sublevel-convolution, an operation analogous to the infimal convolution, which has proved to be of importance in the field of quasiconvex functions. We provide examples delineating the limits of the rules we provide.

Keywords: Harmonic sum, incident derivative, lower subdifferential, normal cone, quasiconvex function, subdifferential set, sublevel-convolution.

MSC: 26B25, 90C26; 2651, 26A27

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