Generalized Convexity of Functions and Generalized Monotonicity of Set-Valued Maps

Generalized Affine Functions and Generalized Differentials

“…Moreover, in such a case, one sees @f and @f are pseudomonotone (in the sense of [26]; see also [20], [25]). Thus we get a multivalued generalization of the concept of PPM map which has been used in [3] to study variational inequalities.…”

Some Remarks on the Class of Continuous (Semi-) Strictly Quasiconvex Functions

“…The definition of quasimonotonicity is classical (see [17], [4], [12], [9], for example) and is associated with the following result, which is a cornerstone of quasiconvex analysis (see [19], [21], [1] or [11, Theorem 2.1], for example).…”

“…of generalized monotonicity of their subdifferential (see for instance [19], [21], [1], [10]). As an outcome, a natural correspondence between notions of generalized convexity and of generalized monotonicity arises, reflecting the known duality between utility functions (typically being semistrictly quasiconvex) and demand correspondences (typically being semistrictly quasimonotone multifunctions) in the consumer's theory (see [4], [14], for example).…”

Generalized convex functions and generalized differentials

“…Among the tools used to de…ne or study these notions are the various subdi¤erentials of nonsmooth analysis ( [1], [17], [16], [24], [29], [32], [34], [39], [35]...), the convexi…cators of [11], the pseudo-di¤erentials of Jeyakumar and Luc ([18]), the normal cones to sublevel sets ( [2], [4], [5], [6]) and the generalized directional derivatives ( [19], [20], [22], [40]). In the present paper we use a concept of generalized derivative which can encompass all these notions but the last one.…”