Near Equality And Almost Convexity Of Functions With Applications To Optimization And Error Bounds

J. Li ; G. Mastroeni

Journal of convex analysis, Tome 26 (2019) no. 3, pp. 785-822

Résumé

We investigate near equality and almost convexity of extended real valued functions defined on finite-dimensional Euclidean spaces. The main result states that an almost convex function (respectively, its domain, lower level set) is nearly equal to (respectively, the domain, lower level set of) its closure, convex hull and closed convex hull. It is proved that almost convexity of an extended real valued function is equivalent to near equality of itself and another almost convex function. Moreover, it is shown that the operations given by sum, scalar multiple, pointwise supremum, epi-sum and epi-multiple of almost convex functions preserve almost convexity, the formulation of the subdifferential of sum and scalar multiple of almost convex functions on the relative interior of their domain is analogous to that of convex functions under suitable additional assumptions and the proximal average of almost convex functions enjoys analogous properties of lower semi-continuous and convex functions.

Détail

Classification : 26B25, 52A20, 90C26
Mots-clés : Near equality, almost convexity, near convexity, proximal average, optimization, error bounds