Applied mathematics

Smooth approximations of nonsmooth convex functions

L. N. Polyakova

St Petersburg State University, 7–9, Universitetskaya nab., St Petersburg, 199034, Russian Federation

Abstract: For an arbitrary convex function, using the infimal convolution operation, a family of continuously differentiable convex functions approximating it is constructed. The constructed approximating family of smooth convex functions Kuratowski converges to the function under consideration. If the domain of the considered function is compact, then such smooth convex approximations are uniform in the Chebyshev metric. The approximation of a convex set by a family of smooth convex sets is also considered.

Keywords: set-valued mapping, semicontinuous mapping, conjugate function, Kuratowski converge, infimal convolution operation, smooth approximation.

Received: July 21, 2022
Accepted: September 1, 2022

Language: English

DOI: 10.21638/11701/spbu10.2022.408