Di erential Equations, Vol. 30, No. 11, 1994, pp. 1703 1713 ON FINITE CONVERGENCE OF PROCESSES Summary: Dierential Equations, Vol. 30, No. 11, 1994, pp. 17031713 ON FINITE CONVERGENCE OF PROCESSES TO A SHARP MINIMUM AND TO A SMOOTH MINIMUM WITH A SHARP DERIVATIVE 12 A.S. Antipin UDC 517.97+519.82 (Revised version 14 February 2003) 1. INTRODUCTION Let us consider an illustrating example. The problem is to minimize the Euclidean norm in a nite-dimensional space x # # Argmin{|x| : x # R n }, (1.1) where the function f(x) = |x| = (x 2 1 + x 2 2 + . . . + x 2 n ) 1/2 is dierentiable everywhere except for the origin and its gradient is #f(x) = x/|x| (this can easily be checked by direct dierentiation of the Euclidean norm). At the point x # = 0 the goal function has a sharp minimum. The gradient #f(x) generates a vector eld at each point of which the unit vector is directed to the origin along the radius vector. The subdierential at the point of minimum is the unit ball centered on the origin. The vector eld has a nite jump at the point of minimum.