 
Summary: Dierential Equations, Vol. 30, No. 11, 1994, pp. 17031713
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
nitedimensional 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 dierentiable everywhere except for the
origin and its gradient is #f(x) = x/x (this can easily be checked by direct dierentiation 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 subdierential 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.
