 
Summary: Dierential Equations, Vol.30, No.9, 1994, pp. 13651375
MINIMIZATION OF CONVEX FUNCTIONS ON CONVEX SETS
BY MEANS OF DIFFERENTIAL EQUATIONS 1
A.S. Antipin UDC 519.82
(Revised version 14 March 2003)
1. INTRODUCTION
Let us consider the problem on minimizing a function on a simple set, namely
x # # Arg min {f(x) : x # Q}, (1.1)
where f(x) is a dierentiable scalar function, x # Q # R n , R n is an Euclidean nitedimension
space, and Q is a simple set, i.e., a set onto which one can easily project. Examples of such sets
are the positive orthant, a parallelepiped, a ball, etc.
The gradient approach to solving problem (1.1) has long been a conventional method, but
its continuous variants are not well investigated. First of all, this pertains to methods in which
one should take into account constraints imposed on the variables. In this paper we consider the
gradient methods of rst and second order, where the constraints are taken into account by means
of projection operators. The asymptotic and exponential stability of such processes is proved.
2. CONTINUOUS METHOD OF GRADIENT PROJECTION
The idea of the method can be presented in the following way. If x # is a minimum point of
problem (1.1), then the necessary and sucient conditions
x # = #Q (x #  ##f(x # )) (2.1)
