Di erential Equations, Vol.30, No.9, 1994, pp. 1365 1375 MINIMIZATION OF CONVEX FUNCTIONS ON CONVEX SETS Summary: Dierential Equations, Vol.30, No.9, 1994, pp. 13651375 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 dierentiable scalar function, x # Q # R n , R n is an Euclidean nite-dimension 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 sucient conditions x # = #Q (x # - ##f(x # )) (2.1)