 
Summary: Dierential Equations, Vol.28, No.11, 1992, pp. 14981510
CONTROLLED PROXIMAL DIFFERENTIAL SYSTEMS
FOR SADDLE PROBLEMS 1
A.S. Antipin UDC 517.977
(Revised version 10 January 2003)
Optimization problems have been successfully applied to mathematical modeling mainly be
cause there is a developed theory for these problems. The theory has several main approaches
involving parametrization concepts (e.g., proximal method and penalty function method), li
nearization (e.g., gradient method), and quadratic approximation (e.g., Newton method).
Under special conditions these methods and their combinations always converge to a solution
of a singular optimization problem.
The situation is quite dierent when we deal with equilibrium problems, where none of these
methods nor modications of them are suitable. A simple equilibrium with a saddle point is a
sucient example of this. Let us consider the search for a saddle point of the function L(x, p) =
= x · p. The saddle point of this function is at the origin (0, 0) and satises the inequality: 0 · p #
# 0 · 0 # x · 0 for all x # R 1 and p # R 1 . The saddle gradient method in one variable is falling
and the other is ascending and has the form
dx
dt
= #p,
