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Di erential Equations, Vol.28, No.11, 1992, pp. 1498 1510 CONTROLLED PROXIMAL DIFFERENTIAL SYSTEMS
 

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,

  

Source: Antipin, Anatoly S. - Dorodnicyn Computing Centre of the Russian Academy of Sciences

 

Collections: Computer Technologies and Information Sciences; Mathematics