Di erential Equations, Vol.28, No.11, 1992, pp. 1498 1510 CONTROLLED PROXIMAL DIFFERENTIAL SYSTEMS Summary: Dierential Equations, Vol.28, No.11, 1992, pp. 14981510 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 dierent when we deal with equilibrium problems, where none of these methods nor modications of them are suitable. A simple equilibrium with a saddle point is a sucient 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 satises 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,