Dierential Equations, Vol.28, No.11, 1992, pp. 14981510 CONTROLLED PROXIMAL DIFFERENTIAL SYSTEMS Summary: Dierential Equations, Vol.28, No.11, 1992, pp. 14981510 CONTROLLED PROXIMAL DIFFERENTIAL SYSTEMS FOR SADDLE PROBLEMS1 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 R1 and p R1 . The saddle gradient method in one variable is falling and the other is ascending and has the form dx