We investigate some properties of an inexact proximal point method for pseudomonotone equilibrium problems in a real Hilbert space. Un- like monotone case, in pseudomonotone case, the regularized subprob- lems may not be strongly monotone, even not pseudomonotone. How- ever, every proximal trajectory weakly converges to the same limit, We use these properties to extend a viscosity-proximal point algorithm de- veloped in  to pseudomonotone equilibrium problems. Then we pro- pose a hybrid extragradient-cutting plane algorithm for approximating the limit point by solving a bilevel strongly convex optimization prob- lem. Finally, we show that by using this bilevel convex optimization, the proximal point method can be used for handling ill-possed pseu- domonotone equilibrium problems.
UNPUBLISHED: 1)report number: 1; 2)Institution address: 1. Bui Van Dinh, Department of Mathematics, Le Quy Don University, No 100, Hoang Quoc Viet, Hanoi, Vietnam; 2. Pham Gia Hung, Nha Trang University, Nha Trang, Vietnam;3. Le Dung Muu, Institute of Mathematics, Hanoi, Vietnam; 3).Month/Year: 01/2013