Finite termination of the proximal point algorithm

where q~ is a c losed, convex func t ion def ined on R n, having values in ~ and S is a c losed, convex set in ~n. We write S for the op t ima l so lu t ion set o f (1), S : = arg minxes cb(x) and assume this set to be non-empty , in o rde r tha t a p ro jec t ion o p e r a t i o n on to this set is well def ined. In o rde r to s impl i fy our analysis , let us define , b ~ ( x ) := ¢~(x )+~, (x lS ) and note tha t this is a c losed convex funct ion , s ince the ind ica to r func t ion of the set S, ~0(. I S) , is respect ive ly c losed and convex i f and only i f S is c losed and convex. We can now rewri te p r o b l e m (1) as min imize Cbs(X). (2) X G ~ n