A New Hybrid Inexact Logarithmic-Quadratic Proximal Method Theorem 3.2. Let x be an arbitrary point of the n solution set of (1). For a given x k R + and k c k > 0 , let x k , x be computed from (10) and (15), respectively, ( k ) be defined by Theorem 3.1, ( x k , x k , k , µ ) and d ( x , , , µ ) x be defined by (13) and (14), respectively, then for any k > 0 we have k k k k k k Therefore, we can get that: k 1 , 2 k ( 1 - 1 - µ 2 ) x k - x k 2 + x k - x k 2 ( ) 2 ( 1 + µ ) 2 x k - x k 2 ( 1 - 1 - µ 2 ) , 4 ( 1 + µ ) ( k ) > ( k ) := k d ( x , , , µ ) . x (28) and ( k ) = The proof can be found in (Auslender et al., 1999) Using Theorems 3.1 and 3.2, we can get ( ) ( ) (29)where ( ) = { x - x k k 2 k { x k - x k 2 + k ( x k , x k , k , µ )} 2 2 ( 1 - 1 - µ 2 ) 2 16 ( 1 + µ ) 2 x k - x k 2 . + k ( x , x , , µ )} k k k (30) The above inequality tells us how to choose a suitable k . Since ( ) is a quadratic function of k , it reaches its maximum at k For fast convergence, we take a relaxation fac- tor [ 1 , 2 ) and the step-size k by k = ( k ) . Simple calculations show that ( ) = { x - k k 2 + ( x , , , µ )} k k k