Step 0: Matrix A n × m , vector b m , parameters λ , γ > 0 , θ [ 1 , 1 ] , t > 0 , s > 0 , t s > 1 4 ( 1 + θ ) 2 λ max ( A T A ) .

Step 1 Solve the variational inequality

y y + ( u u ˜ k ) T { F ( u ˜ k ) + M ( u ˜ k u k ) } 0 (7)

we can get a predicted point u ˜ k .

correct u k + 1 :

u k + 1 = u k γ α k M ˜ ( u k u ˜ k ) (8)

Step 2

y ˜ k = a r g m i n y Δ m λ y + t 2 y ( y k + 1 t A T y k 2 b y k ) 2 (9)

x k = x k 1 s + λ ( A ( 1 + θ ) y ˜ k θ x k ) + y k (10)

Step 3

( y k + 1 x k + 1 ) = ( y k x k ) γ α k ( I n ( θ 1 ) 1 s A I m ) ( y k y ˜ k x k x ˜ k ) (11)

where

α k = ( u k u ˜ k ) T M ( u k u ˜ k ) M ˜ ( u k u ˜ k ) H 2 (12)

Step 4 if A x k b 2 b 2 ε end; else for k = k + 1 , do Step 2