(9.1) (9.2-1) (9.2-2) (9.2-3) jes KEB (9.2-4) jes KEB ndnev = (9.2-5) jes KEB =1, VnE U (9.2-6) jes keB (9.2-7) In P2, the objective is to minimize the normalized QoS degradation of all UEs. Com- pared with the constraints in P1, the only difference is using equalities in constraints (9.2-3)--(9.2-5) to replace the inequalities in (9.1-3)-(9.1-5) by introducing elastic vari- ables. In P2, the optimization variables are binary indicators xik as well as the continuous elastic variables r and dn. Hence, P2 is a mixed integer linear programming (MILP). As we introduce the elastic variables into the MILP, the QoS of UEs can vary with the elastic variables, and thus problem P2 is always feasible. In the following, we solve P2. Using Lagrange decomposition theory [11], we first introduce constraints (9.2-3)-(9.2-5) into the optimization objective by associating Lagrange multipliers An, Un, and Mn. Let A, v, , ir, à, and x be the corresponding vectors of An, Un, Mn,Vn,dn, and xik, respectively. For P2, we give Lagrange dual problem P3 with respect to constraints (9.2-3)-(9.2-5): r,d,x (9.3) s.t. Constraints (9.2-1), (9.2-2), (9.2-6) and (9.2-7) where Lagrangian L(A, v, , r, d,x) is expressed as L(A,v, u,r,d,x) 9.1 A Framework of User Access Control for RAN Slicing (9.4) Here, strong duality holds. Therefore, we first solve P3 with the fixed Lagrange multipliers A, v, and , and then maximize g(A, v, u) to find the optimal solution to P2. Owing to the independence of ir, , ,, and x, L(A, v, , r, d, x) can be decoupled into two sub-functions and L2(A,v, ,), i.e. = + L2(A,v, ,x), where (9.5) (9.6) Hence, P3 is decomposed into two subproblems: P3(1) Il d) (9.7) s.t. Constraints (9.2-1), (9.2-2), (9.2-6) and (9.2-7) (9.8) s.t. Constraints (9.2-1), (9.2-2), (9.2-6) and (9.2-7) When A, v, are fixed, we solve P3(1) and P3(2) respectively. Focusing on P3(1), we find that the constraints (9.2-1), (9.2-2), (9.2-6), and (9.2-7) are unrelated to the optimization variables ir, d. Hence, P3(1) is actually an unconditional opti- mization