8) to P. 10 Robust RAN Slicing Algorithm 10.1: VNS-ROBUST Input: P, an initial feasible solution X*; Output: The optimal solution x*; 1: Add an objective cut: P = (P|cTx<cTx*); 2: Solve the LP(P) to obtain an optimal solution X. while if Xj is binary for all j E B B then break; end if Set Dj Setq={jeBDj(x*,x)#0}l; while Solve the reduced problem P(x, Bk) to obtain x; if . then Local search x' in the neighborhood of x to update x*; Update P with an objective cut P = (P|cTx<cTx*); Solve the LP(P) to obtain an optimal solution x; break; k = k - kstep; - end if end while 22: end while 23: return the best solution x* of P if one is generated. We illustrate our VNS based heuristic algorithm (VNS-ROBUST). There are two loops in VNS-ROBUST: in the outer loop (between lines 3 and 22); we calculate the distance between the feasible solution x* of P and the solution X of LP(P) to decide the set Bk, which repre- sents those integer variables to be fixed in the inner loop. In line 10, we do not fix k, which denotes the size of Bk. Instead, we change k from kmax to kmin with a given step size kstep the inner loop of VNS-ROBUST (between lines 11 and 21), the initial searching point x' is obtained by solving the reduced problem P(x, Bk), using the Matlab solver (i.e. intlinprog). If the solution x' of the reduced problem is better than the incumbent x*, a local search (which will be elaborated in the local search algorithm within the neighborhood of x') is performed to update x*. After updating x*, an objective cut is added to prevent the local search process from revisiting solution x*. The solution x of LP(P) is solved to decide whether the stopping criteria is fulfilled to terminate both inner and outer loops. Otherwise, the solution x' is not better than the incumbent x*; we 10.4 Numerical Results set k = k - kstep to reduce the size of Bk, which means less integer variables will be fixed in the next iteration of the inner loop. The inner and outer loops proceed until the stopping criterion is reached. Algorithm 10.2: Local 