rank operation is not convex; thus, we drop the constraint (5.8h) first, as previously in P3. Additionally, the objective function relies on a realistic non-linear energy harvesting model, and it is not convex either. Essentially, it is a sum-of-ratio problem, and its global optimization is possible by applying the following transformations: (5.19a) (5.19b) After applying the S-Procedure of [25] to (5.19b), it becomes (5.20) Ak E K. Furthermore, according to [24, 92], if P6 has the optimal solutions Wt and V*, there exist two sets of vectors = {M1,M2, MK} and € such that the solu- tions are also optimal for the following equivalent parametric optimization problem: (5.21) The optimal solutions and the vectors should satisfy exp(-ak(r*-bi)))- - Mk = 0, (5.22a) Mk (1 + -exp(-ax(zt-by)))-1= = E K, (5.22b) where 5.4 Maximum Harvested Energy Problem Formulation Now, the objective function has the log-concave form and it can be solved given the sets and €. The iterative update of the vector sets can be carried out in the following way. Let us define the function F(u,€) = [ek (1 + = exp(-ax(7*-bk))) - Mk,...,Mk + exp(- bk))) - 1], Ak E K. The next set of values of and € can be updated by solving F(u, e) = 0. Specifically, in the q-th iteration, we update them as: (5.23) the Jacobian matrix of F(, €), w9 is the largest w' that satisfies||F(wp,ewp1 - l = 1,2 0  ' < 1, and 0<t<1[24,9 92]. A two-loop algorithm is proposed for solving the problem. The outer loop gives and € as the inputs of the inner loop, while the inner loop finds Wt and V*. Observe that in (5.20), there is a coupling variable TK, which is convex with a given p. Therefore, in the inner loop, we have to perform a one-dimensional search for P as well. The detailed algorithm is formulated in Algorithm 5.1. Algorithm 5.1 Robust Precoding Design for EH Maximization Problem 1: Input: Minimum required data rate Rk of SU k, noise power OKS and of channel uncer- tainty Q2 and 42 maximum allowed interference power Pn.p for PU n, maximum BS transmitted power P, and ra