rank of Wk should be 1 is also non-convex. In what follows, we first reformulate C1 in (5.10b) according to the S-Procedure of [25]. Lemma 5.1 C1 in (5.10b) can be reformulated as (5.11) Ak E K,i{k,k+1,...,K}, where Ck = Wk - Yk,min Wj + V) and - Yk,min conditioned on Ai,k Proof: Given = (5.9), we have wivwa (5.12) From the fact that Ahiah and according to the S-Procedure, the lemma is proved. Similarly, C3 in (5.10d) can be transformed into (5.13) where = = V, and Bn 0 is also a slack variable. Next, we apply similar manipulations to 5.10c, which becomes (5.14) where /ak+bk a constant. This condition holds, provided that ak > 0, which is always true in real systems. Then, applying the S-Procedure to (5.14), we have the following OkI+ hiz (5.15) Ak E K, where Ok 0. Therefore, P2 becomes (5.16a) 5.3 Power Minimization-Based Problem Formulation s.t. (5.11), (5.13), (5.15), (5.8e), (5.8f), (5.8g), (5.16b) Ak K, i = {k, k + 1, K}, An E K. (5.16c) Observe that we drop (5.8h), since it is not a convex term. This relaxation is commonly referred to as the semi-definite relaxation (SDR) technique. For the specific problem in P2, the following theorem proves that the optimal Wk has a limited rank. Theorem 5.1 If P2 is feasible, the rank of Wk, E K is always less than or equal to 2. Proof: See Appendix A. The transformed problem P3 is not convex because of the coupling variables p in (5.15) and (1 - p) in the denominator of (5.11). To be able to take advantage of the CVX software package, we introduce a pair of auxiliary variables. Specifically, way, (5.11), (5.13), and (5.15) become convex terms. Then, we have additional constraints for p and q: (5.17) It may be readily verified that this transformation does not change the optimal solution of P3. 5.3.2 Matrix Decomposition Now we proceed to find the solution of the problem P2, after which there is one more step to get the original solution for Wk. If Wk yields rank 1, we can simply write Otherwise, if Rank(Wx) = 2, we have several optional approaches to extract W*. name a 