Joint User Selection, Beamforming and Power Allocation Algorithm 55

According to the result in Proposition 4.1, if at an arbitrary step, the VUL powers are in the setD = D1∪ D2, then, the problem is infeasible. However, it can be easily shown that, the PU power updates based on complementary slackness, in Section 3.1, enforce the renewed VUL powers to be in the setD2. This holds for both the fixed point and subgradient based algorithm, presented in Section 3.

In the case of the subgradient based algorithm in Table 3.2, this can be shown as follows. Let us consider q1(tout+ 1) and{uk(tout+ 1)} as the optimum VUL power vector and beamformers respectively, obtained after the convergence of the inner loop for q₂(t_out).

We then have:

η^Tq₁(t_out+ 1) = η^T(D (t_out+ 1)− G1(t_out+ 1))^−T 1 + G^T₂ (t_out+ 1) q₂(t_out)

(4.8a)

= p^T₁ (t_out+ 1) 1 + p^T₂ (t_out+ 1) q₂(t_out) (4.8b)

= p^T₁ (t_out+ 1) 1 + q^T₂ (t_out+ 1) 1 (4.8c)

> q^T₂ (t_out+ 1) 1. (4.8d)

Thus, {q1(t_out+ 1), q₂(t_out+ 1)} are in D2. A similar result holds for the fixed point algorithm in Table 3.1, if, instead of the updated SU VUL powers, the optima of the eigenvalue problemsEk(q₁(t) , q₂(t) , u_k(t + 1)) are employed, where t denotes the iteration number. This results as follows. Letting q_k, Ek(q1(t), q2(t), u_k(t + 1)), and stacking these elements into the vector q₁, we have

η^Tq₁ ≥ η^T(D (t + 1)− G1(t + 1))^−T 1 + G^T₂ (t + 1) q₂(t)

. (4.9)

The inequality in (4.9) holds due to VUL feasibility, as discussed in the third remark of Section 3.2. From (4.9), similar inequalities hold as in (4.8b)-(4.8d), thus we can conclude that η^Tq₁− q2(t + 1)^T1≥ 0. Note that optimal values of Ek(q₁(t) , q₂(t) , u_k(t + 1)) are already available after solving the eigenvalue problems in Step 2, and are therefore obtained at virtually no cost. Furthermore, the result of the k-th eigenvalue problem can be viewed as an intermediary VUL power of the k-th SU.

In conclusion, to test infeasibility, it is sufficient to test if

{qk(tout+ 1)}_k∈S,{ql(tout+ 1)}_{l∈SP U}

∈ D1, (4.10)

in the case of the subgradient based algorithm in Table 3.2, and similarly that

{Ek(q1(t) , q2(t) , u_k(t + 1))}_k∈S,{ql(t + 1)}_{l∈SP U}

∈ D1, (4.11) in the case of the fixed point algorithm in Table 3.1. If infeasibility is detected, an SU is removed, based on heuristics, which we derive in Section 4.2.2. Moreover, the feasibility test procedure can be terminated once a point in the set F has been found. This check comes at virtually no cost, as it only involves the verification of whether the PU interference constraints after the updates are respected. For completeness, we show in Table 4.1 the beamforming and admission control procedure, based on the subgradient method in Table 3.2. Naturally, the procedure can be adapted for the fixed point algorithm, where the infeasibility test is based on Eq. (4.11).

CHAPTER 4. Joint Downlink Beamforming and User Selection 57

Algorithm 4.1 Heuristic Deflation Algorithm

Step 1. Initialize{qk}_k∈S and{ql}_{l∈SP U} and the corresponding beamform-ing matrix such that we have an SU-feasible point

Step 2. Until convergence of{ql}l∈S_{P U}, iterate the following steps:

2.1 Update the SU uplink powers for fixed beamformers Eq. (3.50) 2.2 Update the unit norm beamformers for fixed powers Eq. (3.51) Step 3. Update the PU virtual uplink powers with Eq. (3.54)

Step 4. Perform the infeasibility test: [q₁, . . . , q_K+L+1] (t_out+ 1)∈ D1. If infeasible, remove user based on an appropriate heuristic, as described in Section 4.2.2 and go to Step 2.

Step 5. Perform convergence check

Table 4.1: Joint deflation based user selection and beamforming control, based on subgra-dient method.

4.2.2 Heuristic Selection

The proposed procedure assumes as initialization the largest SU feasible set. Then, the it-erative beamforming algorithm with feasibility control is started and, whenever infeasibility is detected, one user is removed at a time, based on appropriately chosen heuristics. The structure of the problem and its interpretation in the VUL domain, makes it possible to choose powerful heuristics. Recalling that infeasibility is detected when the VUL powers are in the set D1, it is reasonable to consider the removal of the user, whose elimination

‘moves’ the remaining set of VUL powers the furthest away fromD1. A potential candidate, to satisfy this, is the one with the largest weighted VUL power, i.e., γ_kq_k. The advantages of using this heuristic, in a deflation based scheme, lie in its simplicity and generally good performance, as shown in simulations. We refer to this scheme as ‘Fast Removal’, through the rest of the chapter.

This scheme performs well when there exists an SU with a significantly larger weighted VUL power. There may however occur cases when several SUs have similar such coefficients,

Algorithm 4.2 Look Ahead Step 1. for j=1,2

1.1 Solve PU(S^j(t), SP U) until {qk}_k∈S^j_{(t)∪SP U} converges or infeasibility is detected.

1.2 With the obtained{qk}_k∈S^j_{(t)∪SP U} compute

mj =





 P

k∈S^j(t)γ_kσ_k²q_k−PL+1

l=1 q_K+l if PU(S^j(t),S_{P U}) feasible;

|T | where T ={k ∈ S^j(t)| γkq_k> γ_jq_j} otherwise.

Return (m_j, infeasibility status) Step 2. if both PU S^j(t), S_{P U}

, j = 1, 2 have the same infeasibility status eliminate user j with the smallest m_j

else eliminate user j for which PU(S^j(t), S_{P U}) is feasible

Table 4.2: User removal based on depth one branch search. PU denotes generically the VUL problem and can be eitherPU1(S, S_{P U}) in (3.1) orPU2(S, S_{P U}) in (3.13)

e.g., when two SUs interfere with each other or create interference to the same PU. To overcome this problem we consider an alternative user selection scheme, in which a depth first tree search is performed. More specifically, when there is no user with a distinctively high weighted VUL power, we consider the two users, which attain the highest such values, as candidates for removal. Then, temporarily eliminating each one, we perform the iterative update and assess the evolution of the two candidates among the remaining users. Finally, we remove the SU, which at the next branch level has the poorest performance with respect to a chosen metric, e.g., weighted uplink power or interference created to the PUs. This removal approach, which we term ‘Look-Ahead’ is shown in Table 4.2, where for simplicity we consider only two SUs in the candidate set. For convenience, we use the notation S^j(t) to represent the set of users S, considered at iteration t, from each user j has been removed, i.e., S^j = S(t)\ {j}.

CHAPTER 4. Joint Downlink Beamforming and User Selection 59

5 6 7 8 9 10

0 20 40 60 80 100 120 140

SINR, dB

PercentageofCorrectDecisions

Random Removal Fast Removal Look Ahead

Figure 4.1: Percentage of correct decisions, in point of served number of users when the number of PUs is fixed to 6 and SINR target is increasing