Optimization Based on Dual Decomposition

In this section, we propose a low complexity solution to (3.4) based on the Lagrange dual transformation. For the sake of interpretation, we rewrite (3.4) in a new form:

minimize (3.4) are removed temporarily and will be considered afterwards (in Appendix A.1). Although the original objective function f (·) is not convex, it can be transformed into a dual function, which is always concave regardless of the convexity of f (·). Hence traditional convex opti-mization techniques can be used to solve the transformed problem. We start from the Lagrangian of (3.7), which is

where µ = [µ₁, . . . , µ_K]^T and κ = [κ₁, . . . , κ_NT]^T are the vectors of Lagrange multipliers correspond to the rate and power constraints in (3.7). The dual function is defined as

d (µ, κ) =L (r^∗, µ, κ) , (3.9)

In another word, we let the Lagrange multipliers to be constants temporarily and find the r^∗ which minimizes the LagrangianL (r, µ, κ), this is the definition of the dual function d (µ, κ).

Next, we formulate the dual problem which aims to find the optimal Lagrange multipliers that maximize the dual function. In contrast with the dual problem, the original problem (3.7) is called the primal problem. The dual problem is equivalent to the primal problem if the original objective function f (·) is convex, otherwise, there exists a duality gap between these two prob-lems [18]. In our case, f (·) is not convex since it is a pointwise minimum of several convex functions. Nevertheless, [19] shows that this gap can be reduced by increasing the subcarrier size M . In order to find the r^∗ in (3.9), we first express (3.8) in another form:

Since data rate is a function of power, therefore, finding the r^∗which minimizesL is equivalent to finding p^∗_k,m,l, ∀k, m, l which minimize eL, so we set the latter to be our new goal. After making some arrangements to (3.11), it becomes

L =e

where

In the dual function, µ_k and κ_a are treated as constants temporarily, so the last two terms in (3.12) are unrelated terms. Therefore, minimizing eL is equivalent to minimizing L. On each subcarrier, when the user selection has been determined, we can obtain the BD precoding ma-trices F_k.m, ∀k, m, the minimal power and rate allocated on each spatial stream can be given by

For brevity, the derivations of (3.14) and (3.15) are left in Appendix A.1. We call this solution the competitive water-filling solution. The reason for the name will be explained later.

Since BD can mitigate the inter-user interference on the same subcarrier, multiple users can share the same bandwidth. The optimal user selection on each subcarrier would be to search over 2^Kuser combinations and find the one that minimizes

L (m) =b

For the overall M subcarriers, there would be M 2^K choices. As mentioned above, the duality gap approaches zero when M goes to infinity. However, this will make user selection problem to be computational infeasible. The complexity could be reduced by a suboptimal greedy user selection introduced in [12]: For each subcarrier, allocate the user that minimizes bL (m) on subcarrier m. Next, add another one from the remaining K − 1 users if bL (m) can be further reduced, and so on. Note that if bL (m) ≥ 0, there is no user allocated on this subcarrier, since

positive bL (m) will not minimize eL. As the number of users on this subcarrier increases, the BD precoder will project each user's channel to a more restricted space (see section 2.1.1), which makes the channels weak. Hence, it is not always the best to put all the users on each subcarrier, even though they do not interfere to each other after BD. The suitable number of users that allocated on each subcarrier can be found by the greedy user selection algorithm above. In this way, the maximum combination of users over the total M subcarriers becomes M

K∑−1 j=0

(_K−j

1

)=

M K(K+1)

2 , which is small compared to M 2^K when K is large. As for the globally optimal solution, even if the per-antenna power constraints are ignored and the minimal power which satisfies the user rate constraint is obtained by the water-filling solutions (which are (3.14) and (3.15) after setting κ_a= 0, ∀a), it still needs a search over 2^KM possibilities to find the optimal solution, which is computationally prohibitive.

So far, we have found the p^∗_k,m,l, ∀k, m, l that minimize eL and therefore the dual function d (µ, κ) is obtained. Next, we need to find the optimal µ and κ that maximize d (µ, κ). Since d (µ, κ) is concave, we can update µ and κ along some directions to find the optimal point. We adopt a special searching direction named supergradient [20]. In general, the supergradient at a point α∈ R^n×1 is defined as a vector χ∈ R^n×1which satisfies

d (α)e ≤ d (α) + χ^T (αe − α) , ∀eα̸= α. (3.17)

In our optimization problem (3.7), α comprises the Lagrange multiplier vectors µ and κ, and χ can be decomposed into two directions χ₁and χ₂, which are given by

χ₁ = M r^tar−

∑M m=1

r^∗_m (3.18)

and

χ₂ = g (r^∗)− p^con, (3.19)

where g (·) is the function defined in (3.7). The proofs are shown in Appendix A.2. Without

changing the direction, we divide the first supergradient by M , that is,

The second supergradient remains the same, which is

χ₂ = [χ_2,1, . . . , χ_2,a, . . . χ_2,NT]^T (3.22)

We update the two Lagrange multiplier vectors in an iteration manner:

µⁱ⁺¹_k = max{

where i is the iteration index, δ₁ⁱ and δ₂ⁱ are the two positive step sizes for µ and κ, respectively.

As we can see in (3.21), if the allocated rate to the user k exceeds the its target, the direction becomes negative and µ_kwill be reduced in the next iteration. On the contrary, µ_kwill increase if it falls below the target rate. The similar actions can be observed in (3.23). Since the rate (3.15) is directly proportional to µ_kbut inversely proportional to ϕ_a, consider a case that a user requests so much rate that the allocated power goes beyond the per-antenna power constraints, then χ₂becomes positive and hence increases κ, as a consequence, the power will be dropped.

However, it will be raised again since the target rates are not satisfied due to insufficient power.

This causes a struggle situation in our iterative algorithm and that is why the allocation scheme is named by competitive water-filling solution. On the other hand, if the required rates are not that

much or the power constraints are set to very high, the allocated rate to each user will gradually converge to their targets from an initial point without struggling.