In this section, and later in Chapters 3-5, we focus on the weighted sum of performance functions q7→α0q, q∈ Q, or, equivalently p7→
XK
k=1
αkΨ(γk(p)), p∈ P, (2.16) withα∈ Aand
A={α≥0 :kαk= 1}, (2.17)
as the objective in the optimization of power allocation. It is intuitive to require the norm-constraint in (2.17) to be the1-norm constraint. However, there is no loss in generality when other norms are taken, as is the case e.g. in Chapter 3.
The optimization of weighted aggregated performance given in (2.16) is the most common opti-mization goal under best-eort, or elastic trac [16], [28], [23]. In analogy to the original denition in [16] (for wired trac), best-eort trac comes from applications that are able to modify their QoS according to the achievable limits within the network and trac priorities. Hereby, the link weightsαk,1≤k≤K, in (2.16) are usually determined by the corresponding trac priorities.
With assumed decreasingness of Ψ, the problem of weighted aggregated performance optimiza-tion takes the form
minp∈P
XK
k=1
αkΨ(γk(p)). (2.18)
From geometry it is known that the power allocationpαsolving (2.18) generates the Pareto-optimal QoS vector qα = Ψ(Γ(pα)), which is the vector at which the hyperplane with normal vector α supports the set of all achievable QoS vectors, that is the performance regionQ[47]. In other words, any solution to the problem (2.18) is one-to-one associated, by mapping (2.10), with some solution of the scalarized vector optimization of the form
minq∈Qα0q, (2.19)
Problem (2.19) and Pareto optimality is illustrated in Fig. 2.1.
2.3.1 Global optimizers
We can show that log-convexity of the QoS-SIR mapping ensures the existence of only global optimizers of problem (2.18). The result is a consequence of convexity of the performance region.
Proposition 3 If Φ = Ψ−1 is log-convex, then any local minimizer of problem (2.18) is global as well, and the Kuhn-Tucker conditions are necessary and sucient optimality conditions, provided that P satises constraint qualication.
Proof By (2.16), one can see that any solution to the problem (2.18) is one-to-one associated, by mapping (2.10), with some solution to the problem
minq∈Qα0q,
which is convex due to convexity of Q, implied by log-convexity of Φ (Propositions 1, 2). By contradiction, assume the existence of at least two distinct local minimizersp,˜ pˇ∈ P of (2.18), with
2.3 Optimization of weighted aggregated performance 13
Q
qα= arg minq∈Q˜
P2
k=1αkq˜k
−α
q1 q2
Figure 2.1: An exemplary QoS region in the two-link case with a Pareto-optimal QoS vector qα for some weight vectorα.
only one of them, sayp˜, global. Let the distinct local minimizers of (2.19) uniquely associated with
˜
pandpˇ beq˜∈ Qandˇq∈ Q, respectively. By convexity of the problem (2.19), the local minimizers
˜
q and ˇq and all their convex combinationsq(t) = (1−t)˜q+tˇq,t∈(0,1), are also global solutions to (2.19) [48]. Thus, pˇ is a global minimizer of (2.18) as well, which contradicts the assumption and proves that all local minimizers of (2.18) are also global. The necessity and suciency of the Kuhn-Tucker conditions follows by the standard optimization theory due to satised constraint
qualication [48]. ¤
Existence of only global minimizers of problem (2.18) implies that any locally converging opti-mization routine nds a globally optimal power allocation. Thus, Proposition 3 implies that adaptive online power (re-) allocation according to (2.18) is signicantly facilitated for QoS functions with log-convex QoS-SIR dependence.
2.3.2 Matrix characterization of the solution
The constraint inequalities determining the domain in (2.18) take the form−p≤0,PK
k=1pk−P ≤0 in the downlink case (PP) and−p≤0,p−ˆp≤0in the uplink case (Ppˆ). With the Perron-Frobenius Theory (Section 2.1), the vectorial nonnegativity constraint on the power allocation can be replaced in both cases by the scalar inequality constraint ρ(Γ(p)V) < 1. With this, the Lagrangian of problem (2.18) can be written as
Lα(p, µ, ν) = XK
k=1
αkΨ(γk(p)) +µ(
XK
k=1
pk−P) +ν(ρ(Γ(p)V)−1) (2.20) in the case of sum-power constraint (e.g. downlink) and
Lα(p,µ, ν) = XK
k=1
αkΨ(γk(p)) + XK
k=1
µk(pk−pˆk) +ν(ρ(Γ(p)V)−1) (2.21)
in the case of individual power constraints (e.g. uplink), with µ, µ = (µ1, . . . , µK) and ν as the Lagrangean multipliers. Since the complementary slackness condition ν(ρ(Γ(p)V)−1) = 0 is a necessary optimality condition and we further haveρ(Γ(p)V)→1only ifp→ ∞, it follows that the optimum value of the Lagrange multiplier ν is ν= 0 [49], [48]. This lets us state the Kuhn-Tucker conditions ∇pLα(p, µ,0) = 0and ∇pLα(p,µ,0) = 0in the downlink and uplink, respectively, in a nice compact form. Letting functionp7→g= (g1(p), . . . , gK(p)),p≥0, with
gk(p) =αkΨ0(γk(p))γk(p)
pk , 1≤k≤K, (2.22)
we yield precisely
∇pLα(p, µ,0) =g(p)−V0Γ(p)g(p) +µ1= 0 in the downlink case and
∇pLα(p,µ,0) =g(p)−V0Γ(p)g(p) +µ= 0, in the uplink.
For QoS functions with log-convex QoS-SIR dependence, this yields with the remaining Kuhn-Tucker conditions and Proposition 3 a necessary and sucient matrix equation characterization of the optimal power allocation in (2.18) (not that the constraint qualication is satised in the cases of PP and Ppˆ).
Proposition 4 With Φ as a log-convex function, the power vectorp generating the SIR matrix Γ solves problem (2.18) if and only if it solves
p= (I−ΓV)−1Γσ2 g(p) =−(I −(ΓV)0)−1c, ρ(ΓV)<1
(2.23)
withc=µ1≥0,PK
k=1pk−P ≤0,µ(PK
k=1pk−P) = 0under sum-power constraint andc=µ≥0, p−pˆ≤0,µ0(p−p) = 0ˆ under power constraints per link.
Obviously, under lack of log-convexity of Ψ, Proposition 4 provides a necessary and sucient matrix equation characterization of a local minimizer of (2.18). In some sense, the structural simi-larity of the matrix equations in the optimality conditions (2.23) gives rise to ecient decentralized algorithmic solutions to problem (2.18) (Chapter 4).
2.3.3 Fairness of medium access
The links which are allocated zero transmit power are said to be idle. From the point of view of fairness in the network it is desirable when αk > 0 implies pk > 0 under the optimality in terms of (2.18), that is, when nonzero link priority implies a non-idle link under optimized weighted aggregated performance. Such feature ensures medium access for any nonzero trac priority at the optimum of weighted aggregated performance (a kind of medium access fairness). We can show that the class of QoS functions with log-convex QoS-SIR dependence provides such kind of fairness in medium access.
Proposition 5 Givenα>0 and a log-convex function Φ, any solution to (2.18) is positive.
2.3 Optimization of weighted aggregated performance 15
Proof We rst prove the following crucial Lemma.
Lemma If Φ = Ψ−1 is log-convex, thenΨ0(0) =−∞.
By assumed decreasingness and dierentiability of Φ, we have Ψ0(γ)<0,γ ≥0. Then, by Lemma 1, Ψ has a log-convex inverse if and only if Ψe is convex. Obviously, Ψe is convex if and only if Ψ0e(x) = Ψ0(ex)ex is nondecreasing. Take a series{xn}n∈N⊂R, withlimn→∞xn=−∞and assume by contradiction Ψ0(0) =c > −∞. Then, limn→∞Ψ0e(xn) = limn→∞Ψ0(exn)exn = c·0 = 0, due to the continuity of Ψ(implied by dierentiability [50]). Further, we have Ψ0e(xn) = Ψ0(exn)exn <
0, n ∈ N, due to Ψ0(γ) < 0, γ ≥ 0. Since this holds for any series {xn}n∈N ⊂ R such that limn→∞xn =−∞, we yield by separability of R that limx→−∞Ψ0e(x) = 0 and Ψ0e(x) <0,x ∈ R. But this contradicts nondecreasingness of Ψ0e and completes the proof of the Lemma.
Let now a series of power vectors {pn}n∈N be convergent to p˜ ∈ P and let, by contradiction,
˜
p be a solution to (2.18) such that p˜k = 0, for some k ∈ K. Then, it is clear from (2.1) and the assumption σ2k > 0 that for the k-th SIR function we have limn→∞γk(p(n)) = γk(˜p) = 0. Thus, with the Lemma above we have then
n→∞lim XK
k=1
αkΨ(γk(p(n))) = XK
k=1
αkΨ(γk(˜p)) =∞,
which contradicts the assumption that p˜ is a solution to (2.18) and completes the proof. ¤ 2.3.4 Convex reformulation of the problem
We showed that for QoS functions with log-convex QoS-SIR dependence the online power (re-) allocation is facilitated due to the existence of only global minimizers of problem (2.18). From the point of view of online solvability of problem (2.18) an even more desirable, but more restrictive, property is convexity of the problem statement (that is, convexity of the objective and the opti-mization domain [47]). Under convexity of the problem, powerful tools of convex optiopti-mization, such as interior point methods, can be used in the design of iterative optimization schemes. Convexity of the problem statement ensures good global convergence behavior of applied iterative schemes.
We show that under log-convexity of Φthe optimization problem (2.18) can be translated into an equivalent convex form by logarithmic transformation of the domain.
Proposition 6 Let Φbe log-convex and X ={x= logp:p∈ P}. Then, the function x7→
XK
k=1
αkΨ(γk(ex)), x∈ X, (2.24)
is convex and the optimization problem minx∈X
XK
k=1
αkΨ(γk(ex)) (2.25)
is a convex problem.
Proof With the denition of functionΨe, we can write for each addend in (2.24)
Ψ(γk(ex)) = Ψ(elogγk(ex)) = Ψe(logγk(ex)), 1≤k≤K. (2.26) By the assumption of log-convexity of Φ and by Lemma 1, Ψe is convex and decreasing (due to assumed decreasingness of Ψ). Further, it is known from [10] that the function logγk(ex),
1 ≤ k ≤ K, is concave. Thus, it follows by the standard result from convex analysis that the concatenation Ψe(logγk(ex)), 1 ≤ k ≤ K, is a convex function [50]. Convexity of the objective (2.24) as a sum of convex functions follows then immediately. With convexity of the setX (precisely, setsXP andXpˆ), convexity of the optimization problem (2.25) is implied and the proof is completed.
¤
In the view of the power-QoS mapping (2.10), Proposition 6 implies that the map from loga-rithmic power vectors to performance vectors
x7−→exp p7−→Γ7−→Ψ q is convex wheneverΦ = Ψ−1 is log-convex.
In Fig. 2.2 a simulative comparison of convergence is provided for two dierent QoS parameters with log-convex QoS-SINR map. The advantage of convexity is mirrored in Fig. 2 by the fact that the gradient method applied to the convex problem form performs as well as the more ecient BFGS (Broyden-Fletcher-Goldfarb-Shanno) method applied to the nonconvex problem (2.18). In contrast to the gradient method, the BFGS method utilizes approximative second-order information [47].
It has to be underlined that the reformulation of aggregated performance optimization (2.18) in the form (2.25) is allowable under much more general conditions than under log-convexity of the QoS-SIR dependence. Precisely, the domain in problem (2.18) can be transformed logarithmically when
p= arg min
˜ p∈P
XK
k=1
αkΨ(γk(p))>0
(note, that by Proposition 5 this is satised in particular for QoS functions with log-convex QoS-SIR dependence). In Chapter 4 however, we use the following slightly more restrictive condition allowing us to work with problem form (2.25).
Condition 1 Any local minimizerp of problem (2.18) satises p>0.
Finally, it has to be underlined that the convexity condition and convex problem form from Proposition 6 are essentially dierent (although similar at rst glance) from the ones used e.g. in [29] and relying on geometric programming approach. The reason is that in our case the QoS parameter is a function of link SINR, while in the multi-hop context of [29] the QoS parameters are dependent on source data-rates.