Beam Power Alloation for more than two beams

Forthegeneralaseof

B > 2

^{beams,ananalytialtreatmentof(3.20)doesnot}unfortunately seemtratable, beauseofthe lakofonvexity. Therefore, weproposehereasuboptimal

- yet eient- iterativealgorithm that aims to inrease systemthroughput by alloating

poweroverthebeams. Thealgorithmtriestoidentifytheextremepointsofthesumrateand

ndthepowervetor

P

^{thatmaximizes(3.20). Theextremumofthesumratefuntionan}

be foundanalytially using Lagrangianduality theory and onsidering the

Karush-Kuhn-Tuker(KKT)onditions. LetWloG

B = M

^{anddenetheobjetivefuntion}

G (P) =

Inordertosolvetheoptimizationproblem

max

wemayformulatetheLagrangianfuntion as

L ( p , µ, ν) = G ( p ) +

where

ν ≥ 0

^and

µ ≥ 0

^{aredualvariables. Theostfuntion isneitheronvexnotonave}

withrespetto

{ P m } ^M m=1

^{,thereforeaglobaloptimalsolutionforanyhannelmodelishard}

toobtain. However,KKTonditionsareneessaryforextremum,whether loal orglobal,

of

G (P)

^{. By}dierentiatingwithrespetto

P m

^,wend

∂ G (P)

∂P m

+ ν m − µ = 0, 1 ≤ m ≤ M

^(3.33)

P m ≥ 0, 1 ≤ m ≤ M P − X

m

P m ≥ 0

TheKKT onditionsare neessaryand suient ifand onlyif the Hessian of (3.32) is a

negativedenitematrix. Forsuhlassofhannels,aglobalmaximumisidentiedthrough

theKKTonditionsabove. Forgeneralhannels,theKKTpointsanbeaglobalorloal

maximum,asaddle-point,orevenaglobalorloalminimum.

Iterative BeamPower Control Algorithm

Performingtransformationoftheprimalproblem(3.20)intoitsdualandsolvingthelatter

byKKTonditionsdoesnotguaranteeglobaloptimalprimalsolution. Astheprimalisnot

aonvexoptimizationproblem, thereouldbeaduality gap. Nevertheless,weproposean

iterativealgorithm,inspiredbytheiterativewater-lling(IWF)algorithm[73℄andtheKKT

solutionof (3.31), as ameans to identify the extremepointson the boundary of

P ^M

^{. In}

thisIterativeBeamPowerControlAlgorithm,eahuseriterativelymaximizesitsownrate

byperforming single-userwater-llingand treatingthemultiuserinterferenefromall the

otherusers(beams)asnoise. Clearly,ouralgorithmdoesnotseektondaglobaloptimum,

howeveritanprovidesigniantsum-rateimprovement.

AlgorithmI Let

P ⁽⁰⁾

^{betheinitialpointand}

I (P ⁽ⁱ⁾ ) = σ ² + P

j 6 =m P _j ⁽ⁱ⁾ η k j j

^bethe

inter-ferenefuntionat

i

^{-thiteration. Thestepsof thealgorithmaresummarizedin Table3.1.}

IterativeBeamPowerControlAlgorithm

Step1(Initialization)Set

P ⁽⁰⁾ = 0

Step2Foriteration

i = 1, 2, . . .

^,ompute

∀ k m ∈ S

^:

λ ⁽ⁱ⁾ _{k m} = _{I (P} ^{η km m} (i−1) ) = ^{η kmm}

σ ² + P

j6=m P _j ⁽ⁱ⁻¹⁾ η _{kj j}

Step3(Water-lling): let

π ⁽ⁱ⁾

^{bethesolutionof:}

π ⁽ⁱ⁾ =

^arg

max

π ≥ 0, P

m π m ≤ P

X

k m ∈S

log ₂

1 + π m λ ⁽ⁱ⁾ _{k m}

Step4(Update): let

P ⁽ⁱ⁾ = π ⁽ⁱ⁾

Table3.1: IterativeBeam PowerControlAlgorithmforSum-RateMaximization

Someobservationsareinorder:

At eah iteration

i

^{, one}

λ ⁽ⁱ⁾ _{k m} = ^{η km m}

(σ ² + P

j 6 =m P _j ^{(i − 1)} η _{km j} )

is alulatedfor eah user

k m

^using

P _j ^{(i − 1)} , j 6 = m

^{,itiskeptxedandtreatedasnoise. Giventhetotalpoweronstraint}

P

^,the

`water-llingstep'isaonvexoptimization problem similarto multiuserwater-llingwith

ommonwater-llinglevel. Thus,alltransmitpowersin

P

^{assignedtobeamsarealulated}

simultaneouslyinordertomaintainaonstantwater-llinglevel. Thealgorithmomputes

iteratively thebeam power alloation that leadsto sum rate inreaseand onverges to a

limitvaluegreaterorequaltothesumrateofequalpoweralloation. Formally,thepower

assignedtobeam

m

^atiteration

i

^yields

P m ⁽ⁱ⁾ = [µ − 1/λ ⁽ⁱ⁾ _{k m} ] ⁺

^,with

X

k m ∈S

[µ − 1/λ ⁽ⁱ⁾ _{k m} ] ⁺ = P

^,

where

µ

^{is theommon}water-llinglevel. Thebeampowerontrol for strategy3assigns transmitpowersoverthebeamsaordingto theiterativesolutionwhen theahievedsum

rateishigherthanthatoftheboundarypoints.

Convergene Issues As stated before, no global maximum is guaranteed due to the

lakofonvexityofsum-ratemaximizationproblem. Therefore,wedonotexpetthat the

onvergene pointof theiterativealgorithm begenerallyaglobal optimal powersolution.

Interestingly, it an be shown that the onvergene leads to a Nash equilibrium, when

onsidering that eah userpartiipatesin anon-ooperativegame. Theonvergeneto an

equilibrium pointanbeguaranteedsine

I ( P )

^isastandardinterferenefuntion [73,74℄.

The proof of existene of Nash equilibrium follows from an easy adaptation of the proof

in[75℄. However,theuniquenessoftheseequilibriumpointsannotbeeasilyderivedforthe

aseofarbitraryhannels.

Letusnowderiveanalytiallytheonvergenepointofthe2-beamaseusingthe

itera-tivealgorithmandompareitwiththeoptimalbeampowersolutiongivenbyTheorem3.2.

Atthesteadystate,sayiteration

s

^,wehavethat

( P ₁ ^(s) = µ − 1/λ ^(s) _{k 1}

fromthesumpoweronstraint. Upononvergeneofthealgorithm,

wehave that

P _i ^(s) = P _i ^{(s − 1)} , i = 1, 2

^{, whih results into a systemof equations}

AP ^T = b

It an be observed that (3.35) is dierent from (3.25a). Fortunately, it still provides a

heuristipoweralloationalgorithmandasshownthroughsimulationsinSetion3.9,there

is not a signiant redutionin sum rate by alloating the powerover beams using this

algorithm.

Reinterpretationin termsofSuessiveConvex Approximation

Inthis setion, weresortto GeometriProgramming (GP) [72℄ whih representsthestate

beome averypopular and powerfultehnique asit provides eient solutions in power

ontrol problems with non-linear objetive funtions and spei SINR onstraint, by

re-vealing thehidden onvexitystruture. Furthermore,the proposed solutionsareveryfast

and numerially eient, often exhibiting polynomial time omplexity. In partiular, we

apitalizeontheso-alledsuessive onvexapproximation (SCA)tehnique[72,76℄,whih

isshowntobeonvergentandturnsoutthat itoftenomputesthegloballyoptimalpower

alloation. Interestingly, the heuristi iterative algorithm proposed in Table 3.1 nds an

equivalentinterpretation,sineapplyingSCAtoourbeampowerontrolproblemresultsin

thesameiterativealgorithm. Werstlowerbound

log(1 +

^SINR

)

^{intheobjetivefuntion}

forsome

a

^and

b

^[76℄:

log(1 +

^SINR

) ≥ a log(

^SINR

) + b

^(3.36)

Applying(3.36)into theoptimizationproblem(3.20)resultsintherelaxation

max P

whih stillremainsanon-onvexproblemsinetheobjetivefuntion isnotonavein

P

^.

However,usingthetransformation

P ˜ m = log(P m )

^{wehavethefollowingonave}

maximiza-tionproblem:

DeningtheLagrangianfuntion as

D ( ˜ P , λ) =

weonsiderthedualproblem(3.38)thatis

min

λ max

P ˜ D ( ˜ P, λ)

^{. Thedualsolutionoftheinner}

maximizationproblem is given by the stationary point of the Lagrangian funtion (3.38)

with

λ

^xed. Dierentiatingwrt

P ˜ m

^{andapplyingtheinverse}transformation

P m = e ^{P ˜ m}

^we

formthefollowingxed-pointequation

∂ D

Remarkably, this xed point-equationprovidesthe samepoweralloation algorithm asin

Table3.1for

a m = 1, ∀ m

^{(wlog)where thepowersanbeupdated}iterativelyusing(3.39).

However,wenotethatazerodualitygapannotbeguaranteedformallyduetolakof

on-vexity,implyingthat notheoretialargumentanshowonvergenetotheglobaloptimum

forgenerallassofhannels.

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