[PPT] - Optimal Joint Provision of Backhaul and Radio Access Networks PowerPoint Presentation

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Motivation RAN Backhaul Joint Provision Numerical Results

Optimal Joint Provision of Backhaul and Radio Access Networks

Zhi-Quan (Tom) Luo

Joint work with Wei-cheng Liao, Mingyi Hong, Ruoyu Sun, Meisam Razaviyayn, Hang Zhang, Hadi Baligh University of Minnesota

IEEE CTW, May 25-28, 2014, Curacao

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SLIDE 2

Motivation RAN Backhaul Joint Provision Numerical Results

5G and Beyond: Key Features

Cell-less deployment of radio access network (RAN)
A large number of heterogeneous base stations connected via

a backhaul network

Virtualization: Software-defined, cloud-based provision of the

backhaul and RAN

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Motivation RAN Backhaul Joint Provision Numerical Results

Main Issues: downlink case

RAN: user-base station association, coordinated beamforming

for interference mitigation

Backhaul: multicommodity traffic engineering with

capacitated links

Joint provision and why: user-base station association
affects Backhaul: where to route the flow
affects RAN: direct link vs. interfering links

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Motivation RAN Backhaul Joint Provision Numerical Results

Joint Provision of RAN and Backhaul

In this talk, we describe an algorithmic approach (similar to those used for BIGDATA)

Tailored for large-scale SDN with both wired and wireless links
Achieves high system resource utilization
Well suited for distributed/parallel implementation

Approach: integration of two existing algorithms

The WMMSE algorithm for interference management in RAN
The ADMM algorithm for traffic engineering in Backhaul

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Motivation RAN Backhaul Joint Provision Numerical Results

Road Map

Resource management for RAN
Traffic engineering for Backhaul
Joint provision
Simulations ⇒ joint provision can provide 3x gain

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Motivation RAN Backhaul Joint Provision Numerical Results

Interfering Broadcast Channel (IBC)

K cell MIMO IBC (multicell downlink)
Each base station k serves Ik number of users in cell k
The Tx k uses the beamformer Vik to send the signal to Rx i

in cell k xk =

Ik

∑

i=1

Viksik

The received signal of the ik-th user in cell k:

yik = HikkViksik + ∑

ℓ̸=i

HℓkkVℓksℓk + ∑

j̸=k Ij

∑

ℓ=1

HikjVℓjsℓj + nik

Hikj: the channel matrix from Tx j to the Rx i in cell k
Interacell and intercell interference

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SLIDE 7

Motivation RAN Backhaul Joint Provision Numerical Results

General utility maximization

Sum-utility maximization problem:

max

{V} K

∑

k=1 Ik

∑

ik=1

uik(Rik) s.t.

Ik

∑

i=1

Tr(VikVH

ik) ≤ Pk, ∀ k = 1, 2, . . . , K

(P)

The rate function (define Qik VikVH

ik):

Rik log det   I+HikkQikHH

ikk

  ∑

(ℓ,j)̸=(i,k)

HikjQℓjHH

ikj+σ2 ikI

 

−1

 

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Motivation RAN Backhaul Joint Provision Numerical Results

System Utilities

Consider α-fairness utility functions
For α ≥ 0, it is defined as follows

Uα(R1, · · · , RK) =           

K

∑

k=1

R1−α

k

1 − α if α ̸= 1;

K

∑

k=1

log(Rk) if α = 1. (1)

Special cases: sum-rate, proportionally fair rate, harmonic

mean rate, max-min rate etc

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Motivation RAN Backhaul Joint Provision Numerical Results

Joint BS Association and Transceiver Design

Two Goals:
for each user ik, identify a small set of serving BSs Sik ⊆ Qk;
optimize transmit beamformers {vqk

ik }qk∈Sik ,ik∈Ik

|Sik| is small =

⇒ vik [ (v1

ik)H, · · · , (vQk ik )H]H should

contain only a few nonzero blocks

Sparse utility maximization!

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Motivation RAN Backhaul Joint Provision Numerical Results

Utility Maximization for Joint Clustering/Precoder Design

The beamforming vector vik should be group sparse ⇒

nonsmooth regularization.

A utility maximization problem [Hong et.al. 2013]

max

{v

qk ik }

∑

k∈K

( ∑

ik∈Ik

uik(Rik) ) + λ ∑

k∈K,qk∈Qk

∥vqk

ik ∥2

s.t. ∑

ik∈Ik

(vqk

ik )Hvqk ik ≤ Pqk, ∀ qk ∈ Qk, ∀ k ∈ K

(P1)

User ik served by one BS ⇔ vik has one nonzero block ⇔

|Sik| = 1.

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Motivation RAN Backhaul Joint Provision Numerical Results

Interference Management via Utility Maximization

An area of active research, many algorithms have been

proposed:

Game theory based, best response
Successive convex approximation
Pricing based, uplink-downlink duality
Distributed/parallel, Gauss-Seidel or Jacobi
Geometric programming, sparse optimization, stochastic

incremental

Many many contributors:
S. Barbarossa M. Bengtsson R. Berry
E. Bjornson
H. Boche
M. Chiang
R. Cui
D. Gesbert
G. Giannakis
A. Goldsmith R. Heath
M. Honig
E. Jorswieck
M. Juntti
E. Larsson
V. Lau
K. Ma
M. Moonen
B. Ottersten D. Palomar

J.S. Pang

A. Paulraj
M. Pesavento
A. Petropulu
V. Poor
T. Quek
M. Schubert
G. Scutari
N. Sidiropoulos S. Stanczak
A. Tolli
W. Utschick
L. VandendorpS. Vorobyov
W. Yu
R. Zhang

......

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Motivation RAN Backhaul Joint Provision Numerical Results

Complexity Analysis

1. Sum utility maximization for IBC [L.-Zhang’08]
K = 1, M, N arbitrary: convex opt. (e.g., water-filling)
K arbitrary, min{M, N} ≥ 3: NP-hard
2. Joint user-BS association and precoder design [Hong et.al.’13]

Suppose |Sik| = 1 for all ik and utility function is Uα(·). The system level problem (P1) is NP-hard when

either α = 0 (the Sum-Rate utility function);
or min(M, N) ≥ 3

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Motivation RAN Backhaul Joint Provision Numerical Results

A Polynomial Time Solvable Case

Consider the following network setting
K = B, i.e., the number of BSs and the number of users are

the same

Mb = Nk = 1 ∀ b, k, i.e., both the BSs and users have a

single antenna

Each BS can only serve a single user
The objective: maximize the minimum transmission rate (the

min-rate utility function)

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Motivation RAN Backhaul Joint Provision Numerical Results

A Special Case (Cont.)

In the above setting, the problem becomes a joint user-BS

matching and power allocation problem

Let p = [p1 · · · , pB] denote the BSs’ transmission power
the optimization problem is

max

p,a

min

k=1,...,K{Rk},

s.t. 0 ≤ pb ≤ Pb, b = 1, · · · , B |Hkak|2pak σ2

k + ∑ b̸=ak |Hkb|2pb

≥ 1, k = 1, . . . , K ak ̸= al, ∀ k ̸= l. (2)

Related work: Rashid-Farrokhi et.al.’97, ’98; Hanly’95

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Motivation RAN Backhaul Joint Provision Numerical Results

A Special Case (Cont.)

Result: if (2) is feasible, then
the optimal association can be found via maximum weighted

matching

the weights are {log |Hkb|2}(k,b)∈K×B
Algorithm:

Step 1: solve the maximum weighted matching problem to

btain a∗;

Step 2: fix a = a∗, solve a max-min SIR balancing problem to find optimal p∗

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Motivation RAN Backhaul Joint Provision Numerical Results

Max-min Fair Joint BS Assignment and Power Control

Figure: BS association via Max-log(weight) Matching

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Motivation RAN Backhaul Joint Provision Numerical Results

Two Commonly Used Utilities

Weighted sum–rate maximization:

max

{V} K

∑

k=1 Ik

∑

ik=1

αikRik s.t.

Ik

∑

i=1

Tr(VikVH

ik) ≤ Pk, ∀ k = 1, 2, . . . , K

(3)

Sum–MSE minimization:

min

{U,V} K

∑

k=1 Ik

∑

i=1

αikTr(Eik) s.t.

Ik

∑

i=1

Tr(VikVH

ik) ≤ Pk, ∀ k = 1, 2, . . . , K

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Motivation RAN Backhaul Joint Provision Numerical Results

Two Commonly Used Utilities

Weighted sum–rate maximization:

max

{V} K

∑

k=1 Ik

∑

ik=1

αikRik s.t.

Ik

∑

i=1

Tr(VikVH

ik) ≤ Pk, ∀ k = 1, 2, . . . , K

(4)

Weighted sum–MSE minimization:

min

{U,V} K

∑

k=1 Ik

∑

i=1

αikTr(W∗Eik) s.t.

Ik

∑

i=1

Tr(VikVH

ik) ≤ Pk, ∀ k = 1, 2, . . . , K

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Motivation RAN Backhaul Joint Provision Numerical Results

Two commonly used utilities (cont.)

Eik Es,n [ (ˆ sik − sik)(ˆ sik − sik)H] = (I − UH

ikHikkVik)(I − UH ikHikkVik)H

+ ∑

(ℓ,j)̸=(i,k)

UikHikjVℓjVH

ℓj HH ikjUH ik + σ2 ikUH ikUik,

The well known MMSE receiver:

Ummse

ik

= J−1

ik HikkVik,

where Jik ∑K

j=1

∑Ij

ℓ=1 HikjVℓjVH ℓj HH ikj + σ2 ikI.

Using the MMSE receiver leads to the MMSE matrix:

Emmse

ik

= I − VH

ikHH ikkJ−1 ik HikkVik.

We have Rik = log det

(( Emmse

ik

)−1)

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Motivation RAN Backhaul Joint Provision Numerical Results

A matrix weighted MMSE problem

Theorem: Let Wik ≽ 0 be the weight matrix for receiver ik. Then, the optimization problem min

{W,U,V} K

∑

k=1 Ik

∑

i=1

αik (Tr(WikEik) − log det(Wik)) s.t.

Ik

∑

i=1

Tr(VikVH

ik) ≤ Pk, ∀ k = 1, 2, . . . , K

(5) is equivalent to the weighted sum-rate maximization problem (3).

Equivalence means they have the same local/global

minimizers.

An extension of the WMMSE algorithm for the BC channel

(S. Christensen, R. Agarwal, etc., IEEE TWC’08)

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Motivation RAN Backhaul Joint Provision Numerical Results

The WMMSE algorithm

WMMSE algorithm: solve (5) by the block coordinate

descent algorithm

Closed form updates at each iteration
Subproblems are decomposed completely across users
We prove any limit point of the WMMSE algorithm is a

stationary point of the weighted sum rate maximization problem (3)

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Motivation RAN Backhaul Joint Provision Numerical Results

The pseudocode of the WMMSE algorithm

1 Initialize Vik’s such that Tr ( VikVH

ik

) = pk

Ik

2 repeat 3 W′

ik ← Wik,

∀ ik ∈ I 4 Uik ← (∑

(j,ℓ) HikjVℓjVH ℓjHH ikj + σ2 ikI

)−1 HikkVk, ∀ ik ∈ I 5 Wik ← ( I − UH

ikHikkVik

)−1 , ∀ ik ∈ I 6 Vik ← αik  ∑

(j,ℓ)

αℓjHH

ℓjkUℓjWℓjUH ℓjHℓjk + µ∗ kI

 

−1

HH

ikkUikWik, ∀ ik

7 until

∑

(j,ℓ) log det

( Wℓj ) − ∑

(j,ℓ) log det

( W′

ℓj

)

≤ ϵ

Note: no parameters to tune!

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Motivation RAN Backhaul Joint Provision Numerical Results

Extensions of the WMMSE Approach

WMMSE algorithm is quite flexible. Several extensions are possible.

deal with general utility functions
joint user grouping and transceiver design
joint base station association/activation and transceiver design
partial CSI: stochastic WMMSE for expected sum-rate

maximization

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Motivation RAN Backhaul Joint Provision Numerical Results

TE for the Backhaul

Two formulations for traffic engineering (TE) without

interference [Bertsekas 87,88]

i) Path-flow formulation: Paths are predetermined (dash

lines)

Flow rate on each path is nonnegative, i.e., x1 ∼ x4 ≥ 0
Flow rate for this source-destination pair rw = ∑4

i=1 xi

(Pros) Suitable for small/medium network
(Cons) Possible number of paths grow exponentially

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Motivation RAN Backhaul Joint Provision Numerical Results

Existing Works (cont.)

ii) Link-flow formulation: Paths are implicitly determined
Flow rate on each link is nonnegative, i.e., ∀ rij ≥ 0,

i, j ∈ {A ∼ E}

Each node satisfies flow rate balance equation, e.g., for node D

rBD + rCD = rDE

(Pros) Suitable for large network (# of variables grows linearly)
(Cons) Result in undesirably large number of flow paths

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Motivation RAN Backhaul Joint Provision Numerical Results

Existing Works (cont.)

TE in the presence of wireless interference is much more

challenging because

Link capacity is not fixed
Existence of multiuser interference
Existence of multiple parallel channels (or multiple antennas)
Cross-layer network utility maximization problem considers the

joint routing and resource optimization [Shroff 06][Chiang 07] [Xiao 04]

(Approximate) no interference
Dual decomposition ⇒ slow convergence

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Motivation RAN Backhaul Joint Provision Numerical Results

The Problem

Consider a backhaul with no interference (e.g., only wired

links or highly directional wireless links)

The nodes of this backhaul, V, consist of the set of network

routers N, the set of BSs B

The set of directed links:

L { (s, d) | (s, d) ∈ L, ∀s, d ∈ N ∪ B }

Let rl(m) denote the flow on link l for commodity m.
Each link l ∈ L has a fixed capacity Cl:

M

∑

m=1

rl(m) ≤ Cl (6)

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Motivation RAN Backhaul Joint Provision Numerical Results

The Problem (cont.)

Task: route M commodities using M flows, each with rate

rm, m = 1, . . . , M.

Each commodity m is associated with a source-destination

pair (S(m), D(m))

Flow conservation constraint: per node/flow

∑

l∈In(v)

rl(m) + 1{S(m)}(v)rm

incomming flow

= ∑

l∈Out(v)

rl(m) + 1{D(m)}(v)rm

utgoing flow

, ∀ m, v ∈ V (7) where In(b) (resp. Out(b)) are the set of links that go into (resp. come out of) node b.

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Motivation RAN Backhaul Joint Provision Numerical Results

The Problem (cont.)

Use the minimum rate as our optimization criterion:

max min

m rm,

s.t. (6) and (7) which is equivalent to a large linear program (LP) (P) : max r s.t. r ≥ 0, rm ≥ r, rl(m) ≥ 0, ∀ m, ∀ l ∈ L, (6) and (7)

Variables: M|L|, constraints: |L| + M|V|; general purpose LP

solvers can be quite slow

Customized Algorithm: decompose (P) into simple

subproblems of small sizes, and solve in parallel ⇒ ADMM!

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Motivation RAN Backhaul Joint Provision Numerical Results

Brief Review of ADMM Algorithm

The ADMM was developed in 1970s; recently popular for

large-scale optimization [Boyd 11]

The ADMM solves the following structured convex problem

min f(x) + g(z) (8) s.t. Ax + Bz = c, x ∈ C1, z ∈ C2 (9)

First introduce a quadratic penalization term

(ρ/2)∥Ax + Bz − z∥2, ρ > 0, to the objective function: min f(x) + g(z) + (ρ/2)∥Ax + Bz − c∥2 s.t. Ax + Bz = c, x ∈ C1, z ∈ C2.

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Motivation RAN Backhaul Joint Provision Numerical Results

The ADMM Algorithm

The Lagrangian function for the penalized problem is

Lρ(x, z, y) = f(x) + g(z) + yT (Ax + Bz − c) + (ρ/2)∥Ax + Bz − c∥2

The primal problem is given by

d(y) = min

x∈C1,z∈C2 Lρ(x, z, y)

(10)

The dual problem is

d∗ = max

y

d(y), (11) d∗ equals the optimal value of (8) under mild conditions

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Motivation RAN Backhaul Joint Provision Numerical Results

The Dual Ascent Algorithm

Dual Ascent Algorithm At each iteration t ≥ 1, first update the primal variable x and then update the dual multiplier:

           (xt+1, zt+1) = arg min

x∈C1,z∈C2L(x, z; yt)

= arg min

x∈C1,z∈C2f(x) + g(z) + ⟨yt, Ax + Bz − c⟩ + ρ

2∥Ax + Bz − c∥2, yt+1 = yt + α(Axt+1 + Bzt+1 − c),

where α > 0 is the step size for the dual update.

Note: ∇d(yt) = Axt+1 + Bzt+1 − c ⇒ dual ascent

d(yt+1) ≥ d(yt).

However, the minimization (primal step) can be difficult.
Since the objective is separable, we may perform the primal step

inexactly using block coordinate descent..., ⇒ ADMM!

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Motivation RAN Backhaul Joint Provision Numerical Results

The ADMM Algorithm

Alternating Direction Method of Multipliers (ADMM) At each iteration r ≥ 1, first update the primal variable blocks in the Gauss-Seidel fashion and then update the dual multiplier:

         xt+1 = argmin

x∈C1 L(x, zr; yt),

zt+1 = argmin

z∈C2 L(xt+1, z; yt),

yt+1 = yt + α(Axt+1 + Bzt+1 − c),

where α > 0 is the step size for the dual update.

Inexact primal minimization ⇒ (Axt+1 + Bzt+1 − c) is no

longer the dual gradient!

Dual ascent property d(yt+1) ≥ d(yt) is lost ⇒ complications

in the convergence analysis

Consider α = 0 or ≈ 0 ...

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Motivation RAN Backhaul Joint Provision Numerical Results

The ADMM Algorithm (cont.)

The Alternating Direction Method of Multipliers (ADMM)
ptimizes the augmented Lagrangian function one block

variable at each time [Hong-Luo 12, Bertsekas 10]

Recently found lots of applications in large-scale structured
ptimization; see [Boyd 11] for a survey
Highly efficient, especially when the per-block subproblems are

easy to solve (with closed-form solution)

If the optimal solution set is non-empty, and if AT A and

BT B are invertible, then every limit point of {xk, zk} is an

ptimal solution
The convergence rate of ADMM can be enhanced via

dynamically adjusting ρ or over-relaxation

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Motivation RAN Backhaul Joint Provision Numerical Results

An ADMM Approach for Multi-commodity Routing

Apply ADMM to solve coupled problem (P) with easy and

parallel subproblems

To decouple the elements of V from the conservation

constraints (7), the following slack variables are introduced

}

⇒ The constraints are then decoupled

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Motivation RAN Backhaul Joint Provision Numerical Results

An ADMM Approach for Multi-commodity Routing

Specifically, problem (P) is equivalent to

( ˆ P) : max (r + ˆ r)/2 s.t. r ≥ 0, rm ≥ r, rl(m) ≥ 0, m = 1 ∼ M,

M

∑

m=1

rl(m) ≤ Cl, l ∈ L ∑

l∈In(v)

ˆ rv

l (m) + 1{S(m)}(v)ˆ

rv

m

= ∑

l∈Out(v)

ˆ rv

l (m) + 1{D(m)}(v)ˆ

rv

m, m = 1 ∼ M, ∀v ∈ V,

and (12).

Objective function and constraints (except (12)) are separable
ver r and ˆ

r ⇒ Satisfy the structure of ADMM algorithm!

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Motivation RAN Backhaul Joint Provision Numerical Results

An ADMM Approach for Multi-commodity Routing

ADMM approach Algorithm 1 is outlined as follows

where δ {δ, δS(m)

m

, δD(m)

m

, δs

l (m), δd l (m) | m = 1 ∼ M, ∀ l =

(s, d) ∈ L} is the dual variables for (12).

Theorem Every limit point of the sequence {r(k)} generated

by Algorithm 1 is an optimal solution of problem (P).

Each step of Algorithm 1 will be discussed in details (iteration

index will be dropped for simplicity)

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Motivation RAN Backhaul Joint Provision Numerical Results

Solving r

The first step (solving r) can be decoupled into two parts
{r, rm | m = 1 ∼ M}
{rl(m) | m = 1 ∼ M, ∀ l ∈ L}
Subproblem for {r, rm | m = 1 ∼ M}:

ρ1 )2  s.t. rm ≥ r, m = 1 ∼ M, r ≥ 0.

⇒ Solved by bisection search over r ≥ 0.

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Motivation RAN Backhaul Joint Provision Numerical Results

Solving r (cont.)

Subproblem for {rl(m) | m = 1 ∼ M, ∀ l ∈ L}:

⇒ Decoupled over each link ⇒ For link l = (s, d) ∈ L, the following problem is solved

min

M

∑

m=1

[( ˆ rs

l (m) − rl(m) − δs l (m)

ρ1 )2 + ( ˆ rd

l (m) − rl(m) − δd l (m)

ρ1 )2] s.t.

M

∑

m=1

rl(m) ≤ Cl, rl(m) ≥ 0, m = 1 ∼ M.

⇒ Efficiently solved via by bisection procedure.

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Motivation RAN Backhaul Joint Provision Numerical Results

Solving ˆ r and Updating Dual Variables

The second step (solving ˆ

r) can be decoupled into two parts

{ˆ

rS(m)

m

, ˆ rD(m)

m

, ˆ rs

l (m), ˆ

rd

l (m)} and ˆ

r

Subproblem for {ˆ

rS(m)

m

, ˆ rD(m)

m

, ˆ rs

l (m), ˆ

rd

l (m)}:

⇒ Decouple across nodes. For node v ∈ V, the subproblem is

min ∑

l∈In(v) ∪ Out(v)

( ˆ rv

l (m) − rl(m) − δv l (m)

ρ1 )2 + 1{S(m),D(m)}(v) ( ˆ rv

m − rm − δv m

ρ1 )2 s.t. ∑

l∈In(v)

ˆ rv

l (m) + 1{S(m)}(v)ˆ

rv

m =

∑

l∈Out(v)

ˆ rv

l (m) + 1{D(m)}(v)ˆ

rv

m

⇒ One equality constraint only ⇒ Closed-form solution

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Motivation RAN Backhaul Joint Provision Numerical Results

Solving ˆ r and Updating Dual Variables (cont.)

Subproblem for ˆ

r: An easy unconstraint quadratic problem

max ˆ r/2 − (ρ1/2) ( ˆ r − r − δ ρ1 )2 ⇒ ˆ r⋆ = r + 1 + 2δ 2ρ1 .

The third step (update the Lagrange multipliers δ(t+1)) is

(m)), m = 1 ∼ M, ∀l = (s, d) ∈ L

They can be updated locally by each node

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Motivation RAN Backhaul Joint Provision Numerical Results

Implementation Issues for Algorithm 1

Low complexity, scales well with the network size

Each step is (semi)closed-form and solvable in parallel across links/nodes The per-iteration complexity is in the order of O(|L| + |V|).

Distributed implementation
All computation can be distributed to the nodes
A single master node that can communicate with all source

and destination nodes is needed

The neighboring nodes exchange 2M real symbols in each

iteration

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Motivation RAN Backhaul Joint Provision Numerical Results

Implementation of Algorithm 1 (cont.)

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Motivation RAN Backhaul Joint Provision Numerical Results

Implementation of Algorithm 1 with Zones of Nodes

For SDN, each cloud center manages a subset of nodes within

geographical zone

centralized computation within each zone
distributed computation/message exchange between zones
Denote v ∈ Zi if node v is within the ith zone
Modify variable splitting procedure:

rl(m) = ˆ rs

l (m), rl(m) = ˆ

rd

l (m),

∀ l = (s ∈ Zi, d ∈ Zj) ∈ L, i ̸= j, m = 1 ∼ M, rl(m) = ˆ rl(m), ∀ l = (s ∈ Zi, d ∈ Zj) ∈ L, i = j, m = 1 ∼ M,

i.e., only the link rates on the boundary are split

Links belong to bordering links (BD) or interior links (IT)

BDi {l = (s, v), l = (v, d) ∈ L | ∀ v ∈ Zi, s, d ∈ Zk, k ̸= i} ITi {l = (s, v), l = (v, d) ∈ L | ∀s, d, v ∈ Zi}

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Motivation RAN Backhaul Joint Provision Numerical Results

Implementation of Algorithm 1 with Zones of Nodes

Similar ADMM subproblems except the second step for zone i:

min ∑

l∈BDi

( ˆ rv

l (m) − rl(m) − δv l (m)

ρ1 )2 + ∑

l∈ITi

( ˆ rl(m) − rl(m) − δv

l (m)

ρ1 )2 + 1{S(m),D(m)}(v) ( ˆ rv

m − rm − δv m

ρ1 )2 s.t. ∑

l=(s,v)∈L

( 1{∪

k̸=i Zk}(s)ˆ

rv

l (m) + 1{Zi}(s)ˆ

rl(m) ) + 1{S(m)}(v)ˆ rv(m) = ∑

l=(v,d)∈L

( 1{∪

k̸=i Zk}(d)ˆ

rv

l (m) + 1{Zi}(d)ˆ

rl(m) ) + 1{D(m)}(v)ˆ rv(m), ∀ m = 1 ∼ M.

No closed-form solution – efficient network optimization

algorithms, e.g., relax code [Bertsekas 87]

Pros: faster convergence rate, less info. exchange

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SLIDE 46

Motivation RAN Backhaul Joint Provision Numerical Results

Numerical Experiment

For each commodity, the source (destination) node is a

randomly selected network router (BS)

All simulation results are averaged over 100 realizations
The ADMM stoping criterion is

Maximum constraint violation ≤ 10−4 Maximum relative increase of objective ≤ 10−3

We set the stepsize ρ1 = 0.01 if not stated
Algorithm 1 is implemented in C language

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SLIDE 47

Motivation RAN Backhaul Joint Provision Numerical Results

Parallel Implementation

−1500 −1000 −500 500 −1000 −500 500 1000 1500 2000 2500 3000 (m) BSs Gateway BSs Network routers

The performance of Algorithm 1 is tested using the network

topology provided by Huawei (114 BSs and 12 network routers)

The BS nodes are split into 8 cores, and all RAN nodes

belong to 1 core.

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SLIDE 48

Motivation RAN Backhaul Joint Provision Numerical Results

Parallel Implementation

The computation time vs the # of commodities

(AMD Opteron 8356 2.3GHz)

# of Commodities 50 100 200 300 Time (s) (Sequential) 1.04 2.03 4.73 8.53 Time (s) (Parallel) 0.20 0.37 0.75 1.10 Time (s) (Gurobi) 0.20 0.64 1.65 2.51 # of Variables 1.4×104 2.9×104 5.8×104 8.7×104 # of Constraints 2.1×104 4.2×104 8.4×104 1.3×105

As the # of commodities increases, the efficiency

improvement of parallel implementation increases.

2 times faster than commercial LP solver – Gurobi
Time is approximately linear over # of commodities.

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SLIDE 49

Motivation RAN Backhaul Joint Provision Numerical Results

Comparison of Network Decomposition Algorithms

Compare with i) dual decomposition [Chiang07] and ii) F.

Kelly’s algorithm [Kelly14]

Random connected graph with 100 network routers.
Each network router connects to 3 network routers.
Proportional fairness with path-flow formulation
Path for each commodity is the shortest path
# of Commodities=50: (ρ1 = 0.5)

100 200 300 400 500 10 20 30 40 50 Iteration Objective ADMM

F. Kelly

Dual Decomposition 100 200 300 400 500 10

−4

10

−2

10 10

2

Iteration

Max. Capacity Violation

ADMM

F. Kelly

Dual Decomposition 49 / 74

SLIDE 50

Motivation RAN Backhaul Joint Provision Numerical Results

Comparison of Network Decomposition Algorithms

# of Commodities=100: (ρ1 = 0.5)

100 200 300 400 500 10 20 30 40 50 Iteration Objective ADMM

F. Kelly

Dual Decomposition 100 200 300 400 500 10

−4

10

−2

10 10

2

Iteration

Max. Capacity Violation

ADMM

F. Kelly

Dual Decomposition

Maximum capacity violation metric:
Algorithm 1 has faster convergence rate

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SLIDE 51

Motivation RAN Backhaul Joint Provision Numerical Results

Further Enhancement

−600 −400 −200 200 400 600 −500 500 1000 1500 2000 2500 3000 (m) BSs Gateway BSs Network routers

Acceleration enhancement of Algorithm 1 is tested using the

tree network topology provided by Huawei (57 BSs and 12 network routers)

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SLIDE 52

Motivation RAN Backhaul Joint Provision Numerical Results

Further Enhancement (cont.)

The # of iterations for Algorithm 1 with/without acceleration

enhancement

# of Pairs 30 50 80 100 300 # of Iterations (ρ1 = 0.01) 615 620 669 654 644 # of Iterations w/ dynamically adjust ρ (ρ1 = 0.001) 293 293 318 306 300 # of Iterations w/ dynamically adjust ρ & over-relaxation (ρ1 = 0.001) 285 285 298 291 281

More than 50% reduction in the # of iterations
With relaxation method, up to additional 5% reduction.
CPU times are similarly halved.

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SLIDE 53

Motivation RAN Backhaul Joint Provision Numerical Results

Algorithm 1 with Zoning

−600 −400 −200 200 400 600 −1000 1000 2000 3000 4000 (m) (m) BSs BSs connected to network routers Network routers

Accelerate Algorithm 1 with zones of nodes is tested using the

mesh network topology provided by Huawei (5 zones with 57 BSs and 12 network routers)

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SLIDE 54

Motivation RAN Backhaul Joint Provision Numerical Results

Algorithm 1 with Zoning (cont.)

300 400 500 600 700 800 900 1000 # of ADMM iterations ρ=0.001 # of commodities=20 # of commodities=50 # of commodities=100 # of commodities=200 Decompose to nodes Decompose to zones of nodes

# of ADMM iterations significantly decreases – less slack

variables

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SLIDE 55

Motivation RAN Backhaul Joint Provision Numerical Results

Agenda

The Multi-Commodity Routing Problem

Brief Review of ADMM Algorithm A Distributed ADMM Approach

The Joint Power Allocation and Routing Problem

Algorithm Outline An ADMM Approach for Updating {r, p}

Numerical Results

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SLIDE 56

Motivation RAN Backhaul Joint Provision Numerical Results

Problem Formulation: Wireless Interfering Links

Consider a more general problem that takes wireless

interfering links into consideration

U: the set of mobile users
K: # of orthogonal frequency tones in each wireless link
Lwl = {(s, d, k)|s ∈ B, d ∈ U, k = 1 ∼ K}: the set of wireless

links

pk

ds: scalar transmit precoder on link (s, d, k)

rl(m): rate on link l for commodity m
Assume a per-BS power budget constraint

K

∑

k=1

∑

d:(s,d,k)∈Lwl

|pk

ds|2 ≤ ¯

ps, ∀ s ∈ B (13)

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SLIDE 57

Motivation RAN Backhaul Joint Provision Numerical Results

Problem Formulation: Wireless Interfering Links (cont.)

The rate constraint for a link l = (s, d, k) ∈ Lwl is

M

∑

m=1

rl(m) ≤ ¯ rl(p) log     1 + |hk

ds|2|pk ds|2

∑

(s′,d′,k)∈I(l) (s′,d′,k)̸=l

|hk

ds′|2|pk d′s′|2 + σ2 d

     (14)

where

p {pk

ds | ∀ (s, d, k) ∈ Lwl}

σ2

d: the AWGN at receiver d,

hk

ds ∈ C: the wireless channel for l = (s, d, k)

I(l) ⊆ Lwl: the set of links that interfere link l, i.e.,

I(l) {(s′, d′, k) | hk

ds′ ̸= 0, (s, d, k) = l}.

The rate constraint is nonconvex with respect to p!

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SLIDE 58

Motivation RAN Backhaul Joint Provision Numerical Results

Problem Setting for Wireless Links (cont.)

Task: Jointly perform 1): routing of M commodity flows, and

2) allocating transmit power on each wireless link. (Q) : max r s.t. r ≥ 0, rm ≥ r, rl(m) ≥ 0, m = 1 ∼ M, ∀ l ∈ L (6), (7), (13), and (14).

Difficult joint optimization problem
Wireless link rate constraints (14) are nonconvex
Flow conservation constraints couple all variables
Multiple frequency tones and multiple antenna at BS makes

the problem NP-hard [Razaviyayn 13]

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SLIDE 59

Motivation RAN Backhaul Joint Provision Numerical Results

The Proposed N-MaxMin WMMSE Algorithm

To handle the nonconvex problem (Q), we exploit the

following rate-MSE relationship

Lemma [Razaviyayn 13] For a given link l = (s, d, k) ∈ Lwl,

its rate ¯ rl(p) can be equivalently expressed as ¯ rl(p) = max

ul,wl 1 + log(wl) − wlEl(p, ul)

(15) where

For link l = (s, d, k), the MSE at user :

El(p, ul) 1 + σ2

d|ul|2 − 2Re{u∗ l hk ds}pk ds

+ ∑

(s′,d′,k)∈I(l)

|ul|2|hk

ds′|2|pk d′s′|2

ul: the receive beamformer
wl: the weighting coefficient of MSE

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SLIDE 60

Motivation RAN Backhaul Joint Provision Numerical Results

The Proposed N-MaxMin WMMSE Algorithm (cont.)

Given the rate-MSE relationship, ¯

rl(p) is replaced with fixed u {ul | l ∈ Lwl} and w {wl | l ∈ Lwl}:

( ˆ Q) : max r s.t. r ≥ 0, rm ≥ r, rl(m) ≥ 0, m = 1 ∼ M, ∀ l ∈ L

M

∑

m=1

rl(m) ≤ 1 + log(wl) − wlEl(p, ul), ∀ l ∈ Lwl, (quadratic) (6), (13), and (7)

⇒ Convex for each u, w, or {r, p} while fixing others ⇒ Propose to iteratively update u, w, and {r, p}

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SLIDE 61

Motivation RAN Backhaul Joint Provision Numerical Results

The Proposed N-MaxMin WMMSE Algorithm (cont.)

Outline of the proposed N-MaxMin WMMSE Algorithm

(Algorithm 2)

Theorem The sequence {r(t), p(t)} generated by Algorithm 2:
converges to the stationary solution of problem (Q)
every global optimal solution of problem (Q) corresponds to a

global optimal solution of the reformulated problem ( ˆ Q) with the same objective value

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SLIDE 62

Motivation RAN Backhaul Joint Provision Numerical Results

Updating Steps for Each Variable Set

Fixing p, the optimal u (resp. w) is obtained in closed-form

for any tuple l = (s, d, k) ∈ Lwl:

ul =   ∑

(s′,d′,k)∈I(s,d,k)

| hk

ds′|2|pk d′s′|2 + σ2 d

 

−1

hk

dspk ds,

wl = ( 1 − (hk

dspk ds)∗ul

)−1 .

⇒ Decouple over each wireless link

Fixing u and w problem ( ˆ

Q) is a large convex problem with coupled flow conservation constraints ⇒ Apply ADMM algorithm again!

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SLIDE 63

Motivation RAN Backhaul Joint Provision Numerical Results

An ADMM Approach for Updating {r, p}

To decouple the constraints, the following slack variables

besides (12) are introduced

rl(m) = rs

l (m), rl(m) = rd l (m), m = 1 ∼ M, ∀ l = (s, d, k) ∈ Lwl,

pk

d′s′,ds = pk ds, ∀ (s, d, k), (s′, d′, k) ∈ Lwl, ∀ (s, d, k) ∈ I(s′, d′, k).

With such variable splitting, each variable pk

d′s′,ds will only

appear in a single rate-MSE constraint

The outline of the proposed ADMM approach for {r, p} is

given below where ˆ p {pk

d′s′,ds | ∀(s, d, k) ∈ Lwl, (s, d, k) ∈ I(s′, d′, k)}.

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SLIDE 64

Motivation RAN Backhaul Joint Provision Numerical Results

An ADMM Approach for Updating {r, p} (cont.)

Similar to Algorithm 1, the proposed ADMM approach for

updating {r, p} has the following properties

Efficiency — (semi)closed-form and solvable in parallel
Distributed implementation with local information exchange
Moreover, problem (Q) can be extended to include

per-commodity QoS requirements

Commodity q of a set Q is required to satisfy minimum rate rq
rm ≥ 0, ∀ m, is modified as

{ rm ≥ rq, ∀ m ∈ Q rm ≥ r, ∀ m ∈ Qc .

The proposed Algorithm 2 can then be easily modified.

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SLIDE 65

Motivation RAN Backhaul Joint Provision Numerical Results

Numerical Experiment

The network topology is the same as the tree topology in the

routing problem (57BSs and 12 network routers)

Each frequency tone has 1MHz, and number of K = 3
The channel is generated as CN(0, (200/d)3) where d is the

distance between BS and user

ρ1 is chosen as 0.1
The termination criterion is chosen the same as the routing

problem

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SLIDE 66

Motivation RAN Backhaul Joint Provision Numerical Results

Numerical Experiment (cont.)

For comparison, the following two heuristics are used
Heuristic 1:

(a) Each user chooses the BS having strongest channel and frequency pair with itself as the serving BS (b) Each BS uniformly allocates its power budget to each served user while the power budget for each frequency is uniformly allocated. (c) With this fixed power allocation, the capacity of the wireless links are available (d) Maximize the minimum achievable rate with predetermined optimal routing

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SLIDE 67

Motivation RAN Backhaul Joint Provision Numerical Results

Numerical Experiment (cont.)

Heuristic 2:

(a) Each BS uniformly allocates its power budget to each

rthogonal subcarrier tone

(b) The max-min problem is solved under additional interference-free constraint

max r s.t. rm ≥ r, rl(m) ≥ 0, m = 1 ∼ M, ∀l ∈ L

M

∑

m=1

rl(m) ≤ αl log ( 1 + |hk

ds|2¯

ps/K σ2

d

) , ∀ l = (s, d, k) ∈ Lwl ∑

n∈I(l)

αn = 1, αl ∈ {0, 1}, ∀ l, n ∈ Lwl, (6) and (7).

(c) The integer constraint is hard ⇒ relax to αl = [0, 1]

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SLIDE 68

Motivation RAN Backhaul Joint Provision Numerical Results

Numerical Experiment (cont.)

Each mobile user is served only by BSs within 300 meters

while being interfered by all BSs.

More than twice of minimum rate of the heuristic algorithms.

5 10 15 20 25 30 2 4 6 8 10 # of commodities Min−rate (Mnats/s) N−MaxMin WMMSE (p=10dB) Heuristic1 (p=10dB) Heuristic2 (p=10dB) N−MaxMin WMMSE (p=20dB) Heuristic1 (p=20dB) Heuristic2 (p=20dB) 5 10 15 20 2 4 6 8 10 p (dB) Min−rate (Mnats/s) N−MaxMin WMMSE Heuristic1 Heuristic2

M=5 M=30

⇒ Proper power allocation is needed for problem (Q).

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SLIDE 69

Motivation RAN Backhaul Joint Provision Numerical Results

Numerical Experiment (cont.)

Each mobile user is served only by BSs within 300 meters

while being interfered by BSs within 800 meters.

Power budget for each BS: 10dB, ρ2 = 0.005 (for precoder

variables)

10 20 30 40 0.5 1 1.5 2 2.5 3 3.5 4 # of N−MaxMin iterations Min−rate (Mnats/s) # of Commodities=10 # of Commodities=20 # of Commodities=30 10 20 30 40 100 200 300 400 500 # of N−MaxMin iterations # of ADMM iterations # of Commodities=10 # of Commodities=20 # of Commodities=30

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SLIDE 70

Motivation RAN Backhaul Joint Provision Numerical Results

Numerical Experiment (cont.)

Each mobile user is served only by BSs within 300 meters

while being interfered by BSs within 800 meters.

Power budget for each BS: 20dB, ρ2 = 0.001 (for precoder

variables)

10 20 30 40 1 2 3 4 5 6 # of N−MaxMin iterations Min−rate (Mnats/s) # of Commodities=10 # of Commodities=20 # of Commodities=30 10 20 30 40 100 200 300 400 500 600 700 # of N−MaxMin iterations # of ADMM iterations # of Commodities=10 # of Commodities=20 # of Commodities=30

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SLIDE 71

Motivation RAN Backhaul Joint Provision Numerical Results

Numerical Experiment (cont.)

Computational time for the first 10 N-MaxMinWMMSE

iterations:

10 15 20 25 30 10

2

# of end−to−end commodity pairs Computational time (s) Power budget=10dB Power budget=20dB

⇒ Within 3.5 minutes for all considered scenarios without exploiting parallel programming

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SLIDE 72

Motivation RAN Backhaul Joint Provision Numerical Results

Further Enhancement

Apply the acceleration enhancement of ADMM
Power budget for each BS: 10dB

10 20 30 40 50 1 2 3 4 5 # of N−MaxMin WMMSE iterations Min−rate (Mnats/s) Original, M=10 Acce, M=10 Original, M=20 Acce, M=20 10 20 30 40 50 100 200 300 400 500 # of N−MaxMin WMMSE iterations # of ADMM iterations Original, M=10 Acce, M=10 Original, M=20 Acce, M=20

# of inner iterations decrease – especially in the first few

iterations

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SLIDE 73

Motivation RAN Backhaul Joint Provision Numerical Results

Final Remarks

Joint provision of backhaul and RAN offers great potential to

improve user QoS and network throughput.

For routing only problem, we develop a distributed/parallel

algorithm; two times faster than commercial solvers

For joint routing and power allocation problem, we exploit the

rate-MSE relationship, and develop an algorithm capable of computing a high-quality solution in a distributed/parallel manner

Joint provision can more than double the network

performance.

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SLIDE 74

Motivation RAN Backhaul Joint Provision Numerical Results

Future Directions

A gold mine of challenging optimization problems – much more remains to be done

Reduce the message passing between RAN and Backhaul
Asynchronous updates: Backhaul TE updates at much slower

rate than the RAN

Joint processing among BSs, network caching, network coding
Reduce the CSI requirement: expected sum-utility

maximization

Stochastic formulation of network provisioning to account for

traffic dynamic

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SLIDE 75

Reference

1 [Andrews 13] J. Andrews, ”Seven ways that hetnets are a cellular paradigm shift,” IEEE Communications Mag., vol. 51, no. 3, pp. 136-144, Mar. 2013. 2 [Bertsekas 87] D. P. Bertsekas, P. Hosein, and P. Tseng, ”Relaxation methods for network flow problems with convex arc costs,” SIAM Journal

n Control and Optimization, vol. 25, no. 5, pp. 1219-1243, Sep. 1987.

3 [Bertsekas 88] D. P. Bertsekas and P. Tseng, ”The relax codes for linear minimum cost network flow problem,” Annals of Operations Research,

vol. 13, no. 1, pp. 125-190, 1988.

4 [Bjornson 13] E. Bjornson and E. Jorswieck, ”Optimal resource allocation in coordinated multi-cell systems,” Foundations and Trends in Communications and Information Theory, vol. 9, no. 2-3, pp. 113-381, 2013. 5 [Chiang 07] M. Chiang, S. Low, A. R. Calderbank, and J. C. Doyle, ”Layering as optimization decompositioon: A mathematical theory of network architectures,” Proceedings of the IEEE, vol. 95, no. 1, pp. 255-312, Jan. 2007.

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SLIDE 76

Reference

6 [Chowdhery 11] A. Chowdhery, W. Yu, and J. M. Cioffi, ”Cooperative wireless multicell ofdma network with backhaul capacity constraints,” in

Proc. of IEEE International Conference on Communications, Jun. 2011,
pp. 1-6.

7 [Kelly 14] F. Kelly, ”Global optimization of network flows,” preprint, 2014. 8 [Gesbert 10] D. Gesbert, S. Hanly, H. Huang, S. Shamai (Shitz), O. Simeone, and W. Yu, ”Multi-cell MIMO cooperative networks: A new look at interference,” IEEE Journal on Selected Areas in Communications, vol. 28, no. 9, pp. 1380-1408, Dec. 2010. 9 [Hong 12] M. Hong and Z.-Q. Luo, ”Signal processing and optimal resource allocation for the inter- ference channel,” EURASIP E-Reference Signal Processing, 2012, accepted, available at http://arxiv.org. 10 [Razaviyayn 13] M. Razaviyayn, M. Hong, and Z.-Q. Luo, ”Linear transceiver design for a MIMO interfering broadcast channel achieving max-min fairness,” Signal Processing, vol. 93, no. 12, pp. 3327-3340,

Dec. 2013.

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SLIDE 77

Reference

11 [Shroff 06] N. Shroff and R. Srikant, ”A tutorial on cross-layer

ptimization in wireless networks,” IEEE Journal on Selected Areas in

Communications, vol. 24, no. 8, pp. 1452-1463, Aug. 2006. 12 [Simeone 09] O. Simeone, O. Somekh, H. Poor, and S. Shamai (Shitz), ”Downlink multicell processing with limited-backhaul capacity,” EURASIP Journal on Advanced Signal Processing, vol. 2009, pp. 1-10, Feb. 2009. 13 [Xiao 04] L. Xiao, M. Johansson, and S. P. Boyd, ”Simultaneous routing and resource allocation via dual decomposition,” IEEE Transactions on Communications, vol. 52, no. 7, pp. 1136-1144, Jul. 2004. 14 [Zakhour 11] R. Zakhour and D. Gesbert, Optimized data sharing in multicell MIMO with finite backhaul capacity, IEEE Transactions on Signal Processing, vol. 59, no. 12, pp. 6102-6111, Dec. 2011.

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