Resource Allocation for Wireless Communications: An...

Transcript of Resource Allocation for Wireless Communications: An...

Resource Allocation for Wireless Communications: AnOptimization Perspective

Ya-Feng Liu

State Key Laboratory of Scientific/Engineering Computing

Institute of Computational Mathematics and Scientific/Engineering Computing

Academy of Mathematics and Systems Science

Chinese Academy of Sciences, Beijing, China

Email: [email protected]

Electronic and Information Engineering Young Scholars’ Symposium

November 18–19, 2017, XiDian University

Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 1 / 36
Two Questions: An Optimization Perspective

Given an optimal resource allocation problem in wireless communications,

Q1: Is there any polynomial time algorithm which can solve it to globaloptimality?

Q2: How to design a “good” algorithm for solving it with guaranteedperformance?

A1: Study the computational complexity of the problem and identify polynomialtime solvable subclass (if the general problem is “hard”)!

A2: Design customized algorithms for the problem by fully taking advantage of itsspecial structures such as (hidden) convexity, separability, and nonnegativity!

Two Questions: An Optimization Perspective

Given an optimal resource allocation problem in wireless communications,

Q1: Is there any polynomial time algorithm which can solve it to globaloptimality?

Q2: How to design a “good” algorithm for solving it with guaranteedperformance?

A1: Study the computational complexity of the problem and identify polynomialtime solvable subclass (if the general problem is “hard”)!

A2: Design customized algorithms for the problem by fully taking advantage of itsspecial structures such as (hidden) convexity, separability, and nonnegativity!

Complexity Theory

Convexity versus nonconvexity

Complexity theory: a robust tool to characterize the computationaltractability of an optimization problem.

Once a problem is shown to be “hard”, the search for an efficient, exactalgorithm should often (but NOT always) be accorded low priority or avoided.

Concentrate on other less ambitious approaches

- look for efficient algorithms that solve various special cases of the generalproblem

- look for algorithms that, though not guaranteed to run quickly, seem likely todo so most of the time

- relax the problem somewhat, looking for a fast algorithm that finds a solutionmerely satisfying most of constraints [Luo-Ma-So-Ye-Zhang, IEEE SPM, 2010]

- look for approximation algorithms (with guaranteed performance)

Complexity Theory









Complexity Theory









Algorithm Design

Power control:

min{pk}

K∑k=1

pk

s.t.gkkpk∑

j 6=k

gkjpj + ηk≥ γk , k = 1, 2, ...,K

0 ≤ pk ≤ p̄k , k = 1, 2, ...,K

fmincon ≺ linprog ≺ Foschini-Miljanic Algorithm

↑ ↑ ↑

Avoided Low Priority Preferred

The Foschini-Miljanic algorithm can efficiently solve the above problem byexploiting its special structures [Foschini-Miljanic, IEEE TVT, 1993; Cited by1850 times]

Algorithm Design

Power control:

min{pk}

K∑k=1

pk

s.t.gkkpk∑

j 6=k

gkjpj + ηk≥ γk , k = 1, 2, ...,K

0 ≤ pk ≤ p̄k , k = 1, 2, ...,K

fmincon

≺ linprog ≺ Foschini-Miljanic Algorithm

↑ ↑ ↑



Algorithm Design

Power control:

min{pk}

K∑k=1

pk

s.t.gkkpk∑

j 6=k

gkjpj + ηk≥ γk , k = 1, 2, ...,K

0 ≤ pk ≤ p̄k , k = 1, 2, ...,K

fmincon ≺ linprog

≺ Foschini-Miljanic Algorithm

↑ ↑ ↑



Algorithm Design

Power control:

min{pk}

K∑k=1

pk

s.t.gkkpk∑

j 6=k

gkjpj + ηk≥ γk , k = 1, 2, ...,K

0 ≤ pk ≤ p̄k , k = 1, 2, ...,K


↑ ↑ ↑



Algorithm Design

Power control:

min{pk}

K∑k=1

pk

s.t.gkkpk∑

j 6=k

gkjpj + ηk≥ γk , k = 1, 2, ...,K

0 ≤ pk ≤ p̄k , k = 1, 2, ...,K


↑ ↑ ↑



Outline

Multi-User Interference Channel

Optimal Linear Beamforming Design

Recent Beamforming Design

Concluding Remarks

Multi-User Interference Channel

System Model

K users or links (transmitter-receiver pairs) and denote K = {1, 2, . . . ,K}

Assume each transmitter sends one data stream sk to its intended receiver

Mk and Nj denote the number of antennas at receiver k and transmitter j

Hkj ∈ CMk×Nj is the channel matrix from transmitter j to receiver k

- Mk = 1,Nk = 1−→ SISO IC

- Mk = 1,Nk ≥ 2−→ MISO IC

- Mk ≥ 2,Nk = 1−→ SIMO IC

- Mk ≥ 2,Nk ≥ 2−→ MIMO IC

SINR and Rate

Linear transmission and reception strategy

uk and vj are the BEAMFORMING vectors at receiver k and transmitter j

User k’s received signal:

ŝk = u†kHkkvksk︸︷︷︸signal term

+∑j 6=k

u†kHkjvjsj︸︷︷︸interference term

+ u†knk︸︷︷︸noise term

SINR at receiver k (MIMO):

SINRk =|u†kHkkvk |2∑

j 6=k

|u†kHkjvj |2 + ηk‖uk‖2

Transmission rate: rk = log2 (1 + SINRk)

Coordinated Beamforming

Multicast Beamforming

Max-Min Fairness Linear Transceiver Design

max{uk ,vk}

mink∈K

SINRk :=|u†kHkkvk |2∑

j 6=k

|u†kHkjvj |2 + ηk‖uk‖2

s.t. ‖uk‖ = 1, ‖vk‖2 ≤ p̄k , k ∈ K

m

max{uk ,vk ,ζ}

ζ

s.t. SINRk ≥ ζ, ‖uk‖ = 1, ‖vk‖2 ≤ p̄k , k ∈ K

Strongly NP-Hard: MIMO Case

Theorem (L.-Dai-Luo, IEEE ICC, 2011)

Given a SINR target ζ, the problem of checking the feasibility of problemSINRk ≥ ζ, k ∈ K‖uk‖ = 1, k ∈ K‖vk‖2 ≤ p̄k , k ∈ K

is strongly NP-hard for the MIMO interference channel with

Mk ≥ 3,Nk ≥ 2, ∀ k ∈ K

orMk ≥ 2,Nk ≥ 3, ∀ k ∈ K.

Feasibility −→ Optimization

Complexity Status

Complexity Status of Max-Min Fairness Linear Transceiver Design

HHHHHRx

TxNk = 1 Nk = 2 Nk ≥ 3

Mk = 1 Poly. Time Sol. Poly. Time Sol. Poly. Time Sol.Mk = 2 Poly. Time Sol. ??? Str. NP-hardMk ≥ 3 Poly. Time Sol. Str. NP-hard Str. NP-hard

The complexity is sensitive to the number of antennas!

Hidden convexity =⇒ Polynomial time solvable

SDPBA globally solves the SIMO problem in polynomial time!

The complexity analysis is completed in [L., IEEE TIT, 2017].

Complexity Status


HHHHHRx

TxNk = 1 Nk = 2 Nk ≥ 3






Complexity Status


HHHHHRx

TxNk = 1 Nk = 2 Nk ≥ 3






Complexity Status


HHHHHRx

TxNk = 1 Nk = 2 Nk ≥ 3






Complexity Status


HHHHHRx

TxNk = 1 Nk = 2 Nk ≥ 3






CCAA

CCAA: cyclic coordinate ascent algorithm

ECCAA: polynomial time solvable subproblems

ICCAA: a low-complexity variant

Global convergence guarantee

Good numerical performance

CCAA






CCAA






CCAA






CCAA






SINR VS number of users (MIMO Scenario)

Average minimum SINR versus the number of users with SNR = 15 dB

2 3 4 5 6 7 8 9 10−15

−10

−5

0

5

10

15

20

Number of Users

Ave

rage

Min

imum

SIN

R [d

B]

Upper BoundICCAAECCAABenchmark2Benchamrk1

CPU time VS number of users (MIMOScenario)

Average CPU time versus the number of users with SNR = 15 dB

2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

14

16

18

20

Number of Users

Ave

rage

CP

U T

ime

[s]

ICCAAECCAA

SINR VS number of users (SIMO Scenario)

Average minimum SINR versus the number of users with SNR = 15 dB

2 3 4 5 6 7 8 9 10

−5

0

5

10

15

Number of Users

Ave

rage

Min

imum

SIN

R [d

B]

SDPBAICCAABenchmark

CPU time VS number of users (SIMOScenario)

Average CPU time versus the number of users with SNR = 15 dB

2 3 4 5 6 7 8 9 10

10−1

100

101

Number of Users

Ave

rage

CP

U T

ime

[s]

SDPBAICCAA

Related Works

Ya-Feng Liu, “Dynamic Spectrum Management: A Complete Complexity Characterization,”

IEEE Transactions on Information Theory, vol. 63, no. 1, pp. 392–403, 2017.

Ya-Feng Liu, Yu-Hong Dai, and Zhi-Quan Luo, “Max-Min Fairness Linear Transceiver

Design for a Multi-User MIMO Interference Channel,” IEEE Transactions on Signal

Processing, vol. 61, no. 9, pp. 2413–2423, 2013.

Ya-Feng Liu, Mingyi Hong, and Yu-Hong Dai, “Max-Min Fairness Linear Transceiver

Design Problem for a SIMO Interference Channel is Polynomial Time Solvable,” IEEE

Signal Processing Letters, vol. 20, no. 1, pp. 27–30, 2013.

Ya-Feng Liu, Yu-Hong Dai, and Zhi-Quan Luo, “Coordinated Beamforming for MISO

Interference Channel: Complexity Analysis and Efficient Algorithms,” IEEE Transactions on

Signal Processing, vol. 55, no. 3, pp. 1142–1157, 2011.

Multicast Beamforming

minw

‖w‖2

s.t. |hHk w| ≥ 1, k = 1, 2, . . . ,M

m

minw, c

‖w‖2

s.t. ck = hHk w, k = 1, ...,M − 1|ck | ≥ 1, k = 1, ...,M − 1

hHMw ≥ 1

Argument Cuts

How to deal with the nonconvex constraint |ck | ≥ 1?

Simple convex envelope (relaxation) gives ck ∈ C. Too LOOSE!

A new idea: adding Argument Cuts!

D[lk ,uk ] = {(xk , yk)|ck = xk + yk i, |ck | ≥ 1, arg (ck) ∈ [lk , uk ]}

Conv(D[lk ,uk ]) = {(x , y) | sin(lk )x−cos(lk )y ≤ 0, sin(uk )x−cos(uk )y ≥ 0, akx+bky ≥ a2k +b

2k}

Argument Cuts


Simple convex envelope (relaxation) gives ck ∈ C.

Too LOOSE!




2k}

Argument Cuts






2k}

Argument Cuts






2k}

An Illustration of Argument Cuts

Our Results

Propose a branch-and-bound (BB) algorithm

Guaranteed global optimality

Promising numerical performance

Cheng Lu and Ya-Feng Liu, “An Efficient Global Algorithm for Single-Group Multicast

Beamforming,” IEEE Transactions on Signal Processing, vol. 65, no. 14, pp. 3761–3774,

Jul. 2017. [ACR-BB]

Our Results






Jul. 2017. [ACR-BB]

Our Results






Jul. 2017. [ACR-BB]

Our Results






Jul. 2017. [ACR-BB]

Relative Errors with (N ,M) = (4, 40)

E t1 =|Lt − ν̄|

ν̄, E t2 =

|U t − ν̄|ν̄

, E t3 =|U t − Lt |

Lt

0 2000 4000 6000 8000 10000 12000 14000 1600010

−8

10−6

10−4

10−2

100

102

104

Number of Iterations

Rel

ativ

e E

rror

E1E2E3

Comparison of Our Proposed ACR-BB Algorithm and Baron for Solving 10Randomly Generated Problem Instances with (N,M) = (2, 8)

Instance ACR-BB BaronID Value # of Iter. Time Value # of Iter. Time01 1.2344 90 0.7 1.2344 6297 327.902 1.1412 40 0.2 1.1412 2249 147.403 1.3541 68 0.4 1.3541 2917 94.504 1.4251 49 0.3 1.4251 2847 261.505 1.4136 61 0.4 1.4136 1825 49.806 0.8540 40 0.3 0.8540 1303 31.907 0.9777 48 0.3 0.9777 457 19.008 5.2469 1 0.0 5.2469 4305 58.409 11.9555 16 0.1 11.9555 5358 87.110 1.3258 62 0.4 1.3258 653 26.5

Average 2.6928 47.5 0.3 2.6928 2821.1 110.4

Providing an Important Benchmark

N. D. Sidiropoulos, T. N. Davidson, and Z.-Q. Luo, “Transmit beamforming for

physical-layer multicasting,” IEEE Transactions on Signal Processing, vol. 54, no. 6, pp.

2239–2251, Jun. 2006. [IEEE Signal Processing Society 2011 Best Paper Award] [SDR]

L. N. Tran, M. F. Hanif, and M. Juntti, “A conic quadratic programming approach to

physical layer multicasting for large-scale antenna arrays,” IEEE Signal Process. Lett.,

vol. 21, no. 1, pp. 114–117, Jan. 2014. [SLA]



Jul. 2017. [ACR-BB]







vol. 21, no. 1, pp. 114–117, Jan. 2014. [SLA]



Jul. 2017. [ACR-BB]







vol. 21, no. 1, pp. 114–117, Jan. 2014. [SLA]



Jul. 2017. [ACR-BB]

Average Objective Values and Relative Gaps Obtained by SLA and SDR

Setting Average Objective Value Average Relative Gap

(N,M) ACR-BB SLA SDR SLA SDR

(2,8) 1.629 1.629 1.642 0.02% 0.79%

(2,16) 2.800 2.803 2.827 0.10% 0.96%

(2,24) 3.389 3.398 3.463 0.26% 2.20%

(2,32) 4.457 4.489 4.642 0.73% 4.16%

(2,40) 5.720 5.731 5.871 0.20% 2.65%

(2,48) 5.679 5.739 5.892 1.06% 3.77%

(2,56) 5.461 5.491 5.758 0.55% 5.44%

(2,64) 5.914 5.967 6.291 0.89% 6.37%

(4,8) 0.514 0.525 0.577 2.01% 12.17%

(4,16) 0.837 0.902 1.121 7.83% 33.97%

(4,24) 1.132 1.256 1.710 10.95% 51.03%

(4,32) 1.328 1.587 2.045 19.48% 54.04%

(4,40) 1.525 1.770 2.587 16.08% 69.64%

(6,8) 0.334 0.341 0.393 2.30% 17.75%

(6,16) 0.531 0.578 0.799 8.90% 50.39%

(6,24) 0.667 0.779 1.113 16.77% 66.99%

(8,8) 0.246 0.253 0.278 2.95% 12.93%

(8,16) 0.376 0.420 0.618 11.68% 64.11%

Worst-Case Relative Gaps Obtained by SLA and SDR

(N,M) SLA SDR

(2,8) 1% 6%

(2,16) 6% 10%

(2,24) 4% 18%

(2,32) 13% 30%

(2,40) 8% 22%

(2,48) 10% 28%

(2,56) 12% 25%

(2,64) 14% 25%

(4,8) 34% 42%

(4,16) 39% 82%

(4,24) 65% 100%

(4,32) 70% 103%

(4,40) 74% 104%

(6,8) 22% 92%

(6,16) 62% 126%

(6,24) 73% 128%

(8,8) 23% 46%

(8,16) 51% 124%

Probability of Obtaining Global Solution by SLA and SDR

(N,M) SLA SDR

(2,8) 98% 74%

(2,16) 96% 68%

(2,24) 92% 48%

(2,32) 88% 28%

(2,40) 92% 30%

(2,48) 82% 22%

(2,56) 86% 24%

(2,64) 88% 14%

(4,8) 80% 38%

(4,16) 50% 6%

(4,24) 42% 0%

(4,32) 28% 0%

(4,40) 14% 0%

(6,8) 74% 40%

(6,16) 44% 0%

(6,24) 18% 0%

(8,8) 72% 28%

(8,16) 42% 2%

Continuous and Discrete Argument Cuts

MIMO Detection

Input-output relationship of the MIMO channel:

r = Hx∗ + v,

where each entry x∗i of x∗ belongs to a finite set of symbols, i.e.,

x∗i ∈{e iθ | θ = 2jπ/M, j = 0, 1, . . . ,M − 1

}, i = 1, 2, . . . , n.

MIMO detection problem:

minx∈Cn

1

2‖Hx− r‖22

s.t. |xi | = 1, arg (xi ) ∈ A, i = 1, . . . , n,

whereA = {2jπ/M, j = 0, . . . ,M − 1}.

Tightness of ECSDP for MIMO Detection

CSDP: conventional SDP where discrete argument constraints are dropped

ECSDP: our proposed new and enhanced SDP based on argument cuts

TheoremFor any M ≥ 2 :

Supposeλmin

(H†H

)sin (π/M) >

∥∥H†v∥∥∞holds true, then ECSDP is TIGHT for MIMO detection problem;

But CSDP is generically not tight for MIMO detection problem.

Cheng Lu, Ya-Feng Liu, Wei-Qiang Zhang, and Shuzhong Zhang, Tightness of a new and

enhanced semidefinite relaxation for MIMO detection, 2017. (Available Online)

Cheng Lu, Ya-Feng Liu, and Jing Zhou, An efficient global algorithm for nonconvex

complex quadratic problems with applications in wireless communications, to appear in

IEEE/CIC ICCC. (Invited Paper)

Tightness of ECSDP for MIMO Detection

CSDP: conventional SDP where discrete argument constraints are dropped

ECSDP: our proposed new and enhanced SDP based on argument cuts

TheoremFor any M ≥ 2 :

Supposeλmin

(H†H

)sin (π/M) >

∥∥H†v∥∥∞holds true, then ECSDP is TIGHT for MIMO detection problem;

But CSDP is generically not tight for MIMO detection problem.

Cheng Lu, Ya-Feng Liu, Wei-Qiang Zhang, and Shuzhong Zhang, Tightness of a new and

enhanced semidefinite relaxation for MIMO detection, 2017. (Available Online)

Cheng Lu, Ya-Feng Liu, and Jing Zhou, An efficient global algorithm for nonconvex

complex quadratic problems with applications in wireless communications, to appear in

IEEE/CIC ICCC. (Invited Paper)

Recent Beamforming Techniques

Alamouti scheme based beamforming

Symbol-level beamforming based on both the data information and the CSI

Hybrid digital and analog beamforming to reduce the high cost and highpower consumption

Per-antenna constant envelope beamforming

Discrete-phase constant envelope beamforming

· · ·







· · ·







· · ·







· · ·







· · ·







· · ·

Concluding Remarks

Optimal resource allocation in the multi-user interference channel

- max-min fairness linear transceiver design

- multicast beamforming design

Complexity theory provides us valuable information in directing our effortstoward those approaches that have the greatest potential of leading to usefulalgorithms!

Special structures such as (hidden) convexity, separability, sparsity, andnonnegativity should be judiciously exploited to design effective/efficientalgorithms for optimization problems from real applications!

Application Driven Optimization

Concluding Remarks







Concluding Remarks







Concluding Remarks







My Collaborators

Yu-Hong Dai Tom Luo Mingyi Hong Cheng Lu

Thank You!

http://lsec.cc.ac.cn/˜yafliu/

[email protected]

http://lsec.cc.ac.cn/~yafliu/

Resource Allocation for Wireless Communications: An...

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