Resource Allocation for Wireless Communications: An...
Transcript of Resource Allocation for Wireless Communications: An...
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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
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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!
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 2 / 36
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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!
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 2 / 36
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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)
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 3 / 36
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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)
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 3 / 36
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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)
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 3 / 36
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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]
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 4 / 36
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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]
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 4 / 36
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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]
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 4 / 36
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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]
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 4 / 36
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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]
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 4 / 36
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Outline
Multi-User Interference Channel
Optimal Linear Beamforming Design
Recent Beamforming Design
Concluding Remarks
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 5 / 36
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Multi-User Interference Channel
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 6 / 36
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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
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 7 / 36
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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)
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 8 / 36
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Coordinated Beamforming
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 9 / 36
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Multicast Beamforming
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 10 / 36
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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
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 11 / 36
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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
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 12 / 36
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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].
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 13 / 36
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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].
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 13 / 36
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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].
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 13 / 36
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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].
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 13 / 36
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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].
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 13 / 36
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CCAA
CCAA: cyclic coordinate ascent algorithm
ECCAA: polynomial time solvable subproblems
ICCAA: a low-complexity variant
Global convergence guarantee
Good numerical performance
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 14 / 36
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CCAA
CCAA: cyclic coordinate ascent algorithm
ECCAA: polynomial time solvable subproblems
ICCAA: a low-complexity variant
Global convergence guarantee
Good numerical performance
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 14 / 36
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CCAA
CCAA: cyclic coordinate ascent algorithm
ECCAA: polynomial time solvable subproblems
ICCAA: a low-complexity variant
Global convergence guarantee
Good numerical performance
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 14 / 36
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CCAA
CCAA: cyclic coordinate ascent algorithm
ECCAA: polynomial time solvable subproblems
ICCAA: a low-complexity variant
Global convergence guarantee
Good numerical performance
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 14 / 36
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CCAA
CCAA: cyclic coordinate ascent algorithm
ECCAA: polynomial time solvable subproblems
ICCAA: a low-complexity variant
Global convergence guarantee
Good numerical performance
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 14 / 36
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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
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 15 / 36
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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
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 16 / 36
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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
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 17 / 36
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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
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 18 / 36
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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.
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 19 / 36
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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
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 20 / 36
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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}
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 21 / 36
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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}
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 21 / 36
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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}
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 21 / 36
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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}
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 21 / 36
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An Illustration of Argument Cuts
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 22 / 36
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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]
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 23 / 36
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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]
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 23 / 36
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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]
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 23 / 36
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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]
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 23 / 36
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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
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 24 / 36
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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
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 25 / 36
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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]
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]
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 26 / 36
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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]
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]
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 26 / 36
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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]
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]
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 26 / 36
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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%
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 27 / 36
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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%
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 28 / 36
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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%
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 29 / 36
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Continuous and Discrete Argument Cuts
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 30 / 36
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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}.
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 31 / 36
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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)
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 32 / 36
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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)
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 32 / 36
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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
· · ·
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 33 / 36
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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
· · ·
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 33 / 36
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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
· · ·
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 33 / 36
-
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
· · ·
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 33 / 36
-
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
· · ·
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 33 / 36
-
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
· · ·
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 33 / 36
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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
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 34 / 36
-
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
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 34 / 36
-
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
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 34 / 36
-
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
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 34 / 36
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My Collaborators
Yu-Hong Dai Tom Luo Mingyi Hong Cheng Lu
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 35 / 36
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Thank You!
http://lsec.cc.ac.cn/˜yafliu/
Ya-Feng Liu (AMSS, CAS) Resource Allocation for Wireless Communications Nov. 18, 2017 36 / 36
http://lsec.cc.ac.cn/~yafliu/