MAPEL: Achieving Global Optimality in Nonconvex Power Control Problem

1
MAPEL Achieving Global Optimality in Non-convex
Power Control Problem
Angela Yingjun
Zhang Department of Information Engineering
Joint work with Liping Qian and Jianwei Huang
2
Transmission in Wireless Channel
Data Rate
SNR
3
Power Control in Wireless Networks
Cellular Systems
Ad hoc Networks
SINR of link i
4
Fixed SINR Targets
Objective find a p such that
5
Fixed SINR Targets
Objective
When noise can be neglected
According to Perron-Frobenius Theorem, we can
find a non-negative p satisfying the inequality
if and only if the largest eigenvalue of DB is
smaller than or equal to 1.
6
Fixed SINR Targets
The problem is feasible if and only if the
largest eigenvalue of DB is smaller than or equal
to 1.
Distributed Power Control
7
Power Control in Wireless Networks
8
Throughput Maximizing Power Control
Objective
Constraints
Non-convex optimization problem due to the
non-convexity of feasible SINR set!
9
State of Art

• Sung02, Imhof_Mathar05 Feasible SINR
  region is log-convex
• Implication
• Power control problem is convex when the
  objective function is made concave in log(g)

Objective
10
Geometric Programming

• Assumption for all i

Limitation Optimality is compromised when links
are close to each other Cannot deactivate any
links
M. Chiang et al, Power control by geometric
programming, IEEE Trans. Wireless Commun., 2007
11
Throughput Maximization NP Hard!
Ratio between two posynomials
Equivalent to Signomial Programming Solution
successive convex approximation Limitations May
converge to local optimal solutions
Performance is sensitive to initialization,
network topology, etc No way to control the
degradation from the global optimum
M. Chiang et al, Power control by geometric
programming, IEEE Trans. Wireless Commun., 2007
12
Can we do better?
MAPEL An efficient method to achieve global
optimality!
13
Multiplicative Linear Fractional Programming
(MLFP)

• Problem domain Non-empty polytope in
• fi(p), gi(p) Positive linear affine functions on

14
Equivalent Problem
is an increasing function of z. That is,
occurs at the boundary
15
Feasible SINR Region w/o Rate Constraints
p1
z1
1
p2
0
z2
0

• Lower bounded by 1
• The set could be non-convex

16
Feasible SINR Region with Rate Constraints
z1
1
z2
0

• Lower bounded by
• The set could be non-convex

17
Feasible Region of MLFP
z1
z2
0
The feasible region of z is a union of boxes. It
is normal! The optimal solution must occur at the
upper boundary of the feasible region.
18
MAPEL General Idea

• Increasing functions
• The optimal solution only occurs at the upper
  boundary of the feasible set
• Normal feasible set
• Separation property
• Any point outside the feasible set can be
  separated from the feasible set by a cone

z1
z2
0
The feasible set can be covered by a union of
boxes, i.e., polyblock
19
MAPEL
z1
Mapping
z12(z2)
z22
z1
z21
z11
Shrinking polyblock
0
z2
Termination
20
Mapping Dinkelbach-type Algorithm
Problem
Step 1 for pj, find
Step 2 for , find
Termination
Super-linear convergence speed!
21
Convergence of MAPEL

• Out of the sequence , there exists a
  subsequence such that
• Obviously,
• Hence,

22
Error Tolerance e-optimal solution
23
Example A 4 link network
Tradeoff between optimality and running time
24
Optimality
Global optimum achieved regardless of
initialization and topology!
25
Example Random topology
26
Example With Rate Constraints
27
Example
G111.0, G220.75, G330.5, G440.25
28
Conclusion

• Global optimal solution to a non-convex
  optimization problem
• Tradeoff between complexity and optimality
• Benchmark for evaluating other power control
  algorithms
• MAPEL applicable to other engineering
  optimization problems?

Ref H. Tuy, Monotonic optimization problems
and solution approaches, SIAM, 2000.
29

• We are like dwarfs on the shoulders of giants,
• So that we can see more than they,
• And things at a greater distance,
• Not by virtue of any sharpness on sight on our
  part,
• Or any physical distinction,
• But because we are carried high and raised up by
  their giant size.
• ---Bernard of Chartres

Thank you!