Network optimizations using convexity

1
Network optimizations using convexity

• CISL
• Yoo Han Woong
• 2005. 05. 11

2
Contents

• Use of Convexity in Network Field
• List of Papers
• Background of Convex Optimization(4)
• Applications and Algorithms 3 papers
• System Classification
• System Variables
• System Model
• Design Objective and Performance Index
• Solution
• Motivation Contribution
• Conclusion
• Conclusion

3
Use of Convexity in Network Field

• Application
• Congestion Control (Network Flow)
• Wireless Ad-hoc Network (Routing/RA)
• Cellular Network (CDMA RA)
• MIMO (Multi Input and Multi Output) Channel
• Solution
• Auction Algorithm
• Gossip Algorithm

4
List of Papers

• Application
• Congestion Control (Network Flow)
• Low03 A Duality Model of TCP and Queue
  Management Algorithm
• Wireless Ad-hoc Network (Routing/RA)
• Chi04 To Layer or Not To Layer Balancing
  Transport and Physical Layers in Wireless
  Networks
• Kri04Optimal information extraction in
  energy-limited wireless sensor networks
• Cellular Network (CDMA RA)
• Boy03 Simultaneous routing and power allocation
  in CDMA wireless data networks
• MIMO (Multi Input and Multi Output) Channel
• Lag03 Joint Tx-Rx Beamforming Design for
  Multicarrier MIMO Channel- A Unified Framework
  for Convex Optimization
• Solution
• Auction Algorithm
• Ber98 Auction Algorithm
• Gossip Algorithm
• Boy05 Gossip Algorithms Design, Analysis and
  Applications

5
Background of Convex Optimization(1)

• General Optimization Problem

• Convex Programming problems are those for which f
  is convex and C is convex. (Continuous Problem)

Convexity guarantees some nice property to solve
it!
6
Background of Convex Optimization(2)

• Why Convexity is so special?!
• It have only global minima, which is not local.
• A convex set have nonempty relative interior.
• A convex function is continuous and has nice
  differentiability properties
• A convex set is connected and has feasible
  directions at any point.
• A nonconvex function can be convexified while
  maintaining the optimality of its global minima.
• Duality Theory and Optimality Condition can make
  a problem easier.

7
Background of Convex Optimization(3)

• Convexity and Duality
• A multiplier vector for the problem

Where L is Lagrangian Function

• Dual Function and Dual Problem

Where is concave
dual function
8
Background of Convex Optimization(4)

• Optimal primal value
• Optimal dual value
• We have always g f when convexity in the
  primal problem (strong duality)
• ? Optimal solution of dual problem are muliliers
  for the optimal problem.

9
Applications and Algorithms

• Applications
• Low03 A Duality Model of TCP and Queue
  Management Algorithm
• Chi04 To Layer or Not To Layer Balancing
  Transport and Physical Layers in Wireless
  Networks
• Kri04Optimal information extraction in
  energy-limited wireless sensor networks
• Boy03 Simultaneous routing and power allocation
  in CDMA wireless data networks
• Lag03 Joint Tx-Rx Beamforming Design for
  Multicarrier MIMO Channel- A Unified Framework
  for Convex Optimization
• Algorithms
• Ber98 Auction Algorithm
• Boy05 Gossip Algorithms Design, Analysis and
  Applications

10
Applications- Congestion Control (Network Flow)

• Low, S.H,
• "A Duality Model of TCP and Queue Management
  Algorithms, "
• IEEE/ACM Transaction on Networking,
• Vol 11, No. 4, August, 2003

11
A Duality Model of TCP and Queue Management
Algorithm System System Classification
12
A Duality Model of TCP and Queue Management
Algorithm System System Variables

• Input variables none
• State variables the transmission rate xr, the
  flow rate yl, end to end congestion measure qs,
  link congestion measure pl
• Internal parameters source number s, link
  number l, finite capacity c, routing matrix R,
  TCP Scheme, AQM Scheme
• Measured output variable the Congestion
  measure, the transmission rate xr, total
  Throughput, Total Utilization
• Controlled output variables the transmission
  rate xr,

13
A Duality Model of TCP and Queue Management
Algorithm System System Model

• Basic concept
• Transmission rate and Congestion Measure are dual

where The sets Ls define an L x S routing matrix

• TCP AQM Model
• TCP
• AQM

14
A Duality Model of TCP and Queue Management
Algorithm System System Model

• Assume that (1)(3) has set of equilibria (x,p)
  then equilibrium rate xs and congestion measure
  qs is

Assume that Fs is continuously differentiable and
in the open set A(xs, qs)
xs, qs is positive then by inplicit function
theorem, there exists a unique continuously
differentiable function fs from positive xs and qs

• Def. of Utility function

15
A Duality Model of TCP and Queue Management
Algorithm System System Model

• Maximizing Aggregate Utility

Its dual form is
16
A Duality Model of TCP and Queue Management
Algorithm System Solution- Theorem 1

• Hence, various TCP/AQM protocols can be modeled
  as different distributed primal-dual algorithms
  (F, G, H) to solve the global optimization
  problem (8) and its dual (9), with different
  utility functions Us.

• Theorem 1 characterizes a large class of
  protocols that admits such an interpretation. It
  holds as long as the end-to-end congestion
  measure to which the TCP algorithm reacts is the
  sum of the constituent link congestion measures,
  under some mild assumptions on the TCP and
  AQMalgorithms that are typically satisfied

17
A Duality Model of TCP and Queue Management
Algorithm System Design Objective and
Performance Index

• Find A model of TCP Congestion Control Algorithm
  and AQM scheme

18
A Duality Model of TCP and Queue Management
Algorithm System Solution (F,G,H)
19
A Duality Model of TCP and Queue Management
Algorithm System Motivation Contribution

• Motivation
• To Obtain general Utility function considering
  both TCP and AQM.
• Contribution
• To examine the relation between flow(or tr. rate)
  and congestion measure dual relation
• Find a algorithm model (F,G,H)

20
A Duality Model of TCP and Queue Management
Algorithm System Conclusion

• Significance
• To examine the relation between flow (or
  transmission rate) and congestion measure dual
  problem
• Find a algorithm model (F,G,H)
• Problem
• On the equilibrium property stability and
  dynamics is not mentioned
• Extension
• stability and dynamics of these congestion
  control scheme

21
Applications- Cellular Network (CDMA RA)

• Mikael Johansson, Lin Xiao, Stephen Boyd
• Simultaneous Routing and Power Allocation
• in CDMA Wireless Data Networks
• Proc. IEEE Int. Conf. Communications, Vol.1
  Anchorage. AK, May 2003, pp51-55

22
Simultaneous Routing and Power Allocation in CDMA
Wireless Data NetworksSystem Classification
23
Simultaneous Routing and Power Allocation in CDMA
Wireless Data NetworksSystem Variables

• Controlled Input variables communications
  variable ? (or r) including media access scheme,
  such as transmit power,
  bandwidth, time slot fraction and etc., weights
  on flow
• State variables collection of source sink
  vector s, collection of flow vector x,
• Internal variables none
• Internal parameters total node number N, total
  destination number D, total link number L, media
  access methods, coding and modulation scheme,
  network topology Anl,
• Measured output variable average behavior of
  data transmission, total energy consumption ,
  total Throughput, total utilization

24
Simultaneous Routing and Power Allocation in CDMA
Wireless Data NetworksSystem Model

• Review Multicommodity network flow problem

Where network topology A, collection of
source sink vector s, collection of flow
vector x,
25
Simultaneous Routing and Power Allocation in CDMA
Wireless Data NetworksSystem Model

• Communication Model in CDMA
• Communication Variable and Constraint
• Two Particular class of CDMA system
• The Gaussian broadcast channel
• Interference-limited channels

26
Simultaneous Routing and Power Allocation in CDMA
Wireless Data Networks Design Objective and
Performance Index

• The Generic SRRA Fomulation

• Characteristics of SRRA Problem
• Very General Form so that include many important
  design problem
• Examples the objective can be maximum total
  utility, minimum total power
  (or bandwidth), minimax power
  among the nodes, minimax link
  utilization

27
Simultaneous Routing and Power Allocation in CDMA
Wireless Data Networks Design Objective and
Performance Index

• Design Objective
• Find optimal operation of CDMA network i.e.
• Do SRRA(Simultaneous Routing and Resource
  Allocation)

• Performance IndexMinimize or maximize the
  objective function

28
Simultaneous Routing and Power Allocation in CDMA
Wireless Data NetworksSolution

• The Gaussian broadcast channel
• An equivalent characterization of the rate region
  
• The Convex Formulation of the SRRA Problem

29
Simultaneous Routing and Power Allocation in CDMA
Wireless Data NetworksSolution

• Interference-limited channels
• Convex Optimization
• SINR is defined as
• Approximation log(1SINR) ?? log(SINR)

30
Simultaneous Routing and Power Allocation in CDMA
Wireless Data NetworksSolution

• Interference-limited channels
• SRRA Problem
• where the last constraint (which is convex) is
  (5) in the new variable Q. Here the capacity
  constraints tl lt?l(Q) are jointly convex in t
  and Q.
• Omitted by space constraint in this presentation

31
Simultaneous Routing and Power Allocation in CDMA
Wireless Data NetworksMotivation Contribution

• Motivation
• No Description SRRA problem in CDMA Environment
  (previous paper)
• Contribution
• Solve SRRA Problem considering two type of CDMA
  channel models

32
Simultaneous Routing and Power Allocation in CDMA
Wireless Data NetworksConclusion

• Significance
• New Network model considering CDMA Environment
• Finding Optimal operation of network using SRRA
  Method
• Problem
• Do not mention the important issues in wireless
  network such as QoS, Dynamic Routing, media
  access methods, coding and modulation scheme
• Cannot explain some situations packet loss,
  retransmissions, time varying fading, topology
  change
• Extension
• Dynamic routing and resource allocation
• Dual Decomposition Method previous Paper
• Optimizing Distribution Algorithm in Power
  Control

33
Conclusion -Overall

• Conclusion
• Basic Concept of Convex Optimizaiton
• Two region Application and Algorithm
• Review Two Paper Congestion Control and CDMA
  Resource Allocation
• Extension
• To Find uncovered region, make the convex problem

34
Ref.System Model- Description

• Network Topology

35
Ref. System Model- Description

• Network Flow Model