principles of wireless sensor networkscarlofi/teaching/pwsn-2011/lectures/comment… · carlo...

Carlo Fischione Fall 2011 Principles of Wireless Sensor Networks

Principles of Wireless Sensor Networks

http://www.ee.kth.se/~carlofi/teaching/pwsn-2011/wsn_course.shtml

Lecture 5 Stockholm, October 14, 2011

Carlo Fischione

Royal Institute of Technology - KTH

Stockholm, Sweden e-mail: [email protected]

Fast-Lipschitz Optimization

Carlo Fischione Fall 2011 Principles of Wireless Sensor Networks

7RGD\·V�OHFWXUH

� Previous lecture ¾ Non expansive mappings, agreement, consensus

� Today ¾ F-Lipschitz optimization

Protocol stack

Phy

MAC

Routing

Transport

Session

Application

Presentation

Carlo Fischione

7RGD\·V�OHDUQLQJ�JRDOV

In WRGD\·V lecture, you will learn some of the key aspects of Fast

Lipschitz optmization When a problem is F-Lipschitz? Why should I care of F-Lipschitz? The theory is illustrated by distributed estimation applications Why F-Lipschitz is useful for distributed estimation? Other applications will be mentioned as well

Principles of Wireless Sensor Networks Fall 2011

centralizeddistributed/networked optimizationnetwork optimization

optimization

Carlo Fischione

Optimization over networks

� Optimization needs to be computed by fast algorithms of low complexity ¾ time-varying networks, little time to compute optimal solutions ¾ computations often must be distributed ¾ E.g., cross layer protocol design, distributed detection, estimation,

content distribution, routing, ....

� Parallel and distributed computation ¾ Fundamental theory for optimization over networks ¾ Drowback over energy-constrained wireless networks: the cost for communication not considered

� An alternative theory is needed

¾ In a number of cases, Fast-Lipschitz optimization


going for less and less base-stations but more and more distributed networks

Carlo Fischione

Outline

• Definition of Fast-Lipschitz optimization

• Computation of the optimal solution

• Problems in canonical form

• Examples of application

• Conclusions


Carlo Fischione

Network optimization

� We have seen that a class of distributed estimation problems needs a fast distributed computation of an optimization

� Is there a general formulation of optiomization problmes that can

be solved quickly and efficiently in a distributed manner? ¾ In many cases, Fast-Lipschitz optimization

� C. Fischione, “F-Lipschitz Optimization with Wireless Sensor Networks Applications”, IEEE Transactions on Automatic Control, Accepted for Publication, to appear, 2011.


Carlo Fischione

The Fast-Lipschitz optimization

nonempty compact set “containing” other constraints


ex: x could be quality control action, radio power etc

some box D

Carlo Fischione

Computation of the solution

� Centralized optimization ¾ Problem solved by a central processor

� Distributed optimization

¾ Decision variables and constraints are associated to nodes that cooperate to compute the solution in parallel

Network of n nodes


Carlo Fischione

The F-Lipschitz optimization

Convex Optimization

Geometric Optimization

F-Lipschitz Optimization

Interference Function Optimization

Non-Convex Optimization

F-Lipschitz optimization problems can be convex, geometric, quadratic, interference-function,...


Convex optimization problems require heavier solvers to solve than F-Lipschitz

Carlo Fischione

Pareto Optimal Solution


Example(of(Pareto(Optimal:((

( !∗( !! !∗ =531(

((

((( y( !! ! =460.5

(

((None(is(better(or(worse,(since(not(all(elements(are(the(highest,(hence(it(is(a(Pareto(optimal(solution(

(

White board notes to the previous slide

Carlo Fischione

Notation

Norm infinity: sum along a row

Norm 1: sum along a column

Gradient


Carlo Fischione

Functions may be non-convex

F-Lipschitz qualifying properties


Criteria 1 must always be satisfied

while only one of criteria 2-4 needs to be fulfilled

to be a F-Lipschitz optimization problem

Examples(of(F=Lipschitz(qualifying(properties((1b)(

(∇! = !!!!!!< 1< 1< 1

((

(2a)(

∇! =+ +

0+ 0

!!!!!!< 1< 1< 1

((

(3a)((

( !! ! = !!!!!!!!!!!!! =111(

( (

3c)(

( ∇! =− 0 −0 −− −

!

< 1(

(4a)(

( ∇! =

+ 0 −0 −+ −

!

< !!!∆

(

(


Carlo Fischione

Outline





• Conclusions


Carlo Fischione

Optimal Solution

� The Pareto optimal solution is just given by a set of (in general non-

linear) equations.

� Solving a set of equations is much easier than solving an optimization problem by traditional Lagrangian methods.


Carlo Fischione

Lagrangian methods

Theorem: Consider a feasible F-Lipschitz problem. Then, the KKT conditions are necessary and sufficient.

¾ KKT conditions:

Lagrangian

Lagrangian methods to compute the solution


with F-Lipschitz, we do not need the Lagrangian method to solve the problem

scalar x, lambda is a vector of Lagrangian multipliers

contractive iteration

Lagrangian(method((1) Build(the(Lagrangian((

2) ∇!! !, ! = 0∇!! !, ! = 0!! ((Lagrangian(methods(


Carlo Fischione

Centralized optimization

� The optimal solution is given by iterative methods to solve systems of non-linear equations (e.g., Newton methods)

is a matrix to ensure and maximize convergence speed � Many other methods are available, e.g., second-oder methods. Principles of Wireless Sensor Networks Fall 2011

Carlo Fischione

Distributed optimization


the value can not go outside the box D (otherwise approx. around the max bound of the box)

Carlo Fischione

F-Lipschitz optimization

Inequality constraints satisfy the equality at

the optimum?

Compute the solution by F-

Lipschitz methods

Compute the solution by

Lagrangian methods

yes no

� F-Lipschitz optimization: a class of problems for which all the

constraints are active at the optimum

� Optimum: the solution to the set of equations given by the constraints

� No Lagrangian methods, which are computationally expensive, particularly on wireless networks


Carlo Fischione

Outline





• Conclusions


Carlo Fischione

Problems in canonical form

F-Lipschitz form Canonical form Bertsekas, Non Linear Programming, 2004


possible by doing the following

Carlo Fischione

Problems in canonical form

Carlo Fischione

� The problem is convex, but is also F-Lipschitz:

� The solution is given by the constraints at the equality, trivially

Example 1: from canonical to F-Lipschitz

Off-diagonal monotonicity

Diagonal dominance


Carlo Fischione

Example 2: a hidden F-Lipschitz

� Non F-Lipschitz

� Simple variable transformation, , F-Lipschitz


Carlo Fischione

An F-Lipschitz Matlab Toolbox

M. Leithe, Introducing a Matlab Toolbox for F-Lipschitz optimization, Master Thesis KTH, 2011


Carlo Fischione

Outline





• Conclusions

Fall 2011 Principles of Wireless Sensor Networks

Carlo Fischione

Distributed estimation

� We now study F-Lipschitz optimization for ¾ distributed estimation ¾ distributed detection ¾ distributed radio power control


Carlo Fischione

Estimation

� A. Speranzon, C. Fischione, K. H. Johansson, A. Sangiovanni-Vincentelli, “A Distributed Minimum Variance Estimator for Sensor Networks”, IEEE Journal on Selected Areas in Communications, special issue on Control and Communications, Vol. 26, N. 4, pp. 609—621, May 2008.

� A. Speranzon, C. Fischione, K. H. Johansson, “Distributed and Collaborative Estimation over Wireless Sensor Networks”, IEEE CDC 2006.

Centralized estimation: No/little intelligence on nodes

phenomena

1 2 N

Fusion Center

sensors

phenomena

1 2 N

Distributed estimation: no central coordination

sensors


Carlo Fischione

� Nodes perform a noisy measurement of a common time-varying signal

� Communication subject to space-time varying packet losses

Network and signals

d(t)


Carlo Fischione

Distributed estimator

� Nodes exchange local measurements and estimates

� Goal: find locally the coefficients and that minimize the variance of the estimation error

Local estimate

Global vector of the estimates


Carlo Fischione

Estimation coefficients

� Estimation coefficients minimizing the average estimation error, under stability constraints

Small Bias

Stable estimation error

Estimation error

� A centralized optimization problem � How to distribute the computation of the optimal solution?

1. Cost function and first constraint easy to distribute 2. Second constraint is difficult to distribute


Carlo Fischione

How to distribute the second constraint

Node j

Node i

The 1-norm and max-norm are easy to distribute, but give infeasibility


Carlo Fischione

How to distribute the second constraint

A global constraint is translated into a local one by using some thresholds


Carlo Fischione

Distributed estimator

� By using the thresholds: from centralized to distributed optimization


Carlo Fischione

Estimation coefficients

� How to compute the thresholds? � What is the performance of the estimator?


Carlo Fischione

How to compute the thresholds?

� The higher the thresholds the lower the estimation error

� A Lipschitz optimization problem ¾ See second part of the lecture


Carlo Fischione

Performance

The error variance of the estimator that makes a simple average of the received measurements is an upper bound to the error variance of the proposed distributed estimator.


Carlo Fischione

Simulation example

� Network with 30 nodes randomly deployed.

� Signal to track:

� Variance of the additive noise:


Carlo Fischione

Simulation Example (2)

Packet loss probability

“Laplacian” Estimator

Instantaneous Average Estimator Proposed Distributed Estimator


Carlo Fischione

Remarks

� A class of distributed estimators for WSNs ¾ Optimal distributed estimator ¾ Stability conditions ¾ Performance analysis

� Open issues ¾ Model-based estimator ¾ Estimator for signal with spatial and temporal correlation


Carlo Fischione

Optimal thresholds

� Optimal solution: the fixed point of a contraction mapping

� Each node updates asynchronously its threshold after receiving those of neighboring nodes.


Carlo Fischione

Distributed binary detection

measurements at node i

hypothesis thesting with S

measurements and threshold xi

probability of false alarm probability of misdetection � A threshold minimizing the prob. of false alarm maximizes the

prob. of misdetection. � How to choose optimally the thresholds when nodes exchange

opinions? Principles of Wireless Sensor Networks Fall 2011

Carlo Fischione

Threshold optimization in distributed detection

� How to solve the problem by parallel and distributed operations among the nodes?

� The problem is convex ¾ Lagrangian methods (interior point methods) could be applied ¾ Drowback: too many message passing (Lagrangian multipliers) among nodes to

compute iteratively the optimal solution

� An alternative method: F-Lipschitz optimization


done in ex. cognitive radios

Carlo Fischione

5

Distributed detection: F-Lipschitz vs Lagrangian methods

31 36

231

Number of iterations Number of function evaluations

F-Lipschitz Lagrangian methods (interior point)

10 nodes network


can be used also in estimation or radio power control

Carlo Fischione

Interference function theory

� vector of radio powers � e interference that the radio power has to overcome

� Properties of the (Type-I) interference function

nodes

Tx

Rx

Foschini, Miljanic, “A simple distributed autonomous power control algorithm and its convergence,” IEEE Trans. Veh. Technol., 1993.


Carlo Fischione

Example: Radio Power Control with unreliable components

� Unreliable transceivers introduce intermodulation powers difficult to compensate

� A difficult optimization problem � How to distribute the computation?

SINR

Outages

Tx

Rx


Carlo Fischione

� A change of variables and a redefinition

� F-Lipschitz qualifying properties are much more general than the interference function ones.

� More radio power control problems can be solved than traditional ones based on the Interference Function.

Power control as an F-Lipschitz problem


Carlo Fischione

Conclusions

� Existing methods for optimization over networks are too

expensive

� Studied the Fast-Lipschitz optimization ¾ Application to distributed estimation, and other cases

� F-Lipschitz optimization is a panacea for many cases, but still

there is a lack of a theory for fast parallel and distributed computations


principles of wireless sensor networkscarlofi/teaching/pwsn-2011/lectures/comment… · carlo...

Documents