Principles of Wireless Sensor Networkscarlofi/teaching/pwsn-2011/lectures/lec5.pdf · Lipschitz...

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

Today’s lecture

Previous lecture Non expansive mappings, agreement, consensus

Today F-Lipschitz optimization

Protocol stack

Phy

MAC

Routing

Transport

Session

Application

Presentation

Carlo Fischione

Today’s learning goals

In today’s 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

Carlo Fischione

Optimization over networks

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

content distribution, routing, ....

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

An alternative theory is needed

In a number of cases, Fast-Lipschitz optimization


Carlo Fischione

Outline

• Definition of Fast-Lipschitz optimization

• Computation of the optimal solution

• Problems in canonical form

• Examples of application

• Conclusions


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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? 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.


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The Fast-Lipschitz optimization

nonempty compact set “containing” other constraints


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Computation of the solution

Centralized optimization Problem solved by a central processor

Distributed optimization

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

Network of n nodes


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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,...


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Pareto Optimal Solution


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Notation

Norm infinity: sum along a row

Norm 1: sum along a column

Gradient


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Functions may be non-convex

F-Lipschitz qualifying properties


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Outline





• Conclusions


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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.


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Lagrangian methods

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

KKT conditions:

Lagrangian

Lagrangian methods to compute the solution


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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


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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


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Outline





• Conclusions


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Problems in canonical form

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


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Problems in canonical form

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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


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Example 2: a hidden F-Lipschitz

Non F-Lipschitz

Simple variable transformation, , F-Lipschitz


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An F-Lipschitz Matlab Toolbox

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


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Outline





• Conclusions

Fall 2011 Principles of Wireless Sensor Networks

Carlo Fischione

Distributed estimation

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


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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


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

Communication subject to space-time varying packet losses

Network and signals

d(t)


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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


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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


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How to distribute the second constraint

Node j

Node i

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


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How to distribute the second constraint

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


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Distributed estimator

By using the thresholds: from centralized to distributed optimization


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Estimation coefficients

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


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How to compute the thresholds?

The higher the thresholds the lower the estimation error

A Lipschitz optimization problem See second part of the lecture


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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.


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Simulation example

Network with 30 nodes randomly deployed.

Signal to track:

Variance of the additive noise:


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Simulation Example (2)

Packet loss probability

“Laplacian” Estimator

Instantaneous Average Estimator Proposed Distributed Estimator


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Remarks

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

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


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Optimal thresholds

Optimal solution: the fixed point of a contraction mapping

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


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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 Lagrangian methods (interior point methods) could be applied Drowback: too many message passing (Lagrangian multipliers) among nodes to

compute iteratively the optimal solution

An alternative method: F-Lipschitz optimization


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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


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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.


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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


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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


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Conclusions

Existing methods for optimization over networks are too

expensive

Studied the Fast-Lipschitz optimization 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


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