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# On p-Order Cone Relaxation of Sensor Network Localization

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## Localization

Paul Tseng

Mathematics, University of Washington Seattle

BIRS Workshop, Banff November 11-16, 2006

Abstract This is a talk given at Banff, 2006.

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### Talk Outline

• Problem description

• pOCP relaxations

• Properties of pOCP relaxations

• Performance of SOCP relaxation and efficient solution methods

• Conclusions & Future Directions

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### Sensor Network Localization Basic Problem

:

• n pts in <d (d = 1, 2, 3).

• Know last n − m pts (‘anchors’) xm+1, ..., xn and `p-dist. (1 < p < ∞) estimate for pairs of ‘neighboring’ pts

dij ≥ 0 ∀(i, j) ∈ A with A ⊆ {(i, j) : 1 ≤ i < j ≤ n}.

• Estimate first m pts (‘sensors’).

History? Graph realization, position estimation in wireless sensor network, determining protein structures, ...

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### Optimization Problem Formulation

υopt := min

x1,...,xm

X

(i,j)∈A

|kxi − xjkp − dij|

• Objective function is nonconvex.

6. .

_

• Problem is NP-hard (reduction from PARTITION).

6. .

_

• Use a convex (but not SDP) relaxation.

Other formulations: “|kxi − xjkp − dij|2” or “

kxi − xjkpp − dpij

” or ...

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### p-Order Cone Program Relaxation

υopt = min

x1,...,xm,yij

X

(i,j)∈A

|yij − dij|

s.t. yij = kxi − xjkp ∀(i, j) ∈ A

Relax “=” to “≥” constraint:

υsocp := min

x1,...,xm,yij

X

(i,j)∈A

|yij − dij|

s.t. yij ≥ kxi − xjkp ∀(i, j) ∈ A

(p=2: Doherty,Pister,El Ghaoui ’03)

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### Properties of pOCP Relaxation

d = 2, n = 3, m = 1, d12 = d13 = 2

### Problem

:

0 = min

x1∈<2

|kx1 − (0, 1)kp − 2| + |kx1 − (0, −1)kp − 2|

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### pOCP Relaxation

:

0 = min

x1∈<2 y12,y13∈<

|y12 − 2| + |y13 − 2|

s.t. y12 ≥ kx1 − (0, 1)kp y13 ≥ kx1 − (0, −1)kp

If solve pOCP by IP method, then likely get analy. center of soln set.

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### Analytic Center Solution

Define

Aactive := (i, j) ∈ A | yij = kxi − xjkp ∀ solns x1, ..., xm, (yij)(i,j)∈A of pOCP

Relative − interior soln := soln x1, ..., xm, (yij)(i,j)∈A of pOCP with yij > kxi − xjkp ∀ (i, j) ∈ A\Aactive

Analytic center soln := arg min

rel.−int. soln x1,...,xm,(yij)(i,j)∈A

X

(i,j)∈A\Aactive

− log yijp − kxi − xjkpp

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### Properties of pOCP Relaxations Key Property

: If p = 2 or d = 2, then

arg min

x

k

X

i=1

kx − xikp ∈ conv {x1, ..., xk}

for any k and x1, ..., xk ∈ <d. (Remains true if kx − xikp is raised to pth power.)

Q: What if d ≥ 3 and p 6= 2? (False if kx − xikp is raised to pth power. (S. Zhang

’05, C.-K. Sim ’03))

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### Fact 1

: If (x1, ..., xm, yij)(i,j)∈A is the analytic center soln of pOCP, then xi ∈ conv {xj}j∈N (i) ∀i ≤ m

with N (i) := {j | (i, j) ∈ A}.

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True soln (m = 900, n = 1000, nhbrs if `2-dist< .06)

SOCP soln found by IP method (SeDuMi)

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### Fact 2

: If (x1, ..., xm, yij)(i,j)∈A is a relative-interior pOCP soln (e.g., analytic center), then for each i ≤ m,

kxi − xjkp = yij for some j ∈ N (i) ⇐⇒ xi appears in every pOCP soln.

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### Error Bounds

What if distances have errors?

dij = dtrueij + δij,

where δij ∈ < and dtrueij := kxtruei − xtruej kp (xtruei = xi ∀ i > m).

### Fact 3

: If (x1, ..., xm, yij)(i,j)∈A is a relative-interior pOCP soln corresp.

(dij)(i,j)∈A and X

(i,j)∈A

ij| ≤ δ, then for each i,

kxi − xjkp = yij for some j ∈ N (i) =⇒ kxi − xtruei kp = O( p

s X

(i,j)∈A

ij|).

### Fact 4

: As X

(i,j)∈A

ij| → 0, (analytic center pOCP soln corresp. (dij)(i,j)∈A)

→ (analytic center pOCP soln corresp. (dtrueij )(i,j)∈A).

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Error bounds for the analytic center pOCP soln when distances have small errors.

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### Solving pOCP Relaxation I: IP Method

x1,...,xminm,yij

X

(i,j)∈A

|yij − dij|

s.t. yij ≥ kxi − xjkp ∀(i, j) ∈ A

Put into conic form:

min X

(i,j)∈A

uij

s.t. xi − xj − wij = 0 ∀(i, j) ∈ A

yij − uij = dij ∀(i, j) ∈ A

uij ≥ 0, (yij, wij) ∈ p−order cone ∀(i, j) ∈ A.

Solve by an IP method, e.g., SeDuMi or Mosek for p = 2.

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### Solving pOCP Relaxation II: Smoothing + CoordinateGradient Descent

miny≥z |y − d| = max{0, z − d}

So pOCP relaxation:

x1min,...,xm

X

(i,j)∈A

max{0, kxi − xjkp − dij}

This is an unconstrained nonsmooth convex program.

• Smooth approximation:

max{0, t} ≈ hµ(t) := (t2 + µ2)1/2 + t

2 (µ > 0)

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pOCP approximation:

min fµ(x1, .., xm) := X

(i,j)∈A

hµ (kxi − xjkp − dij + µ)

Add a smoothed log-barrier term −µ X

(i,j)∈A

log (µ + hµ (dij − µ − kxi − xjkp))

Solve the smooth approximation using coordinate gradient descent (SCGD):

• If k∇xifµk = Ω(µ), then update xi by moving it along the Newton direction

−[∇2x

ixifµ]−1xifµ, with Armijo stepsize rule, and re-iterate.

• Decrease µ when k∇xifµk = O(µ) ∀i.

µinit = 1e − 5. µend = 1e − 9. Decrease µ by a factor of 10.

Code in Fortran. Computation easily distributes.

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### Simulation Results for p = 2

• Uniformly generate xtrue1 , ..., xtruen in [0, 1]2, m = .9n, two pts are nhbrs if dist< radioR.

Set

dij = kxtruei − xtruej k2 · |1 + ij · nf |,

ij ∼ N (0, 1) (Biswas, Ye ’03)

• Solve SOCP using SeDuMi 1.05 or SCGD.

• Sensor i is uniquely positioned if

kxi − xjk2 − yij

≤ 10−7dij for some j ∈ N (i).

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SeDuMi “SCGD”

n m nf cpu/mup/Errup cpu/mup/Errup 1000 900 0 1.0/327/4.6e-5 .2/357/3.8e-5 1000 900 .001 0.9/438/1.6e-3 .4/442/1.5e-3 1000 900 .01 1.1/555/1.5e-2 1.6/518/1.1e-2 2000 1800 0 33.7/1497/4.3e-4 0.8/1541/3.3e-4 2000 1800 .001 38.0/1465/3.3e-3 1.8/1466/3.6e-3 2000 1800 .01 16.2/1704/6.3e-2 2.9/1707/5.1e-2 4000 3600 0 17.8/2758/3.0e-4 1.6/2844/3.2e-4 4000 3600 .001 20.4/2907/3.2e-3 5.1/2894/3.0e-3 4000 3600 .01 17.2/3023/9.1e-3 6.1/3020/9.1e-3

Table 1: radioR = .06(.035) for n = 1000, 2000(4000)

• cpu (minutes) times are on a HP DL360 workstation, running Linux 3.5.

• mup := number of uniquely positioned sensors.

• Errup := maxi uniq. pos.kxi − xtruei k2.

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◦ = anchors ∗ = true positions of sensors

• = SOCP soln found by SCGD (m = 900, n = 1000, radioR = .06, nf = .01)

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### Conclusions & Future Directions

• Finding an analytic center soln of pOCP is key.

• Exploit network structures of pOCP? Distributed computation?

• Other objective functions, e.g., X

(i,j)∈A

|kxi − xjkp − dij|2 ?

• How to localize sensors not uniquely positioned? Smoothing + local descent on original objective function...

• Extend error bound result to SDP and ESDP relaxations (Biswas, Ye ’04, ..., Wang et al. ’06)) of min

x1,...,xm

X

(i,j)∈A

kxi − xjk22 − d2ij ?

Error bounds for the analytic center SOCP soln when distances have small errors... Decrease µ by a factor

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