On p-Order Cone Relaxation of Sensor Network Localization

Localization

Paul Tseng

Mathematics, University of Washington Seattle

BIRS Workshop, Banff November 11-16, 2006

Abstract This is a talk given at Banff, 2006.

Talk Outline

• Problem description

• pOCP relaxations

• Properties of pOCP relaxations

• Performance of SOCP relaxation and efficient solution methods

• Conclusions & Future Directions

Sensor Network Localization Basic Problem

• n pts in <^d (d = 1, 2, 3).

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

d_ij ≥ 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, ...

Optimization Problem Formulation

υ_opt := min

x₁,...,x_m

X

(i,j)∈A

|kx_i − x_jk_p − d_ij|

• Objective function is nonconvex.

6. .

_

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

6. .

_

• Use a convex (but not SDP) relaxation.

Other formulations: “|kx_i − x_jk_p − d_ij|²” or “

kx_i − x_jk^p_p − d^p_ij

” or ...

p-Order Cone Program Relaxation

υ_opt = min

x₁,...,x_m,y_ij

X

(i,j)∈A

|y_ij − d_ij|

s.t. y_ij = kx_i − x_jk_p ∀(i, j) ∈ A

Relax “=” to “≥” constraint:

υ_socp := min

x₁,...,x_m,y_ij

X

(i,j)∈A

|y_ij − d_ij|

s.t. y_ij ≥ kx_i − x_jk_p ∀(i, j) ∈ A

(p=2: Doherty,Pister,El Ghaoui ’03)

Properties of pOCP Relaxation

d = 2, n = 3, m = 1, d₁₂ = d₁₃ = 2

Problem

0 = min

x₁∈<²

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

pOCP Relaxation

0 = min

x1∈<2 y12,y13∈<

|y₁₂ − 2| + |y₁₃ − 2|

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

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

Analytic Center Solution

Define

A_active := (i, j) ∈ A | y_ij = kx_i − x_jk_p ∀ solns x₁, ..., x_m, (y_ij)_(i,j)∈A of pOCP

Relative − interior soln := soln x₁, ..., x_m, (y_ij)_(i,j)∈A of pOCP with y_ij > kx_i − x_jk_p ∀ (i, j) ∈ A\A_active

Analytic center soln := arg min

rel.−int. soln x1,...,xm,(yij)(i,j)∈A

X

(i,j)∈A\A_active

− log y_ij^p − kx_i − x_jk^p_p

Properties of pOCP Relaxations Key Property

arg min

x

k

X

i=1

kx − x_ik_p ∈ conv {x₁, ..., x_k}

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

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

’05, C.-K. Sim ’03))

Fact 1

with N (i) := {j | (i, j) ∈ A}.

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

SOCP soln found by IP method (SeDuMi)

Fact 2

kx_i − x_jk_p = y_ij for some j ∈ N (i) ⇐⇒ x_i appears in every pOCP soln.

Error Bounds

What if distances have errors?

d_ij = d^true_ij + δ_ij,

where δ_ij ∈ < and d^true_ij := kx^true_i − x^true_j k_p (x^true_i = x_i ∀ i > m).

Fact 3

(d_ij)_(i,j)∈A and X

(i,j)∈A

|δ_ij| ≤ δ, then for each i,

kx_i − x_jk_p = y_ij for some j ∈ N (i) =⇒ kx_i − x^true_i k_p = O( _p

s X

(i,j)∈A

|δ_ij|).

Fact 4

(i,j)∈A

|δ_ij| → 0, (analytic center pOCP soln corresp. (d_ij)_(i,j)∈A)

→ (analytic center pOCP soln corresp. (d^true_ij )_(i,j)∈A).

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

Solving pOCP Relaxation I: IP Method

x₁,...,xmin_m,y_ij

X

(i,j)∈A

|y_ij − d_ij|

s.t. y_ij ≥ kx_i − x_jk_p ∀(i, j) ∈ A

Put into conic form:

min X

(i,j)∈A

u_ij

s.t. x_i − x_j − w_ij = 0 ∀(i, j) ∈ A

y_ij − u_ij = d_ij ∀(i, j) ∈ A

u_ij ≥ 0, (y_ij, w_ij) ∈ p−order cone ∀(i, j) ∈ A.

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

Solving pOCP Relaxation II: Smoothing + Coordinate Gradient Descent

miny≥z |y − d| = max{0, z − d}

So pOCP relaxation:

x₁min,...,x_m

X

(i,j)∈A

max{0, kx_i − x_jk_p − d_ij}

This is an unconstrained nonsmooth convex program.

• Smooth approximation:

max{0, t} ≈ h_µ(t) := (t² + µ²)^1/2 + t

2 (µ > 0)

pOCP approximation:

min f_µ(x₁, .., x_m) := X

(i,j)∈A

h_µ (kx_i − x_jk_p − d_ij + µ)

Add a smoothed log-barrier term −µ X

(i,j)∈A

log (µ + h_µ (d_ij − µ − kx_i − x_jk_p))

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

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

−[∇²_x

ix_if_µ]⁻¹∇_xif_µ, with Armijo stepsize rule, and re-iterate.

• Decrease µ when k∇_xif_µk = O(µ) ∀i.

µ^init = 1e − 5. µ^end = 1e − 9. Decrease µ by a factor of 10.

Code in Fortran. Computation easily distributes.

Simulation Results for p = 2

• Uniformly generate x^true₁ , ..., x^true_n in [0, 1]², m = .9n, two pts are nhbrs if dist< radioR.

Set

d_ij = kx^true_i − x^true_j 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 

kx_i − x_jk₂ − y_ij    

≤ 10⁻⁷d_ij for some j ∈ N (i).

SeDuMi “SCGD”

n m nf cpu/m_up/Err_up cpu/m_up/Err_up 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.

• m_up := number of uniquely positioned sensors.

• Err_up := maxi uniq. pos.kx_i − x^true_i k₂.

◦ = anchors ∗ = true positions of sensors

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

Conclusions & Future Directions

• Finding an analytic center soln of pOCP is key.

• Exploit network structures of pOCP? Distributed computation?

• Additional linear constraints?

• Other objective functions, e.g., X

(i,j)∈A

|kx_i − x_jk_p − d_ij|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,...,x_m

X

(i,j)∈A

kx_i − x_jk²₂ − d²_ij  ?