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Enclosing Ellipsoids of Semi-algebraic Sets and Error Bounds in Polynomial Optimization. Makoto Yamashita Masakazu Kojima Tokyo Institute of Technology. Motivation from Sensor Network Localization Problem. If positions are known, computing distances is easy

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Enclosing Ellipsoids of Semi-algebraic Sets and Error Bounds in Polynomial Optimization

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## Enclosing Ellipsoids of Semi-algebraic Sets and Error Bounds in Polynomial Optimization

Makoto Yamashita

Masakazu Kojima

Tokyo Institute of Technology

### Motivation from Sensor Network Localization Problem

• If positions are known, computing distances is easy

• Reverse is difficult

• To obtain the positions of sensors, we need to solve

Anchor

6

7

2

5

1

4

3

Sensors

8

9

### SDP relaxation (by Biswas&Ye,2004)

Edge sets

Lifting

SDP Relaxation determines locations uniquelyunder some condition.

3’

### Region of solutions

• SNL sometimes has multiple solutions

• Interior-Point Methods generate a center point

• We estimate the regions of solutionsby SDP

4

5

2

3

3

mirroring

1

6

7

### Example of SNL

• Input network

• SDP solution

• Ellipsoids

difficult sensors

Difference of true locationand SDP solution

solved by SFSDP (Kim et al, 2008) http://www.is.titech.ac.jp/~kojima/SFSDP/SFSDP.html

with SDPA 7 (Yamashita et al, 2009)

http://sdpa.indsys.chuo-u.ac.jp/sdpa/

SDP relaxation

(convex region)

SDP solution

### General concept in Polynomial Optimization Problem

Semi-algebraic Sets

Optimal solutions exist in this ellipsoid.

We compute this ellipsoid by SDP.

(Polynomials)

Feasible region

for feasible region

min

Optimal

### Ellipsoid research

• .

• MVEE (the minimum volume enclosing ellipsoid)

• Our approach by SDP relaxation

• Solvable by SDP

• Small computation cost⇒We can execute multiple times changing

### Mathematical Formulation

• .

• Ellipsoidwith

• We want to compute

By some steps, we consider SDP relaxation

• .

• .

• Note that

• Furthermore

(convex hull)

linear (easier)

Still difficult

• .

• .

relaxation

### Inner minimization

• .

• .

• Optimal attained at

• .

• Cover

### Numerical Results on SNL

• We solvefor each sensor by

• Each SDP is solved quickly.

• #anchor = 4, #sensor = 100, #edge = 366

• 0.65 second for each (65 seconds for 100 sensors)

• #anchor = 4, #sensor = 500, #edge = 1917

• 5.6 second for each (2806 seconds for 500 sensors)

• SFSDP & SDPA on Xeon 5365(3.0GHz, 48GB)

• Sparsity technique is very important

### Results (#sensor = 100)

Ellipsoids cover true locations

### More edges case

If SDP solution is good, radius is very small.

### Example from POP

• ex9_1_2 from GLOBAL library(http://www.gamsworld.org/global/global.htm)

• We use SparsePOP to solve this by SDP relaxation

SparsePOP

http://www.is.titech.ac.jp/~kojima/SparsePOP/SparsePOP.html

### Reduced POP

Optimal Solutions:

Optimal Solutions:

Very tight bound

### Results on POP

• Very good objective values

• ex_9_1_2 & ex_9_1_8 have multiple optimal solutions ⇒ large radius

### Conclusion & Future works

• An enclosing ellipsoid by SDP relaxation

• Bound the locations of sensors

• Improve the SDP solution of POP

• Very low computation cost

• Ellipsoid becomes larger for unconnected sensors

• Successive ellipsoid for POP sometimes stops before bounding the region appropriately

This talk is based on the following technical paper

Masakazu Kojima and Makoto Yamashita,

“Enclosing Ellipsoids and Elliptic Cylinders of Semialgebraic Sets

and Their Application to Error Boundsin Polynomial Optimization”,

Research Report B-459,

Dept. of Math. and Comp. Sciences,Tokyo Institute of Technology,

Oh-Okayama, Meguro, Tokyo 152-8552,January 2010.