1 / 23

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

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

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about 'Enclosing Ellipsoids of Semi-algebraic Sets and Error Bounds in Polynomial Optimization' - tavi

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

### 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

• 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/

(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

• .

• MVEE (the minimum volume enclosing ellipsoid)

• Our approach by SDP relaxation

• Solvable by SDP

• Small computation cost⇒We can execute multiple times changing

• .

• Ellipsoidwith

• We want to compute

By some steps, we consider SDP relaxation

• .

• .

• Note that

• Furthermore

(convex hull)

linear (easier)

Still difficult

• .

• .

relaxation

• .

• .

• Optimal attained at

• .

• Cover

• 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

Ellipsoids cover true locations

If SDP solution is good, radius is very small.

• 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

Optimal Solutions:

Ellipsoids for Reduced SDP

Very tight bound

• Very good objective values

• ex_9_1_2 & ex_9_1_8 have multiple optimal solutions ⇒ large radius

• 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

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.