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

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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

- 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

Edge sets

Lifting

SDP Relaxation determines locations uniquelyunder some condition.

3’

- 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/

SDP relaxation

(convex region)

SDP solution

Semi-algebraic Sets

Optimal solutions exist in this ellipsoid.

We compute this ellipsoid by SDP.

(Polynomials)

Feasible region

Local adjustment

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)

quadratic

linear (easier)

Still difficult

- .
- .

relaxation

- .
- .
- Gradient
- 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:

Optimal Solutions:

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

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.