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

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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

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

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

SDP relaxation (by Biswas&Ye,2004)

Edge sets

Lifting

SDP Relaxation determines locations uniquelyunder some condition.


Region of solutions

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

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/


General concept in polynomial optimization problem

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

Local adjustment

for feasible region

min

Optimal


Ellipsoid research

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

Mathematical Formulation

  • .

  • Ellipsoidwith

  • We want to compute

By some steps, we consider SDP relaxation


Lifting

Lifting

  • .

  • .

  • Note that

  • Furthermore

(convex hull)

quadratic

linear (easier)

Still difficult


Sdp relaxation

SDP relaxation

  • .

  • .

relaxation


Inner minimization

Inner minimization

  • .

  • .

  • Gradient

  • Optimal attained at

  • .

  • Cover


Relations of

Relations of


Numerical results on snl

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

Results (#sensor = 100)


Diff v s radius

Diff v.s. Radius

Ellipsoids cover true locations


More edges case

More edges case

If SDP solution is good, radius is very small.


Example from pop

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


Region of the solution

Region of the Solution


Reduced pop

Reduced POP

Optimal Solutions:


Ellipsoids for reduced sdp

Optimal Solutions:

Ellipsoids for Reduced SDP

Very tight bound


Results on pop

Results on POP

  • Very good objective values

  • ex_9_1_2 & ex_9_1_8 have multiple optimal solutions ⇒ large radius


Conclusion future works

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


Enclosing ellipsoids of semi algebraic sets and error bounds in polynomial optimization

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.


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