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

Makoto YamashitaMasakazu Kojima Tokyo Institute of Technology

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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

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Anchor

3

4

2

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Sensors

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SDP relaxation (by Biswas&Ye,2004)

Lifting

SDP Relaxation determines locations uniquely under some condition.

Edge sets

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Region of solutions SNL sometimes

has multiple solutions

Interior-Point Methods generate a center point

We estimate the regions of solutions by SDP

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1

2

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3’

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mirroring

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Example of SNL1. Input network2. SDP solution3. 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.htmlwith SDPA 7 (Yamashita et al, 2009)http://sdpa.indsys.chuo-u.ac.jp/sdpa/

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General concept in Polynomial Optimization Problem

min

Optimal

SDP relaxation(convex region)

SDP solution

Local adjustmentfor feasible region

Optimal solutions exist in this ellipsoid.We compute this ellipsoid by SDP.

Feasible region

Semi-algebraic Sets

(Polynomials)

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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

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Mathematical Formulation . Ellipsoid

with

We want to compute

By some steps, we consider SDP relaxation

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Lifting

.

.

Note that Furthermore

⇒

quadratic

linear (easier)

Still difficult

(convex hull)

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SDP relaxation

. .

relaxation

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. .

Gradient Optimal attained at

.

Cover

Inner minimization

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Relations of

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Numerical Results on SNL We solve

for 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

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Results (#sensor = 100)

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Diff v.s. Radius

Ellipsoids cover true locations

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More edges case

If SDP solution is good, radius is very small.

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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

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

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Region of the Solution

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Reduced POP

Optimal Solutions:

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Ellipsoids for Reduced SDP

Optimal Solutions:

Very tight bound

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Results on POP

Very good objective values ex_9_1_2 & ex_9_1_8 have multiple optimal

solutions ⇒ large radius

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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

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This talk is based on the following technical paperMasakazu 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.