1 Enclosing Ellipsoids of Semi-algebraic Sets and Error Bounds in Polynomial Optimization Makoto Yamashita Masakazu Kojima Tokyo Institute of Technology
Jan 15, 2016
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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.