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
Tokyo Institute of Technology
SDP Relaxation determines locations uniquelyunder some condition.
3’Region of solutions
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)
SDP solutionGeneral concept in Polynomial Optimization Problem
Optimal solutions exist in this ellipsoid.
We compute this ellipsoid by SDP.
for feasible region
By some steps, we consider SDP relaxation
Ellipsoids cover true locations
If SDP solution is good, radius is very small.
Very tight bound
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.