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

  2. 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

  3. SDP relaxation (by Biswas&Ye,2004) Edge sets Lifting SDP Relaxation determines locations uniquelyunder some condition.

  4. 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

  5. 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/

  6. 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

  7. 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

  8. Mathematical Formulation • . • Ellipsoidwith • We want to compute By some steps, we consider SDP relaxation

  9. Lifting • . • . • Note that • Furthermore ⇒ (convex hull) quadratic linear (easier) Still difficult

  10. SDP relaxation • . • . relaxation

  11. Inner minimization • . • . • Gradient • Optimal attained at • . • Cover

  12. Relations of

  13. 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

  14. Results (#sensor = 100)

  15. Diff v.s. Radius Ellipsoids cover true locations

  16. More edges case If SDP solution is good, radius is very small.

  17. 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

  18. Region of the Solution

  19. Reduced POP Optimal Solutions:

  20. Optimal Solutions: Ellipsoids for Reduced SDP Very tight bound

  21. Results on POP • Very good objective values • ex_9_1_2 & ex_9_1_8 have multiple optimal solutions ⇒ large radius

  22. 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

  23. 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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