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 Sensors

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

4 Region of solutions  SNL sometimes has multiple solutions  Interior-Point Methods generate a center point  We estimate the regions of solutions by SDP ’3’ 3 mirroring

5 Example of SNL 1.Input network 2.SDP solution 3.Ellipsoids difficult sensors Difference of true location and SDP solution solved by SFSDP (Kim et al, 2008) with SDPA 7 (Yamashita et al, 2009)

6 General concept in Polynomial Optimization Problem min Optimal SDP relaxation (convex region) SDP solution Local adjustment for feasible region Optimal solutions exist in this ellipsoid. We compute this ellipsoid by SDP. Feasible region Semi-algebraic Sets (Polynomials)

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 .  Ellipsoid with  We want to compute By some steps, we consider SDP relaxation

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

10 SDP relaxation .... relaxation

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

12 Relations of

13 Numerical Results on SNL  We solve for each sensor by  Each SDP is solved quickly. #anchor = 4, #sensor = 100, #edge = second for each (65 seconds for 100 sensors) #anchor = 4, #sensor = 500, #edge = 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 (  We use SparsePOP to solve this by SDP relaxation SparsePOP

18 Region of the Solution

19 Reduced POP Optimal Solutions:

20 Ellipsoids for Reduced SDP Optimal Solutions: 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 ,January 2010.

24 SDP Formulation .. SDP (SOCP) SNL is one of such cases.