1. 1 Ellipsoid-type Confidential Bounds on Semi-algebraic Sets via SDP Relaxation Makoto Yamashita Masakazu Kojima Tokyo Institute of Technology

2. 2 Confidential Bounds in Polynomial Optimization Problem min Optimal SDP relaxation (convex region) SDP solution Local adjustment for feasible region We compute this ellipsoid by SDP. Optimal solutions exist in this ellipsoid. Feasible region Semi-algebraic Sets (Polynomials)

3. 3 Ellipsoid research ..  MVEE (the minimum volume enclosing ellipsoid)  Our approach based on SDP relaxation Solvable by SDP Small computation cost ⇒ We can execute multiple times changing

4. Outline 1.Math Form of Ellipsoids 2.SDP relaxation 3.Examples of POP 4.Tightness of Ellipsoids 4

5. 5 Mathematical Formulation ..  Ellipsoid We define .. By some steps, we consider SDP relaxation

6. .. ..  Note that  Furthermore 6 Lifting ⇒ quadratic linear (easier) Still difficult (convex hull)

7. 7 SDP relaxation .. .. relaxation

8. 8 .. ..  Gradient  Optimal attained at ..  Cover Inner minimization

9. 9 Relations of SDP

10. 10 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://sparsepop.sourceforge.net

11. 11 Region of the Solution

12. 12 Reduced POP Optimal Solutions:

13. 13 Ellipsoids for Reduced SDP Optimal Solutions: Very tight bound

14. 14 Results on POP  Very good objective values  ex_9_1_2 & ex_9_1_8 have multiple optimal solutions ⇒ large radius

15. Tightness of Ellipsoids  Target set  6 Shape Matricies  We draw 2D picture, 15

16. The case p=2 (2 constraints)  The ellipsoids are tight. 16 Target set 6 ellipsoids by SDP

17. More constraints 17 Ellipsoids shrink. But its speed is slower than the target set. p=2 p=32 p=128

18. 18 Conclusion & Future works  An enclosing ellipsoid by SDP relaxation Improve the SDP solution of POP Very low computation cost  Successive ellipsoid for POP sometimes stops before bounding the region appropriately  Ellipsoids may become loose in the case of many constraints

19. 19  Thank you very much for your attention. 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.