1. Privacy-Preserving Linear Programming Olvi Mangasarian UW Madison & UCSD La Jolla UCSD – Center for Computational Mathematics Seminar January 11, 2011 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.:  A A AAA A A A A A A AAA

2. Problem Statement Entities with related data wish to solve a linear program based on all the data The entities are unwilling to reveal their data to each other –If each entity holds a different set of variables for all constraints, then the data is said to be vertically partitioned –If each entity holds a different set of constraints with all variables, then the data is said to be horizontally partitioned Our approach: privacy-preserving linear programming (PPLP) using random matrix transformations –Provides exact solution to the total linear program –Does not reveal any private information

3. Vertically Partitioned Matrix Horizontally Partitioned Matrix A A1A1 A2A2 A3A3 A¢1A¢1 A¢2A¢2 A¢3A¢3 Linear Programming Constraint Matrix Variables 1 2..………….…………. n Constraints 12........m12........m

4. Outline Vertically (horizontally) partitioned linear program Secure transformation via a random matrix Privacy-preserving linear program solution Computational results Summary

5. Vertically Partitioned Data: Each entity holds different variables for the same constraints A¢1A¢1 A¢3A¢3 A¢2A¢2 A¢1A¢1 A¢2A¢2 A¢3A¢3

6. LP with Vertically Partitioned Data We consider the linear program :

7. Secure Linear Program Generation

8. Why Secure Linear Program?

9. Original & Secure LPs Are Equivalent

10. PPLP Algorithm

11. PPLP Algorithm (Continued)

14. Computatinal Results Example 1 (k=n=1000)

15. Computatinal Results Example 2 (k=n=100)

16. Horizontally Partitioned Constraint Matrix: Entities hold different constraints with the same variables A1A1 A2A2 A3A3 A3A3 A2A2 A1A1

17. LP with Horizontally Partitioned Data We consider the linear program :

18. Secure Linear Program Generation

19. Why Secure Linear Program?

20. Original & Secure LPs Are Equivalent

21. PPHPLP Algorithm

22. PPHPLP Algorithm (Continued)

23. Computatinal Results Example 1 (k=1000)

24. Computatinal Results Example 2 (k=1000)

25. Summary & Outlook –Based on a transformation using a random matrix B –Get exact solution to the original linear program without revealing privately held data Possible extensions to: horizontally partitioned inequality constraints, complementarity problems and nonlinear programs Privacy preserving linear programming for vertically or horizontally partitioned data

26. References ftp://ftp.cs.wisc.edu/pub/dmi/tech-reports/10-01.pdf ftp://ftp.cs.wisc.edu/pub/dmi/tech-reports/10-02.pdf Optimization Letters, to appear