1. Iterative LP and SOCP-based approximations to semidefinite and sum of squares programs Georgina Hall Princeton University Joint work with: Amir Ali Ahmadi (Princeton University) Sanjeeb Dash (IBM)

2. IOS 2016 Semidefinite programming: definition

3. IOS 2016 Semidefinite programming: two applications Polynomial optimizationStability number of a graph (De Klerk, Pasechnik)

4. IOS 2016 Semidefinite programming: pros and cons +: high-quality bounds for non convex problems. - : can take too long to solve if psd constraint is too big. Polynomial optimizationStability number of a graph Typical laptop cannot minimize a degree-4 polynomial in more than a couple dozen variables. (PC: 3.4 GHz, 16 Gb RAM)

5. IOS 2016 LP and SOCP-based inner approximations (1/2) Problem SDPs do not scale well Idea Replace psd cone by LP or SOCP representable cones Candidate (Ahmadi, Majumdar)

6. IOS 2016 LP and SOCP-based inner approximations (2/2) Replacing psd cone by dd/sdd cones: +: fast bounds - : not always as good quality (compared to SDP) Iteratively construct a sequence of improving LP/SOCP-based cones Initialization: Start with the DD/SDD cone Method 2: Column GenerationMethod 1: Cholesky change of basis

7. IOS 2016 Method 1:Cholesky change of basis (1/4) psd but not dd dd in the “right basis”

8. IOS 2016 Method 1: Cholesky change of basis (2/4) Step 1Initialize Step 2

9. IOS 2016 Method 1: Cholesky change of basis (3/4) Example 1: minimizing a degree-4 polynomial in 4 variables

10. IOS 2016 Method 1: Cholesky change of basis (4/4) Example 2: stability number of the Petersen graph

11. IOS 2016 Method 2: Column generation DD coneSDD cone Facet description Extreme ray description + + ++ +

12. IOS 2016 Method 2: Column Generation Initial SDPTake the dualFind a new “atom”Take the dualFind a new “atom”

13. IOS 2016 Method 2: Column Generation

14. IOS 2016 Method 2: Column Generation (3/3) Example 1: stability number of the Petersen graph Petersen Graph Apply Column Generation technique to this SDP

15. IOS 2016 Method 2: Column Generation (3/3) Example 2: minimizing a degree-4 polynomial bdt(s)bdt(s)bdt(s)bdt(s)bdt(s) DSOS-10.960.38-18.010.74-26.4515.51-36.527.88-62.3010.68 SDSOS-10.430.53-17.331.06-25.798.72-36.045.65-61.2518.66 -5.5731.19-9.02471.39-20.08600-32.28600-35.14600 SOS-3.265.60-3.5882.22-3.711068.66NA bdt(s)bdt(s)bdt(s)bdt(s)bdt(s) -6.2010.96-12.3870.70-20.08508.63NA 1SDSOS-8.9714.39-15.2982.30-23.14538.54NA

16. IOS 2016 Main messages Can improve on initial DD and SDD inner approximations to the PSD cone. This is done with LPs and SOCPs and iteratively. We presented two methods for this: cholesky change of basis and column generation.

17. Thank you for listening Questions? Want to learn more? http://scholar.princeton.edu/ghall/home http://scholar.princeton.edu/ghall/home 17