Efficient Parallel Software for Large-Scale Semidefinite Programs

1. Efficient Parallel Software for Large-Scale Semidefinite Programs Makoto Yamashita @ Tokyo-Tech Katsuki Fujisawa @ Chuo University MSC 2010 @ Yokohama [2010/09/08]

2. Outline • SemiDefinite Programming • Conversion of stability condition for differential inclusions to an SDP • Primal-Dual Interior-Point Methods and its parallel implementation • Numerical Results

3. Many Applications of SDP • Control Theory • Stability Condition for Differential Inclusions • Discrete-Time Optimal Control Problem • Via SDP relaxation • Polynomial Optimization Problem • Sensor Network Problem • Quadratic Assignment Problem • Quantum Chemistry/Information • Large SDP ⇒ Parallel Solver

4. Standard form of SDP

5. Stability condition for differential inclusions to standard SDP Boyd et al • . • Does the solution remain in a bounded region? • i.e., • Yes, if

6. Conversion to SDP • . • To hold this inequality, • Bounding the condition number⇒SDP.

7. SDP from SCDI • . • Feasible solution ⇒ Boundness of the solution • Some translation for standard SDPby e.g. YALMIP [J. Löfberg].

8. Discrete-Time Optimal Control Problems • This Problem [Coleman et al] can be formulated as SDP via SparsePOP [Kim et al].

9. Primal-Dual Interior-Point Methods • Both Primal and Dual simultaneously in Polynomial-time • Many software are developed • SDPA [Yamashita et al] • SDPT3 [Toh et al] • SeDuMi [Sturm et al] • CSDP [Borcher et al]

10. Algorithmic Framework of Primal-Dual Interior-Point Methods Feasible Region of Central Path Initial Point Step Length to keep interior property Target Point Search Direction The most computational time is consumed by the Search Direction Optimal Solution

11. Bottlenecks in PDIPMand SDPARA • To obtain the direction, we solve • ELEMENTS • CHOLESKY • In SDPARA, parallel computation is applied to these two bottlenecks Xeon 5460,3.16GHz

12. Nonzero pattern ofSchur complement matrix (B) • Sparse Schur complement matrix • Fully dense Schur complement matrix DTOC SCDI

13. Exploitation of Sparsityin SDPA • We change the formula by row-wise • We keep this scheme on parallel computation F1 F2 F3

14. Row-wise distribution for dense Schur complement matrix • 4 CPU is available • Each CPU computes only their assigned rows • . • No communication between CPUs • Efficient memory management

15. Fomula-Cost Based distribution for sparse Schur complement

16. Parallel Computation for CHOLESKY • We employ • ScaLAPACK [Blackford et.al] ⇒ Dense • MUMPS [Amestoy et.al] ⇒ Sparse • Different data storage enhance the parallel Cholesky factorization

17. Problems for Numerical Results • 16 nodes • Xeon X5460 (3.16GHz) • 48GB memory

18. Total 15.02 times ELEMENTS 15.67 times CHOLESKY 14.20 times Computation time on SDP [SCDI1] Xeon X5460(3.16GHz) 48GB memory/node ELEMENTS attains high scalability

19. Total 4.85 times ELEMENTS 13.50 times CHOLESKY 4.34 times Computation time on SDP [DTOC1] Xeon X5460(3.16GHz) 48GB memory/node • Parallel Sparse Cholesky is difficult • ELEMENTS is still enhanced

20. Comparison with PCSDP [Ivanov et al] • SDPARA is faster than PCSDP • The scalability of SDPARA is higher • Only SDPARA can solve DTOC Time is second, O.M.:out of memory

21. Concluding Remarks & Future works • SDP has many applications including control theory • SDPARA solves Larse-scale SDPs effectively by parallel computation • Appropriate parallel computations are the key of SDPARA implementation • Improvement on Multi-Threading for sparse Schur complement matrix