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Parallel Computation for SDPs Focusing on the Sparsity of Schur Complements Matrices

Parallel Computation for SDPs Focusing on the Sparsity of Schur Complements Matrices. Makoto Yamashita @ Tokyo Tech Katsuki Fujisawa @ Chuo Univ Mituhiro Fukuda @ Tokyo Tech Kazuhide Nakata @ Tokyo Tech Maho Nakata @ RIKEN.

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Parallel Computation for SDPs Focusing on the Sparsity of Schur Complements Matrices

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  1. Parallel Computation for SDPs Focusing on the Sparsity of Schur Complements Matrices Makoto Yamashita @ Tokyo Tech Katsuki Fujisawa @ Chuo Univ Mituhiro Fukuda @ Tokyo Tech Kazuhide Nakata @ Tokyo Tech Maho Nakata @ RIKEN INFORMS Annual Meeting @ Charlotte 2011/11/15(2011/11/13-2011/11/16) INFOMRS 2011 @ Charlotte

  2. Key phrase • SDPARA:The fastest solver for large SDPs SemiDefinite Programming Algorithm paRAllel veresion available at http://sdpa.sf.net/ INFOMRS 2011 @ Charlotte

  3. SDPA Online Solver • Log-in the online solver • Upload your problem • Push ’Execute’ button • Receive the result via Web/Mail http://sdpa.sf.net/ ⇒ Online Solver INFOMRS 2011 @ Charlotte

  4. Outline • SDP applications • Standard form and Primal-Dual Interior-Point Methods • Inside of SDPARA • Numerical Results • Conclusion INFOMRS 2011 @ Charlotte

  5. SDP Applications 1.Control theory • Against swing,we want to keep stability. • Stability Condition⇒ Lyapnov Condition⇒ SDP INFOMRS 2011 @ Charlotte

  6. SDP Applications2. Quantum Chemistry • Ground state energy • Locate electrons • Schrodinger Equation⇒Reduced Density Matrix⇒SDP INFOMRS 2011 @ Charlotte

  7. SDP Applications3. Sensor Network Localization • Distance Information⇒Sensor Locations • Protein Structure INFOMRS 2011 @ Charlotte

  8. Standard form • The variables are • Inner Product is • The size is roughly determined by Our target INFOMRS 2011 @ Charlotte

  9. Primal-Dual Interior-Point Methods Central Path Target Optimal Feasible region INFOMRS 2011 @ Charlotte

  10. Schur Complement Matrix Schur Complement Equation Schur Complement Matrix where 1. ELEMENTS (Evaluation of SCM) 2. CHOLESKY (Cholesky factorization of SCM) INFOMRS 2011 @ Charlotte

  11. Computation time on single processor • SDPARA replaces these bottleneks by parallel computation Time unit is second, SDPA 7, Xeon 5460 (3.16GHz) INFOMRS 2011 @ Charlotte

  12. Dense & Sparse SCM Fully dense SCM (100%) Quantum Chemistry Sparse SCM (9.26%) POP SDPARA can select Dense or Sparse automatically. INFOMRS 2011 @ Charlotte

  13. Different Approaches INFOMRS 2011 @ Charlotte

  14. Three formulas for ELEMENTS dense sparse All rows are independent. INFOMRS 2011 @ Charlotte

  15. Server1 Server2 Server3 Server4 Server1 Server2 Server3 Server4 Row-wise distribution • Assign servers in a cyclic manner • Simple idea⇒Very EFFICINENT • High scalability INFOMRS 2011 @ Charlotte

  16. Numerical Results on Dense SCM • Quantum Chemistry (m=7230, SCM=100%), middle size • SDPARA 7.3.1, Xeon X5460, 3.16GHz x2, 48GB memory ELEMENTS 15x speedup Total 13x speedup Very fast!! INFOMRS 2011 @ Charlotte

  17. Drawback of Row-wise to Sparse SCM dense sparse • Simple row-wise is ineffective for sparse SCM • We estimate cost of each element INFOMRS 2011 @ Charlotte

  18. Formula-cost-based distribution Good load-balance INFOMRS 2011 @ Charlotte

  19. Numerical Results on Sparse SCM • Control Theory (m=109,246, SCM=4.39%), middle size • SDPARA 7.3.1, Xeon X5460, 3.16GHz x2, 48GB memory ELEMENTS 13x speedupCHOLESKY 4.7xspeedup Total 5x speedup INFOMRS 2011 @ Charlotte

  20. Comparison with PCSDPby SDP with Dense SCM • developed by Ivanov & de Klerk Time unit is second SDP: B.2P Quantum Chemistry (m = 7230, SCM = 100%)Xeon X5460, 3.16GHz x2, 48GB memory SDPARA is 8x faster by MPI & Multi-Threading INFOMRS 2011 @ Charlotte

  21. Comparison with PCSDPby SDP with Sparse SCM • SDPARA handles SCM as sparse • Only SDPARA can solve this size INFOMRS 2011 @ Charlotte

  22. Extremely Large-Scale SDPs • 16 Servers [Xeon X5670(2.93GHz) , 128GB Memory] Other solvers can handle only The LARGEST solved SDP in the world INFOMRS 2011 @ Charlotte

  23. Conclusion • Row-wise & Formula-cost-based distribution • parallel Cholesky factorization • SDPARA:The fastest solver for large SDPs • http://sdpa.sf.net/ & Online solver Thank you very much for your attention. INFOMRS 2011 @ Charlotte

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