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

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### 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

Key phrase

- SDPARA:The fastest solver for large SDPs

SemiDefinite Programming Algorithm paRAllel veresion

available at http://sdpa.sf.net/

INFOMRS 2011 @ Charlotte

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

Outline

- SDP applications
- Standard form and Primal-Dual Interior-Point Methods
- Inside of SDPARA
- Numerical Results
- Conclusion

INFOMRS 2011 @ Charlotte

SDP Applications 1.Control theory

- Against swing,we want to keep stability.
- Stability Condition⇒ Lyapnov Condition⇒ SDP

INFOMRS 2011 @ Charlotte

SDP Applications2. Quantum Chemistry

- Ground state energy
- Locate electrons
- Schrodinger Equation⇒Reduced Density Matrix⇒SDP

INFOMRS 2011 @ Charlotte

SDP Applications3. Sensor Network Localization

- Distance Information⇒Sensor Locations
- Protein Structure

INFOMRS 2011 @ Charlotte

Standard form

- The variables are
- Inner Product is
- The size is roughly determined by

Our target

INFOMRS 2011 @ Charlotte

Schur Complement Matrix

Schur Complement Equation

Schur Complement Matrix

where

1. ELEMENTS (Evaluation of SCM)

2. CHOLESKY (Cholesky factorization of SCM)

INFOMRS 2011 @ Charlotte

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

Dense & Sparse SCM

Fully dense SCM (100%) Quantum Chemistry

Sparse SCM (9.26%) POP

SDPARA can select Dense or Sparse automatically.

INFOMRS 2011 @ Charlotte

Different Approaches

INFOMRS 2011 @ Charlotte

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

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

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

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

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

Comparison with PCSDPby SDP with Sparse SCM

- SDPARA handles SCM as sparse
- Only SDPARA can solve this size

INFOMRS 2011 @ Charlotte

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

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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