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.

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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

INFORMS Annual Meeting @ Charlotte 2011/11/15(2011/11/13-2011/11/16)

INFOMRS 2011 @ Charlotte


Key phrase
Key phrase Schur Complements Matrices

  • SDPARA:The fastest solver for large SDPs

SemiDefinite Programming Algorithm paRAllel veresion

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

INFOMRS 2011 @ Charlotte


Sdpa online solver
SDPA Online Solver Schur Complements Matrices

  • 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
Outline Schur Complements Matrices

  • SDP applications

  • Standard form and Primal-Dual Interior-Point Methods

  • Inside of SDPARA

  • Numerical Results

  • Conclusion

INFOMRS 2011 @ Charlotte


Sdp applications 1 control theory
SDP Applications Schur Complements Matrices 1.Control theory

  • Against swing,we want to keep stability.

  • Stability Condition⇒ Lyapnov Condition⇒ SDP

INFOMRS 2011 @ Charlotte


Sdp applications 2 quantum chemistry
SDP Applications Schur Complements Matrices 2. Quantum Chemistry

  • Ground state energy

  • Locate electrons

  • Schrodinger Equation⇒Reduced Density Matrix⇒SDP

INFOMRS 2011 @ Charlotte


Sdp applications 3 sensor network localization
SDP Applications Schur Complements Matrices 3. Sensor Network Localization

  • Distance Information⇒Sensor Locations

  • Protein Structure

INFOMRS 2011 @ Charlotte


Standard form
Standard form Schur Complements Matrices

  • The variables are

  • Inner Product is

  • The size is roughly determined by

Our target

INFOMRS 2011 @ Charlotte


Primal dual interior point methods
Primal-Dual Interior-Point Methods Schur Complements Matrices

Central Path

Target

Optimal

Feasible region

INFOMRS 2011 @ Charlotte


Schur complement matrix
Schur Complement Matrix Schur Complements Matrices

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
Computation time on single processor Schur Complements Matrices

  • SDPARA replaces these bottleneks by parallel computation

Time unit is second, SDPA 7, Xeon 5460 (3.16GHz)

INFOMRS 2011 @ Charlotte


Dense sparse scm
Dense & Sparse SCM Schur Complements Matrices

Fully dense SCM (100%) Quantum Chemistry

Sparse SCM (9.26%) POP

SDPARA can select Dense or Sparse automatically.

INFOMRS 2011 @ Charlotte


Different approaches
Different Approaches Schur Complements Matrices

INFOMRS 2011 @ Charlotte


Three formulas for elements
Three formulas for ELEMENTS Schur Complements Matrices

dense

sparse

All rows are independent.

INFOMRS 2011 @ Charlotte


Row wise distribution

Server1 Schur Complements Matrices

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
Numerical Results on Dense SCM Schur Complements Matrices

  • 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
Drawback of Row-wise Schur Complements Matrices to Sparse SCM

dense

sparse

  • Simple row-wise is ineffective for sparse SCM

  • We estimate cost of each element

INFOMRS 2011 @ Charlotte


Formula cost based distribution
Formula-cost-based distribution Schur Complements Matrices

Good load-balance

INFOMRS 2011 @ Charlotte


Numerical results on sparse scm
Numerical Results on Sparse SCM Schur Complements Matrices

  • 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 pcsdp by sdp with dense scm
Comparison with PCSDP Schur Complements Matrices by 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 pcsdp by sdp with sparse scm
Comparison with PCSDP Schur Complements Matrices by SDP with Sparse SCM

  • SDPARA handles SCM as sparse

  • Only SDPARA can solve this size

INFOMRS 2011 @ Charlotte


Extremely large scale sdps
Extremely Large-Scale SDPs Schur Complements Matrices

  • 16 Servers [Xeon X5670(2.93GHz) , 128GB Memory]

Other solvers can handle only

The LARGEST solved SDP in the world

INFOMRS 2011 @ Charlotte


Conclusion
Conclusion Schur Complements Matrices

  • 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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