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
  • 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
  • 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
  • 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 1.Control theory
  • Against swing,we want to keep stability.
  • Stability Condition⇒ Lyapnov Condition⇒ SDP

INFOMRS 2011 @ Charlotte

sdp applications 2 quantum chemistry
SDP Applications2. Quantum Chemistry
  • Ground state energy
  • Locate electrons
  • Schrodinger Equation⇒Reduced Density Matrix⇒SDP

INFOMRS 2011 @ Charlotte

sdp applications 3 sensor network localization
SDP Applications3. Sensor Network Localization
  • Distance Information⇒Sensor Locations
  • Protein Structure

INFOMRS 2011 @ Charlotte

standard form
Standard form
  • 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

Central Path

Target

Optimal

Feasible region

INFOMRS 2011 @ Charlotte

schur complement matrix
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
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
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
Different Approaches

INFOMRS 2011 @ Charlotte

three formulas for elements
Three formulas for ELEMENTS

dense

sparse

All rows are independent.

INFOMRS 2011 @ Charlotte

row wise distribution
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
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
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

formula cost based distribution
Formula-cost-based distribution

Good load-balance

INFOMRS 2011 @ Charlotte

numerical results on sparse scm
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 pcsdp by sdp with dense scm
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 pcsdp by sdp with sparse scm
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
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
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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