1. High Performance Solvers for Semidefinite Programs
This talk is supported by Ewha University
High Performance Solvers for Semidefinite Programs
Makoto Yamashita @ Tokyo Tech
Katsuki Fujisawa @ Chuo Univ
Mituhiro Fukuda @ Tokyo Tech
Kazuhiro NMRI
Kazuhide Nakata @ Tokyo Tech
Maho Nakata @ RIKEN
KSIAM Annual Jeju 2011/11/25 (2011/11/ /11/26)

2. Our interests & SDPA Family
How fast can we solve SDPs?
How large SDP can we solve?
How accurate can we solve SDPs?
Parallel
SDPA
SDPARA
SDPA-M
SDPARA-C
SDPA-C
SDPA-GMP
Matlab
Base solver
Multiple precision
Strucutural Sparsity
SDPA Homepage
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3. SDPA Online Solver http://sdpa.sf.net/ ⇒ Online Solver
Log-in the online solver
Upload your problem
Push ’Execute’ button
Receive the result via Web/Mail
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4. Outline SDP Applications Primal-Dual Interior-Point Methods
Inside of SDPARA (Large & Fast)
Inside of SDPA-GMP (Accurate)
Conclusion

5. SDP Applications Control Theory Quantum Chemistry
Sensor Network Localization Problem
Polynomial Optimization
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6. SDP Applications 1.Control theory
Against swing, we want to keep stability.
Stability Condition ⇒ Lyapnov Condition ⇒ SDP
INFOMRS Charlotte 6

7. SDP Applications 2. Quantum Chemistry
Ground state energy
Locate electrons
Schrodinger Equation ⇒Reduced Density Matrix ⇒SDP
INFOMRS Charlotte 7

8. SDP Applications 3. Sensor Network Localization
Distance Information ⇒Sensor Locations
Protein Structure
INFOMRS Charlotte 8

9. SDP Applications 4. Polynomial Optimization
For example,
NP-hard in general
Very good lower bound by SDP relaxation method
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10. How Large & How Fast & How Accurate
SDP Applications
Control Theory
Quantum Chemistry
Polynomial Optimization
Sensor Network Localization Problem
Many Applications 
How Large & How Fast & How Accurate
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11. Standard form Our target The variables are Inner Product is
The size is roughly determined by
Ordinal solver
Our target
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12. Primal-Dual Interior-Point Methods
Central Path
Target
Optimal
Feasible region
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13. Schur Complement Matrix
Schur Complement Equation
Schur Complement Matrix
where
1. ELEMENTS (Evaluation of SCM)
2. CHOLESKY (Cholesky factorization of SCM)
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14. Computation time on single processor
Time unit is second, SDPA 7, Xeon 5460 (3.16GHz)
Control
POP
ELEMENTS
22228
668
CHOLESKY
1593
1992
Total
23986
2713
Row-wise distribution
Two-dimensional block-cyclic distribution
SDPARA replaces these bottleneks by parallel computation
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15. Row-wise distribution
Example
All rows are independent
Assign processors in a cyclic manner
Simple idea ⇒Very EFFICIENT
High scalability
Processor1
Processor2
Processor3
Processor4
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16. Block Algorithm for Cholesky factorization
Triangular Factorization
(U: upper triangular matrix)
Small Cholesky factorizaton
Block Updates
Parallel Computing

17. Two-dimensional block-cyclic distribution
Example
Scalapack library
From the row-wise to TDBCD requires network communication
Cholesky on TDBCD is much faster than the on row-wise
Processor1
Processor2
Processor3
Processor4 1 2 3 4
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18. Numerical Results of SDPARA
Quantum Chemistry (m=7230, SCM=100%), middle size
SDPARA 7.3.1, Xeon X5460, 3.16GHz x2, 48GB memory
ELEMENTS 15x speedup
CHOLESKY 12x speedup
Total x speedup
Very FAST!!
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19. Acceleration by Multiple Threading
Modern Processors have multi-cores
Multiple Threading is becoming common
Processor1:Thread1
Processor2:Thread1
Processor1:Thread2
Processor2:Thread2
2 Processors
x2 Threads on each processor
Two-level Parallel Computing
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20. (Two-level parallization)
Comparison with PCSDP
developed by Ivanov & de Klerk
SDP: B.2P Quantum Chemistry (m = 7230, SCM = 100%) Xeon X5460, 3.16GHz x2 (8core), 48GB memory
Time unit is second
Servers 1 2 4 8 16
PCSDP
53,768
27,854
14,273
7995
4050
SDPARA
5983
3002
1680
901
565
SDPARA is 8x faster by MPI & Multi-Threading
(Two-level parallization)
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21. Extremely Large-Scale SDPs
Other solvers can handle only m
SCM
time
Esc32_b(QAP)
198,432
100%
129,186
second
(1.5days)
16 Servers [Xeon X5670(2.93GHz) , 128GB Memory]
The LARGEST solved SDP in the world
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22. Numerical Accuracy One weakpoint of PDIPM . PDIPM requires
Eventually, numerical trouble (often, Cholesky fails)
for example,
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23. c c Numerical Precision b b a a SDPA-GMP
Ordinal double precision in C or C++
arbitrary precision in GMP library b c a
64bit = 1bit(sign) + 11bit(exponent)+53bit(fraction);
accuracy = b c a
We can arbitrary set the bit number of fraction part.
(for example, 200bit = )
Replace BLAS(Basic Linear Algebra Sytems) by MPLAPACK (Multiple precision LAPACK)
SDPA-GMP

24. Numerically Hard problem
Test Problem
PDIPM is stable if Slater’s condition
Graph Partition Problem has no interior
Small ⇒ Numerically Hard
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25. Numerical Results of SDPA-GMP
Small ⇒ Numerically Hard
Solver
Accuracy
Time(second)
1.0e-1
SDPA
1.08e-8
2.03
SDPA-GMP
4.80e-48
1.0e-15
1.63e-7
2.26
2.97e-48
5.26e-9
2.36
7.29e-24
24digits for even no-interior case
SDPA-GMP uses 300 digits
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26. Conclusion SDPARA ⇒ How Fast & How Large 100times &
SDPA-GMP ⇒ How Accurate
& Online solver
Thank you very much for your attention.
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