SDPA:Leading-edge Software for SDP

2008/10/14 @ Informs ’08

Tokyo Institute of Technology Makoto YamashitaMituhiro FukudaMasakazu KojimaKazuhide Nakata

Chuo University Katsuki Fujisawa

National Maritime Research Institute Kazuhiro Kobayashi

RIKEN Maho Nakata

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SDPA (SemiDefinite Programming Algorithm) Project

Open Source Software to solveSemiDefinite Programming

Since the 1st release in 1995, it has kept high quality

In 2008, the latest version SDPA 7 was released and has been updated continuously

Many software for more advantage

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SDPA Family

SDPA

SDPARA(Parallel with MPI)

SDPA-C(Matrix Completion)

SDPA-M(Matlab Interface)

SDPA-GMP(Multiple Precision)

SDPARA-C

accessible onSDPA Online Solveras Web service

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Outline of this talk

1. SDP and the improvements of SDPA72. Parallel with MPI3. Multiple Precision4. Online Solver5. Future Works

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Standard form of SDP

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Applications of SDP

Control Theory Lyapnov condition

Combinatorial Optimization Max Cut Theta function

Quantum Chemistry Reduced Density Matrix

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Primal-Dual Interior-Point Methods

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Computation for Search Direction

Schur complement matrix ⇒ Cholesky Factorizaiton

Exploitation of Sparsity in

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SDPA 7

SDPA7 resolves bottlenecks of SDPA6 Introduce sparse Cholesky factorization

for the Schur complement matrix Adopt new data structure Reduce memory space for temporary

variables Introduce configure script for easier

installation

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Sparsity pattern of Schur complement matrix

Fully dense Schur complement matrixFully dense Schur complement matrixSparse Schur complement matrixSparse Schur complement matrixminimum degree ordering minimum degree ordering to minimize to minimize the number of fill-in the number of fill-in

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New Data Structure For multiple diagonal

structure SDPA6

stores nonzero elements of each block

stores all blocks SDPA 7

stores nonzero elements of each block

stores only nonzero blocks

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r2S_broydenTri300.dat-s

SDPA 6 SDPA 7

Computing B 1009.0s 0.8s

Cholesky B 5179.5s 0.5s

Total CPU Time 6204.4s 2.8s

Input data 90MB 1MB

Matrix B 272MB 4MB

Dense Matrix 7MB 3MB

Total Memory 380MB 19MB

Sparse Schur

New Data Structure

Efficient Temporary

Xeon 2.80GHz, 2GB memory, Linux 2.4

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Configure script

Easier installation$./configure –with-blas=”-lblas”

–with-lapack=”-llapack”$ make$ make install

We can link with Optimized BLAS, i.e., ATLAS, GotoBLAS, Intel MKL

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Matlab Interface

SDPA-M is the Matlab interface [mDIM,nBLOCK,bLOCKsTRUCT,c,F] =

read_data(’example1.dat-s’); [objVal,x,X,Y,INFO] =

sdpam(mDIM,nBLOCK,bLOCKsTRUCT,c,F,OPTION);

SeDuMi Input interface [At,b,c,K] = fromsdpa(’example1.dat-s’); [x,y,info] = sedumiwrap(At,b,c,K,[],pars);

Current version is for only LP and SDP cones Parameter control is based on SDPA

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Extremely Large Problem and Bottlenecks

The largest size requires 8.6GB memory

We replace these two bottlenecks by parallel computation

Opteron 246 2.0GHz6GB Memory

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Exploitation of Sparsityin SDPA

We change the formula by row-wise

We keep this scheme on parallel computation

F1

F2

F3

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Row-wise distribution for evaluation of the Schur complement matrix

4 CPU is availableEach CPU computes only their assigned rows

. No communication between CPUsEfficient memory management

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Parallel Cholesky factorization We adopt Scalapack for the Cholesky

factorization of the Schur complement matrix We redistribute the matrix from row-wise to

two-dimensional block-cyclic-distribtuion

Redistribution

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Computation Time for NH3

93883235

1077460

9150

2973

796

201

88

239.4

3.71

10

100

1000

10000

1 4 16 64#processors

seco

nd TOTALComp BChol B

TSUBAME@Tokyo-TechOpteron 880 (2.4GHz)32GB memory/node

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Scalability for LiF

1

10

100

1 4 16 64#processors

scal

abili

ty TOTALComp BChol B

Total 28 times

Comp B 43 times

Chol B 46 times

Row-wise distribution for Comp B is very effective

TSUBAME@Tokyo-TechOpteron 880 (2.4GHz)32GB memory/node

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Multiple Precision SDPA uses ‘double’ precision

53 significant bit almost 8 digit

SDPA 7 result (gpp124-1 from SDPLIB) Objective Function (Only 5 digits)

-7.3430761748645921e+00(Primal)-7.3430800814821620e+00(Dual)

Feasibility5.45696821063e-12 (Primal)1.68252292320e-07 (Dual)

Some applications requires more accuracy

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SDPA-GMP

GMP: Gnu Multiple Precision Library Arbitrary fixed precision ‘double’ precision is replaced by GMP Ultra High Accuracy

by long computation time

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Ultra Accuracy of SDPA-GMP

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Comparison on SDPA and SDPA-GMP(384bit)

gpp124-1(SDPLIB)

SDPA-GMP(7.1.0)Relative gap 1.7163710368162993e-26Objective Function -7.3430762652465377e+00 (Primal) -7.3430762652465377e+00 (Dual)Feasibility 2.0710194844721e-57 (Primal) 1.2329417039702e-29 (Dual)Computation time　　　 228.95 sec. (59 iterations)

SDPA(7.1.0)Relative gap 5.3201361904260111e-07Objective Function -7.3430761748645921e+00 (Primal) -7.3430800814821620e+00 (Dual)Feasibility 5.45696821063e-12 (Primal) 1.68252292320e-07 (Dual)Computation time　　 0.14 sec. (20 iterations)

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SDPA Online Solver

SDPA Online Solver will offer SDPA/SDPARA/SDPARA-C via the Internet.

Internet

InterfaceUser1.Input 2.Ninf-G

3.SDPARA on PC cluster

4.Solution

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To use Online Solver Users without parallel environment can

use SDPARA/SDPARA-C. No Charge. Registration via the Internet is required

so that passwords to protect users data will be generated automatically.

Access SDPA Project Home Page.[SDPA Online for your future.]http://sdpa.indsys.chuo-u.ac.jp/sdpa/

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Online Solver Interface

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Online Solver Usage

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Conclusion The latest version 7 attains higher

performance than version 6 Parallel Solver enables us to solve

extremely large SDPs Matrix Completion is useful for Structural

Sparsity SDPA-GMP generates ultra high accuracy

solution Online Solver provides powerful

computation resources via the Internet

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Future works

Callable Library of SDPA7 Automatic Selection from

SDPA/SDPARA/SDPARA-C