INFOMRS Charlotte1 Parallel Computation for SDPs Focusing on the Sparsity of Schur Complements Matrices Makoto Tokyo Tech Katsuki Chuo Univ Mituhiro Tokyo Tech Kazuhide Tokyo Tech Maho RIKEN INFORMS Annual Charlotte 2011/11/15 (2011/11/ /11/16)

INFOMRS Charlotte2 Key phrase  SDPARA: The fastest solver for large SDPs available at SemiDefinite Programming Algorithm paRAllel veresion

INFOMRS Charlotte3 SDPA Online Solver 1.Log-in the online solver 2.Upload your problem 3.Push ’ Execute ’ button 4.Receive the result via Web/Mail ⇒ Online Solver

INFOMRS Charlotte4 Outline 1.SDP applications 2.Standard form and Primal-Dual Interior-Point Methods 3.Inside of SDPARA 4.Numerical Results 5.Conclusion

INFOMRS Charlotte5 SDP Applications 1.Control theory  Against swing, we want to keep stability.  Stability Condition ⇒ Lyapnov Condition ⇒ SDP

INFOMRS Charlotte6  Ground state energy  Locate electrons  Schrodinger Equation ⇒ Reduced Density Matrix ⇒ SDP SDP Applications 2. Quantum Chemistry

INFOMRS Charlotte7 SDP Applications 3. Sensor Network Localization  Distance Information ⇒ Sensor Locations  Protein Structure

INFOMRS Charlotte8 Standard form  The variables are  Inner Product is  The size is roughly determined by Our target

INFOMRS Charlotte9 Primal-Dual Interior-Point Methods Feasible region Optimal Central Path Target

INFOMRS Charlotte10 Schur Complement Matrix where Schur Complement Equation Schur Complement Matrix 1. ELEMENTS (Evaluation of SCM) 2. CHOLESKY (Cholesky factorization of SCM)

INFOMRS Charlotte11 Computation time on single processor  SDPARA replaces these bottleneks by parallel computation ControlPOP ELEMENTS CHOLESKY Total Time unit is second, SDPA 7, Xeon 5460 (3.16GHz)

INFOMRS Charlotte12 Dense & Sparse SCM SDPARA can select Dense or Sparse automatically. Fully dense SCM (100%) Quantum Chemistry Sparse SCM (9.26%) POP

INFOMRS Charlotte13 Different Approaches DenseSparse ELEMENTSRow-wise distribution Formula-cost-based distribution CHOLESKYParallel dense Cholesky (Scalapack) Parallel sparse Cholesky (MUMPS)

INFOMRS Charlotte14 Three formulas for ELEMENTS densesparse All rows are independent.

INFOMRS Charlotte15 Row-wise distribution  Assign servers in a cyclic manner  Simple idea ⇒ Very EFFICINENT  High scalability Server1 Server2 Server3 Server2 Server3 Server4 Server1 Server4

INFOMRS Charlotte16 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 Charlotte17 Drawback of Row-wise to Sparse SCM densesparse  Simple row-wise is ineffective for sparse SCM  We estimate cost of each element

INFOMRS Charlotte18 Formula-cost-based distribution Server1190 Server2185 Server3188 Good load-balance

INFOMRS Charlotte19 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 speedup CHOLESKY 4.7xspeedup Total 5x speedup

INFOMRS Charlotte20 Comparison with PCSDP by SDP with Dense SCM  developed by Ivanov & de Klerk Servers PCSDP SDPARA 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 Charlotte21 Comparison with PCSDP by SDP with Sparse SCM  SDPARA handles SCM as sparse  Only SDPARA can solve this size #sensors 1,000 (m=16450; density=1.23%) #Servers PCSDPO.M SDPARA #sensors 35,000 (m=527096; density=6.53 × 10−3%) #Servers PCSDPOut of Memory SDPARA

INFOMRS Charlotte22 Extremely Large-Scale SDPs  16 Servers [Xeon X5670(2.93GHz), 128GB Memory] mSCMtime Esc32_b(QAP)198,432100%129,186 second (1.5days) Other solvers can handle only The LARGEST solved SDP in the world

INFOMRS Charlotte23 Conclusion  Row-wise & Formula-cost-based distribution  parallel Cholesky factorization  SDPARA: The fastest solver for large SDPs  & Online solver Thank you very much for your attention.