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Efficient Parallel Software for Large-Scale Semidefinite Programs

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##### Efficient Parallel Software for Large-Scale Semidefinite Programs

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**Efficient Parallel Software for Large-Scale Semidefinite**Programs Makoto Yamashita @ Tokyo-Tech Katsuki Fujisawa @ Chuo University MSC 2010 @ Yokohama [2010/09/08]**Outline**• SemiDefinite Programming • Conversion of stability condition for differential inclusions to an SDP • Primal-Dual Interior-Point Methods and its parallel implementation • Numerical Results**Many Applications of SDP**• Control Theory • Stability Condition for Differential Inclusions • Discrete-Time Optimal Control Problem • Via SDP relaxation • Polynomial Optimization Problem • Sensor Network Problem • Quadratic Assignment Problem • Quantum Chemistry/Information • Large SDP ⇒ Parallel Solver**Stability condition for differential inclusions to standard**SDP Boyd et al • . • Does the solution remain in a bounded region? • i.e., • Yes, if**Conversion to SDP**• . • To hold this inequality, • Bounding the condition number⇒SDP.**SDP from SCDI**• . • Feasible solution ⇒ Boundness of the solution • Some translation for standard SDPby e.g. YALMIP [J. Löfberg].**Discrete-Time Optimal Control Problems**• This Problem [Coleman et al] can be formulated as SDP via SparsePOP [Kim et al].**Primal-Dual Interior-Point Methods**• Both Primal and Dual simultaneously in Polynomial-time • Many software are developed • SDPA [Yamashita et al] • SDPT3 [Toh et al] • SeDuMi [Sturm et al] • CSDP [Borcher et al]**Algorithmic Framework of Primal-Dual Interior-Point Methods**Feasible Region of Central Path Initial Point Step Length to keep interior property Target Point Search Direction The most computational time is consumed by the Search Direction Optimal Solution**Bottlenecks in PDIPMand SDPARA**• To obtain the direction, we solve • ELEMENTS • CHOLESKY • In SDPARA, parallel computation is applied to these two bottlenecks Xeon 5460,3.16GHz**Nonzero pattern ofSchur complement matrix (B)**• Sparse Schur complement matrix • Fully dense Schur complement matrix DTOC SCDI**Exploitation of Sparsityin SDPA**• We change the formula by row-wise • We keep this scheme on parallel computation F1 F2 F3**Row-wise distribution for dense Schur complement matrix**• 4 CPU is available • Each CPU computes only their assigned rows • . • No communication between CPUs • Efficient memory management**Parallel Computation for CHOLESKY**• We employ • ScaLAPACK [Blackford et.al] ⇒ Dense • MUMPS [Amestoy et.al] ⇒ Sparse • Different data storage enhance the parallel Cholesky factorization**Problems for Numerical Results**• 16 nodes • Xeon X5460 (3.16GHz) • 48GB memory**Total 15.02 times**ELEMENTS 15.67 times CHOLESKY 14.20 times Computation time on SDP [SCDI1] Xeon X5460(3.16GHz) 48GB memory/node ELEMENTS attains high scalability**Total 4.85 times**ELEMENTS 13.50 times CHOLESKY 4.34 times Computation time on SDP [DTOC1] Xeon X5460(3.16GHz) 48GB memory/node • Parallel Sparse Cholesky is difficult • ELEMENTS is still enhanced**Comparison with PCSDP [Ivanov et al]**• SDPARA is faster than PCSDP • The scalability of SDPARA is higher • Only SDPARA can solve DTOC Time is second, O.M.:out of memory**Concluding Remarks & Future works**• SDP has many applications including control theory • SDPARA solves Larse-scale SDPs effectively by parallel computation • Appropriate parallel computations are the key of SDPARA implementation • Improvement on Multi-Threading for sparse Schur complement matrix