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Optimization Issues for Huge Datasets and Long Computation Michael Ferris University of Wisconsin, Computer Sciences ferris@cs.wisc.edu Qun Chen, Jin-Ho Lim, Jeff Linderoth, Miron Livny, Todd Munson, Mary Vernon, Meta Voelker

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Update on Gamma Knife In use at U. Maryland Hospitals Covered by Business Week (Apr 2001) Better models, faster solution Requires less user input Skeletonization is key improvement

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Skeleton Starting Points 1020304050 10 20 30 40 50

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Run Time Comparison Average Run Time Size of Tumor SmallMediumLarge Random (Std. Dev) 2 min 33 sec (40 sec) 17 min 20 sec (3 min 48 sec) 373 min 2 sec (90 min 8 sec) SLSD (Std. Dev) 1 min 2 sec (17 sec) 15 min 57 sec (3 min 12 sec) 23 min 54 sec (4 min 54 sec)

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Data Mining & Optimization Prediction, Categorization, Separation Equations, LP, QP, MIP, NLP GAMS, Matlab, so/dll Serial, Parallel, Condor

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Optimization Global Exact Constrained Stochastic Large scale Fast convergence CPU + Memory + Smarts Local Approximate Unconstrained Deterministic Small scale Termination

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MIP formulation minimize c T x subject toAx b l x u and some x j integer Problems are specified by application convenient format - GAMS, AMPL, or MPS

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Data delivery: pay-per-view Optimization model for regional caches: minimize: C remote + P C regional over all possible cached objects/segments subject to: – C regional N channels regional storage N segments regional server stores 0, k or K segments of each object MIP (large number of objects/segments)

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Branch-and-Bound Algorithm 0 Top node Integer infeasible Integer feasible incumbent = Z LP relaxation Z lp > Z LP infeasible 2 1 x f 1x f 0 3 4 x g 0x g 1

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The “Seymour Problem” Set covering problem used in proof of four color theorem CPLEX 6.0 and Condor (2 option files) Running since June 23, 1999 Currently >590 days CPU time per job (13 million nodes; 2.4 million nodes)

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FAT COP FAT - large # of processors –opportunistic environment (Condor) COP - Master Worker control –fault tolerant: task exit, host suspend –portable parallel programming Mixed Integer Program Solver –Branch and Bound: LP relaxations –MPS file, AMPL or GAMS input

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GAMSAMPLMPS FATCOP MWCondor-PVM CPLEX OSL SOPLEX MINOS... Application Problem PVMInternet Protocol LPSOLVER INTERFACE

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MIP Technology Each task is a subtree, time limit –Diving heuristic –Cutting planes (global) –Pseudocosts –Preprocessing Master checkpoint Worker has state, how to share info?

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FATCOP Daily Log Note machine reboot at approx 3:00 am (night)

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Back to Seymour Schmieta, Pataki, Linderoth and MCF –explored to depth 8 in tree –applied cuts at each of these 256 nodes –solved in parallel, using whatever resources available (CPLEX, FATCOP,...) Problem solved with over 1 year CPU –over 10 million nodes, 11,000 hours

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Seymour Node 319 FATCOP – 47.0 hrs with 2,887,808 nodes –average number of machine used is 108 CPLEX –12 days, 10 hrs with 356,600 nodes –single machine, clique cuts useful

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Large datasets Enormous computational resources can sometimes facilitate solution X-validation, slice modeling What about the data? In particular, what if the problem does not fit in core?

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NCP functions Definition: Example: Componentwise definition:

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Implementation (n = 60M) All vectors stored out-of-core (480 MB per vector) –15% degradation –Footprint is 91 MB (constant in k) Asynchronous I/O (overlap computation and I/O) –8 hour wall clock time for 60M points –200,000 elements cached

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Semismooth results

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How can you use these? Specialized codes –Asynchronous I/O Specialized platforms –Condor (executable per architecture) Specific input formats –GAMS, Matlab Handholding operation

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Model centric toolbox GAMS optimization model Solvers LP,QP,MIP, NLP,MINLP Other model formats gms2xx Matlab programming environment Model data exchange Condor Resource Manager Data warehouse Specialized input

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