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CUG 2009 May 7, 2009 * National Center for Computational Sciences + - - PowerPoint PPT Presentation

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Solution of Mixed-Integer Programming Problems on the XT5 Rebecca J. Hartman-Baker * Ingrid K. Busch + Michael R. Hilliard + Richard S. Middleton + Michael S. Schultze + Oak Ridge National Laboratory CUG 2009 May 7, 2009 * National Center for

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Rebecca J. Hartman-Baker* Ingrid K. Busch+ Michael R. Hilliard+ Richard S. Middleton+ Michael S. Schultze+ Oak Ridge National Laboratory

Solution of Mixed-Integer Programming Problems on the XT5

*National Center for Computational Sciences +Center for Transportation Analysis

CUG 2009

May 7, 2009

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Outline

  • Motivation
  • Problem Formulation

– Linear Programming – Mixed-Integer Linear Programming

  • Problem Solution

– Linear Programming – Mixed-Integer Linear Programming – Parallelization of MILP solvers

  • Existing MILP Solvers
  • Results
  • Future Work
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Motivation

  • Mixed-integer linear programs arise in many

applications

– Logistics – Supply-chain analysis – Data mining

  • Problems that restrict some solution

variables to integer values

– E.g., can’t place 0.6 biorefineries

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Linear Programming

  • Optimization problem in which objective

function and constraints are linear functions

  • f unknowns

min cT x subject to Ax ≤ b

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Mixed-Integer Linear Programming

  • Linear program in which some variables

restricted to integer values min cT x subject to Ax ≤ b, x j ∈ Z ∀j ∈ D

D = set of indices of integer variables in x

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Solving Linear Programs

  • Simplex method

– Iteratively examine all vertices of polytope – Worst-case performance (n variables, m constraints): – Pathological examples exist but average performance O(min{(m-n)2, n2})

n m      

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Solving Mixed-Integer Linear Programs (MILPs)

  • MILPs are NP-

complete – use heuristics

  • Solvers rely on

branching concept

  • Combine branching

with means to compute lower bounds, to develop branch and bound method

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Branch and Bound

  • Improve upper and lower bounds by

systematic bounding and branching of subproblems

  • Use relaxation of MILP to LP as bounding

function

  • Eliminate nonviable pieces of domain by

applying bounds

  • Systematically search domain

– Best-first search (smallest lower bounds first) – Depth-first search (deepest branch first)

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Branch and Cut

  • Lower bound

generated by LP relaxation often too loose for efficient solution

  • Improve bound by

adding valid inequalities to problem as computation proceeds

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Parallelization of MILP Solvers

  • Algorithm can be broken into 3 parts

– Ramp-Up – Search and Process – Ramp-Down

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Ramp-Up Phase

  • Break feasible region into enough pieces for

all processes to share

  • Uninvolved processors are idle
  • Can occupy uninvolved processors with

auxiliary tasks such as preprocessing and computation of upper bounds, but may not be worth time

  • Ideally, shorten ramp-up phase
  • No good ramp-up acceleration techniques

developed

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Search and Process Phase

  • Search and process branch nodes, and use

and share info (knowledge management)

  • Two approaches

– Centralized control: manager-worker paradigm, not scalable, but clearer global picture of problem – Decentralized control: more scalable, local hubs serve as local centralized control, may perform more work

  • Search strategy must be considered in terms
  • f global and local scope
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Ramp-Down Phase

  • Symmetric to ramp-up phase
  • Face similar issues
  • Generally ignored but can lead to serious

scalability issues

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Existing MILP Solvers: Commercial

  • CPLEX

– Leading commercial package – Shared-memory parallel version available – ParaLEX, distributed memory version, developed by researchers, and shows limited promise

  • Gurobi

– Just released (Spring 2009) – Parallelizes across multicore processors

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Existing MILP Solvers: Open Source

  • COIN-OR

– Repository of operations research-related open source software – Repository includes SYMPHONY and CHiPPS – CHiPPS contains BLIS, distributed parallel MILP solver

  • PICO

– Distributed parallel package – Capable of scaling to thousands of processors

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BLIS Parallelization Strategy

  • Decentralized approach
  • Global list of candidate nodes spread across

all processes

  • Every process selects nodes from local pool
  • Load balancing uses 3-level master/hub/

worker paradigm

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PICO Parallelization Strategy

  • Hybrid knowledge management scheme,

similar to BLIS

  • Idle processes given work from
  • verburdened processes by hub processes
  • Communication occurs between hub and its

workers and between hubs

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Results

  • Two sets of test problems run on Jaguar

Cray XT5 at ORNL

  • Standard Test Problems

– Problems of varying sizes that came with BLIS distribution – 100-7200 unknowns, 90-1600 constraints

  • Canadian Cities Problems

– Placement of facilities in Canadian cities – 26,000-2.4 million unknowns and constraints

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Results: Standard Test Problems

BLIS

!" !#" !##" !###" !####" !$%" &'" ($" !&" %" '" $"

!"##$%&'($)*(+,-.*/$ 01'2(3$,4$53,+(**,3*$

)*+#,-./0" 1233,-./0" 1*2405!-./0" 67-&#-$#-(-./0"

PICO

!"#$ #$ #!$ #!!$ #!!!$ #!!!!$ %#&$ &%'$ #&($ ')$ *&$ #'$ ($ )$ &$

!"##$%&'($)*(+,-.*/$ 01'2(3$,4$53,+(**,3*$

+,-!%"./0$ 1233%"./0$ 1,2405#"./0$ 4260&"./0$ 7,8"./0$ 0/9(,-"./0$ :/.&"./0$

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Results: Standard Test Problems

!" !#" !##" !###" !####" $!%" %$&" !%'" &(" )%" !&" '" (" %"

!"##$%&'($)*(+,-.*/$ 01'2(3$,4$53,+(**,3*$

*+,#$"-./012" *+,#$"-30452" 6+789:!"-./012" 6+789:!"-30452"

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Results: Canadian Cities

BLIS

!"#$ #$ #!$ #!!$ #!!!$ #!!!!$ #%&$ '($ )%$ #'$ &$ ($ %$ !"##$%&'($)*(+,-.*/$ 01'2(3$,4$53,+(**,3*$ *+,-./01$ *2+1,34$ 56-01/01$ 5/037809:$ ;,<7=,>$ ;,-7</01$ ?7/8@:1:+$ A,B,<$ A01601$ C7DD7DD,2E,$ F0+/@G40+9$ H//,I,$ J2:3:8$ K:E71,$ K78@-016$ L,D9,/001$ L8,+30+02E@$ L2++:4$ M0+01/0$ N,1802B:+$ O716D0+$ O7117.:E$ P0+9$

PICO

!"#$ #$ #!$ #!!$ #!!!$ #!!!!$ %&'$ #%($ ')$ *%$ #'$ ($ )$ %$

!"##$%&'($)*(+,-.*/$ 01'2(3$,4$53,+(**,3*$

+,-./012$ +3,2-45$ 60147819:$ ;-<7=->$ ;-.7<012$ ?708@:2:,$ A-B-<$ A12C12$ D00-E-$ F3:4:8$ G:H72-$ G78@.12C$ I-J9-0112$ I8-,41,13H@$ I3,,:5$ K1,1201$ L72CJ1,$ L7227/:H$ M1,9$

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Results: Analysis

  • Some scalability for standard test problems
  • No scaling for Canadian cities problems

– Majority of compute time spent in ramp-up phase – No room to scale!

  • Both PICO and BLIS had trouble solving some

problems

– Segfaults, OOM, hangs – Insufficiently robust for petascale

  • In principle, parallel MILP solvers show promise;

in practice, need improvement

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Future Work

  • More code development needed
  • One idea: use existing parallel framework to

parallelize MILP solvers

– MADNESS (Multiscale Adaptive Numerical Environment for Scientific Simulation) good candidate – Use MADNESS to distribute branches across processes, paired with well-established branching and bounding/cutting methods

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Acknowledgments

  • This research was sponsored by the

Laboratory Directed Research and Development Program of Oak Ridge National Laboratory, managed by UT-Battelle, LLC, for the U.S. Department of Energy.

  • This research used resources of the National

Center for Computational Sciences at Oak Ridge National Laboratory, which is supported by the Office of Science of the U.S. Department of Energy under Contract

  • No. DE-AC05-00OR22725.
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Resources

  • BLIS

Available for download from http://www.coin-or.org/

  • PICO

Jonathan Eckstein, Cynthia A. Phillips and William

  • E. Hart, “PICO: An object-oriented framework for

parallel branch-and-bound,” in Proc Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, Elsevier Scientific Series

  • n Studies in Computational Mathematics, pp. 219
  • - 265, 2001.
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Resources

  • Books and articles about MILP

Steven G. Nash and Ariela Sofer, Linear and Non- Linear Programming, New York: McGraw-Hill, 1996.

  • T. Ralphs, ``Parallel Branch and Cut,'' in Parallel

Combinatorial Optimization, Wiley, 2006.

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Questions?