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Background Exact Branch-and-Bound Conclusions and Future Work
Background Exact Branch-and-Bound Conclusions and Future Work
Exact solutions to mixed-integer linear programming problems Dan - - PowerPoint PPT Presentation
Exact solutions to mixed-integer linear programming problems Dan - - PowerPoint PPT Presentation
Background Exact Branch-and-Bound Conclusions and Future Work Exact solutions to mixed-integer linear programming problems Dan Steffy Zuse Institute Berlin and Oakland University Joint work with Bill Cook, Thorsten Koch and Kati Wolter
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Background Exact Branch-and-Bound Conclusions and Future Work
Floating-point Computation for MIP
What can go wrong? Suboptimal solution returned Infeasible solution returned Feasible problem declared infeasible
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Background Exact Branch-and-Bound Conclusions and Future Work
Deviation in Integer Programming Software Results
Default Settings
Presolve, cuts and heur. disabled
Example SCIP CPLEX SCIP CPLEX aim-200-1 6-yes1-3 200 200 200 177 alu10 7 91 86 84 83 alu8 9 2.341e5 91 2.341e5 90 neos-585467 399.373 399.373 397.363 392.805 neos-619167 [1.643, 2.611] 2.141 2.141 1.664 ns1637403 46 46 46 14 ns2122603 [94, 7e5] infeasible [3.3,∞] [1.7,∞]
- pti 157 0
8.593e3 infeas./unbd. infeasible infeasible
- pti 24 1
8.210e2 infeas./unbd. infeasible infeasible ran14x18.disj-8 [3595, 3714] 3761 [3574, 3735] [3618, 3797] transportmoment infeasible infeas./unbd. unbounded [-5e10, -3e9] Results of FP based solvers (2 hour time limit) Numerically difficult instances (from MIPLIB 2010 and others)
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Background Exact Branch-and-Bound Conclusions and Future Work
Deviation in Integer Programming Software Results
Default Settings
Presolve, cuts and heur. disabled
Example SCIP CPLEX SCIP CPLEX aim-200-1 6-yes1-3 200 200 200 177 alu10 7 91 86 84 83 alu8 9 2.341e5 91 2.341e5 90 neos-585467 399.373 399.373 397.363 392.805 neos-619167 [1.643, 2.611] 2.141 2.141 1.664 ns1637403 46 46 46 14 ns2122603 [94, 7e5] infeasible [3.3,∞] [1.7,∞]
- pti 157 0
8.593e3 infeas./unbd. infeasible infeasible
- pti 24 1
8.210e2 infeas./unbd. infeasible infeasible ran14x18.disj-8 [3595, 3714] 3761 [3574, 3735] [3618, 3797] transportmoment infeasible infeas./unbd. unbounded [-5e10, -3e9] Results of FP based solvers (2 hour time limit) Numerically difficult instances (from MIPLIB 2010 and others)
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Background Exact Branch-and-Bound Conclusions and Future Work
Necessity of Exact Solutions
Where are exact MIP results needed? VLSI chip design verification Combinatorial auctions Numerically difficult problems Verifying results of test libraries
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Background Exact Branch-and-Bound Conclusions and Future Work
Rational Arithmetic for Linear Programming
http://gmplib.org
Applegate, Cook, Dash and Espinoza [2007] tested rational simplex implementation. It was hundreds or thousands of times slower than floating-point code.
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Background Exact Branch-and-Bound Conclusions and Future Work
Rational Arithmetic for Linear Programming
http://gmplib.org
Applegate, Cook, Dash and Espinoza [2007] tested rational simplex implementation. It was hundreds or thousands of times slower than floating-point code. → try hybrid symbolic-numeric computation
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Background Exact Branch-and-Bound Conclusions and Future Work
Exact Linear Programming
QSopt ex: Exact Rational LP Solver1 Simplex method performed limited/fixed precision Final basic solution checked exactly Precision increased if needed Only 2-5x slower than FP-codes
1 Developed by Applegate, Cook, Dash and Espinoza [2007]
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Background Exact Branch-and-Bound Conclusions and Future Work
Branch-and-Bound Procedure
max
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Background Exact Branch-and-Bound Conclusions and Future Work
Branch-and-Bound Procedure
f
max
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Background Exact Branch-and-Bound Conclusions and Future Work
Branch-and-Bound Procedure
f
max
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Background Exact Branch-and-Bound Conclusions and Future Work
Branch-and-Bound Procedure
f
max
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Background Exact Branch-and-Bound Conclusions and Future Work
Branch-and-Bound Procedure
f
max
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Background Exact Branch-and-Bound Conclusions and Future Work
Branch-and-Bound Procedure
x
max
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Background Exact Branch-and-Bound Conclusions and Future Work
Branch-and-Bound Procedure
x
max
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Background Exact Branch-and-Bound Conclusions and Future Work
Branch-and-Bound Procedure
x
max
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Background Exact Branch-and-Bound Conclusions and Future Work
Exact Mixed Integer Programming
What happens if we use a fast exact LP solver at every node?
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Background Exact Branch-and-Bound Conclusions and Future Work
Exact Mixed Integer Programming
What happens if we use a fast exact LP solver at every node? Applegate, Dash, Cook, Espinoza [2007] found this considerably slower than FP code. Warm starting the LPs in branch-and-bound means lots of LPs with less pivots per LP.
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Background Exact Branch-and-Bound Conclusions and Future Work
Exact Mixed Integer Programming
Hybrid Approach for Exact MIP: Branch & Bound
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Background Exact Branch-and-Bound Conclusions and Future Work
Exact Mixed Integer Programming
Hybrid Approach for Exact MIP: Branch & Bound1 Store exact representation of problem Perform many operations on approximation or relaxation Exact or safe methods must be used for:
Computing feasible solutions Computing LP bounds
1 see Cook, Koch, Steffy and Wolter [2011]
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Background Exact Branch-and-Bound Conclusions and Future Work
Exact Mixed Integer Programming
Hybrid Approach for Exact MIP: Branch & Bound1 Store exact representation of problem Perform many operations on approximation or relaxation Exact or safe methods must be used for:
Computing feasible solutions ← use exact LP solver Computing LP bounds ← many choices here
1 see Cook, Koch, Steffy and Wolter [2011]
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Background Exact Branch-and-Bound Conclusions and Future Work
Valid Dual Bounds: Exact LP and Basis Verification
Exact LP Solve each node LP exactly using QSopt ex Produces tightest possible bound
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Background Exact Branch-and-Bound Conclusions and Future Work
Valid Dual Bounds: Exact LP and Basis Verification
Exact LP Solve each node LP exactly using QSopt ex Produces tightest possible bound Basis Verification Recompute basic sol. from floating-point LP solver exactly If dual feasible → return valid dual bound Otherwise → return infinite bound, branch (Hopefully branching can resolve numerical troubles)
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Background Exact Branch-and-Bound Conclusions and Future Work
Valid Dual Bounds: Primal-Bound-Shift
Linear Program Primal: max cT x s.t. Ax ≤ b l ≤ x ≤ u Dual: min bT y − lT zl + uT zu s.t. AT y − Izl + Izu = c y, zl, zu ≥ 0 Idea: Use dual variables from primal bounds to correct approximate dual solution Applegate et al. [2006], Neumaier and Shcherbina [2004]
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Background Exact Branch-and-Bound Conclusions and Future Work
Valid Dual Bounds: Primal-Bound-Shift
Linear Program Primal: max cT x s.t. Ax ≤ b l ≤ x ≤ u Dual: min bT y − lT zl + uT zu s.t. AT y − Izl + Izu = c y, zl, zu ≥ 0 Rigorous objective bound: Let ˆ y, ˆ zl, ˆ zu ≥ 0 be an approximate dual solution Set r = c − AT ˆ y + ˆ zl − ˆ zu (y, zl, zu) = (ˆ y, ˆ zl + r+, ˆ zu + r−) is a valid dual solution
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Background Exact Branch-and-Bound Conclusions and Future Work
Valid Dual Bounds: Primal-Bound-Shift
Linear Program Primal: max cT x s.t. Ax ≤ b l ≤ x ≤ u Dual: min bT y − lT zl + uT zu s.t. AT y − Izl + Izu = c y, zl, zu ≥ 0 Good: Bound is trivial to compute Floating-point computation with directed rounding can be used Bad: Strength of the bound depends on tightness of primal variable bounds
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Background Exact Branch-and-Bound Conclusions and Future Work
Valid Dual Bounds: Project-and-Shift
Linear Program Primal: max cT x s.t. Ax ≤ b Dual: min bT y s.t. AT y = c y ≥ 0 A more general procedure:
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Background Exact Branch-and-Bound Conclusions and Future Work
Valid Dual Bounds: Project-and-Shift
Linear Program Primal: max cT x s.t. Ax ≤ b Dual: min bT y s.t. AT y = c y ≥ 0 A more general procedure: Main Idea: Find approximate dual solution ˜ y Project ˜ y to satisfy AT y = c Maintaining feasibility, shift to satisfy y ≥ 0
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Background Exact Branch-and-Bound Conclusions and Future Work
Valid Dual Bounds: Project-and-Shift
Linear Program Primal: max cT x s.t. Ax ≤ b Dual: min bT y s.t. AT y = c y ≥ 0 Comments: Under some assumptions, most expensive computations can be done only once at root node and reused through the tree More general than Primal-Bound-Shift (but not entirely general) Some exact computation still required
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Background Exact Branch-and-Bound Conclusions and Future Work
Problem Structure in Tree
Root Primal: max cT x s.t. Ax ≤ b Root Dual: min bT y s.t. ATy = c y ≥ 0 Node Primal: max cT x s.t. Ax ≤ b ¯ Ax ≤ ¯ b Node Dual: min bT y + ¯ bT z s.t. ATy + ¯ ATz = c y, z ≥ 0
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Background Exact Branch-and-Bound Conclusions and Future Work
Valid Dual Bounds: Project-and-Shift
Components of Project-and-Shift for MIP Root Node: Setup Phase Compute exact LU factorization of AT Determine (exact) corrector point y∗ AT y∗ = c, y∗ > 0
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Background Exact Branch-and-Bound Conclusions and Future Work
Valid Dual Bounds: Project-and-Shift
Components of Project-and-Shift for MIP Root Node: Setup Phase Compute exact LU factorization of AT Determine (exact) corrector point y∗ AT y∗ = c, y∗ > 0 Node Bound Computation Start with approximate solution Project into equality space (using LU = AT ) Take convex combination with y∗ to ensure feasibility
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Background Exact Branch-and-Bound Conclusions and Future Work
Implementation
Implementation of Hybrid Branch-and-Bound Method Branch and bound framework: SCIP 1.2.0.8 Exact LP solver: QSopt ex 2.5.5 Floating-point LP solver: CPLEX 12.10 Multiple precision arithmetic package: GMP
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Background Exact Branch-and-Bound Conclusions and Future Work
Computational Tests
Dual Bounding Methods
Exact LP Basis Verification Primal-Bound-Shift Project-and-Shift Combination of above methods (automatic selection at each node)
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Background Exact Branch-and-Bound Conclusions and Future Work
Computational Tests
Dual Bounding Methods
Exact LP Basis Verification Primal-Bound-Shift Project-and-Shift Combination of above methods (automatic selection at each node)
Pure Branch-and-Bound First Fractional Branching Results reported on 57 test problems selected from the MIPLIB and Mittelmann test libraries that were solved by (floating-point) SCIP branch-and-bound within 1 hour
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Background Exact Branch-and-Bound Conclusions and Future Work
Overall Computation Time
# solved in 24h. Nodes∗ Time∗ (s.) Inexact 57 18,030 59.4 Exact LP 51 18,076 550.7 VerifyBasis 51 18,078 461.8 BoundShift 43 24,994 110.4 ProjectShift 49 18,206 369.3 Automatic 55 18,226 91.4
∗Nodes and time reported are the geometric means over 37/57 problems
solved by all methods within 24h. time limit
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Background Exact Branch-and-Bound Conclusions and Future Work
Future Plans
Some Future Plans: Cutting planes and pre-solve Certification of results Solve test library problems
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Background Exact Branch-and-Bound Conclusions and Future Work
East Coast Computer Algebra Day 2012
ECCAD 2012 Where: Oakland University, Rochester, Michigan When: Saturday May 12, 2012 ... more details to come
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Background Exact Branch-and-Bound Conclusions and Future Work