Conflict Analysis in Mixed Integer Programming

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Conflict Analysisin Mixed Integer Programming
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General Idea

• Problem is divided into smaller subproblems
• branching tree
• Some subproblems are infeasible
• Analyzing the infeasibilities can yield
  information about the problem
• Information can be used at other subproblems to
  prune the search tree

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Example
x1 1
c1 x1 x2 x3 ? 0
c2 x1 x2 x3 ? 1
x2 0
x3 0

• a subproblem is infeasible

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Example
x1 1
c1 x1 x2 x3 ? 0
c2 x1 x2 x3 ? 1
x2 0
c3 x1 x3 ? 0
x3 0

• a subproblem is infeasible
• conflict analysis yields feasible constraint,
  that cuts off other nodes in the tree

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Outline

• Conflict Analysis in SAT
• Conflict Analysis in MIP
• Implementation
• Computational Results

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Outline

• Conflict Analysis in SAT
• Conflict Analysis in MIP
• Implementation
• Computational Results

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Satisfiability Problem (SAT)

• Boolean variables x1,...,xn ? 0,1
• Clauses
• Task
• find assignment x ? 0,1n that satisfies all
  clauses, or prove that no such assignment exists

C1 l11 ? ... ? l1k1
...
Cm ln1 ? ... ? lmkm
with literals lik xj or lik ?xj 1 xj
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Binary Constraint Propagation

• clause

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Binary Constraint Propagation

• clause
• fixings

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Binary Constraint Propagation

• clause
• fixings
• deduction

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Conflict Graph

• decision variables
• deduced variables
• conflict

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Conflict Graph
conflict detecting clause

• decision variables
• deduced variables
• conflict

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Conflict Graph Conflict Cuts

• choose cut that separates decision variables from
  conflict vertex

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Conflict Graph Conflict Cuts
reason side
conflict side

• choose cut that separates decision variables from
  conflict vertex

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Conflict Graph Conflict Cuts
reason side
conflict side

• choose cut that separates decision variables from
  conflict vertex
• conflict clause

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Conflict Graph Trivial Cuts

• ?-cut
• decision cut

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Conflict Graph First-UIP

• Unique Implication Point lies on all paths from
  the last decision vertex to the conflict vertex

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Conflict Graph First-UIP

• Unique Implication Point lies on all paths from
  the last decision vertex to the conflict vertex
• First UIP the UIP closest to ? (except ? itself)

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Conflict Graph First-UIP Cut

• Put all vertices fixed after First-UIP to the
  conflict side, remaining vertices to the reason
  side
• First-UIP cut

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Conflict Analysis Algorithm (FUIP)

• BCP detected a conflict

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Conflict Analysis Algorithm (FUIP)

• Initialize conflict queue with the variables
  involved in the conflict

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Conflict Analysis Algorithm (FUIP)

• Initialize conflict queue with the variables
  involved in the conflict

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Conflict Analysis Algorithm (FUIP)

• As long as there is more than one variable of the
  last depth level in the queue, resolve the last
  deduction

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Conflict Analysis Algorithm (FUIP)

• As long as there is more than one variable of the
  last depth level in the queue, resolve the last
  deduction

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Conflict Analysis Algorithm (FUIP)

• As long as there is more than one variable of the
  last depth level in the queue, resolve the last
  deduction

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Conflict Analysis Algorithm (FUIP)

• As long as there is more than one variable of the
  last depth level in the queue, resolve the last
  deduction

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Conflict Analysis Algorithm (FUIP)

• As long as there is more than one variable of the
  last depth level in the queue, resolve the last
  deduction

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Conflict Analysis Algorithm (FUIP)

• As long as there is more than one variable of the
  last depth level in the queue, resolve the last
  deduction

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Conflict Analysis Algorithm (FUIP)

• As long as there is more than one variable of the
  last depth level in the queue, resolve the last
  deduction

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Conflict Analysis Algorithm (FUIP)

• As long as there is more than one variable of the
  last depth level in the queue, resolve the last
  deduction

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Conflict Analysis Algorithm (FUIP)

• As long as there is more than one variable of the
  last depth level in the queue, resolve the last
  deduction

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Conflict Analysis Algorithm (FUIP)

• If there is only one variable of the last depth
  level left, stop

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Conflict Analysis Algorithm (FUIP)

• If there is only one variable of the last depth
  level left, stop
• The remaining variables define the conflict set

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Conflict Analysis Algorithm (FUIP)

• The conflict clause consists of all (negated)
  assignments in the conflict set

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Outline

• Conflict Analysis in SAT
• deductions lead to conflict ? conflict graph
• cut in conflict graph ? conflict clause
• Conflict Analysis in MIP
• Implementation
• Computational Results

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Outline

• Conflict Analysis in SAT
• deductions lead to conflict ? conflict graph
• cut in conflict graph ? conflict clause
• Conflict Analysis in MIP
• Implementation
• Computational Results

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Mixed Integer Program
max cTx
s.t. Ax ? b
x ? Rp?Zq

• linear objective function c
• linear constraints Ax ? b
• real or integer valued variables x

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Conflict Analysis for MIP

• Two main differences to SAT
• non-binary variables
• conflict graph bound changes instead of fixings
• conflict clause ? conflict constraint

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Conflict Analysis for MIP

• Two main differences to SAT
• non-binary variables
• conflict graph bound changes instead of fixings
• conflict clause ? conflict constraint

technical issue
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Conflict Analysis for MIP

• Two main differences to SAT
• non-binary variables
• conflict graph bound changes instead of fixings
• conflict clause ? conflict constraint
• main reason for infeasibility LP relaxation
• conflict graph has no link from the decision and
  deduction vertices to the conflict vertex

technical issue
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Conflict Analysis for MIP

• Two main differences to SAT
• non-binary variables
• conflict graph bound changes instead of fixings
• conflict clause ? conflict constraint
• main reason for infeasibility LP relaxation
• conflict graph has no link from the decision and
  deduction vertices to the conflict vertex

technical issue
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Back to SAT Conflict Graph
conflict detecting clause

• One clause detected theconflict
• Only a few variables linked to the conflict vertex

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Infeasible LP Conflict Graph

• The LP as a whole is responsible for the conflict
• All local bound changes are linked to the
  conflict vertex

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Infeasible LP Conflict Graph

• LP analysis selects some of these local bounds

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Infeasible LP Conflict Graph

• LP analysis selects some of these bounds
• cut yields conflict constraint

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Conflict Analysis for infeasible LPs

• if the LP relaxation is infeasible, the whole
  relaxation is involved in the conflict
• all constraints
• all global bounds
• all local bounds
• try to find a small subset of the local bounds
  that still leads to an infeasible LP relaxation
• variant of minimal infeasible subsystem
  problem(see Amaldi, Pfetsch, Trotter)
• heuristic use dual ray to relax local bounds

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Infeasible LP Dual Ray Heuristic

• LP relaxation

max cTx
s.t. Ax ? b
0 ? x ? ? ? u
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Infeasible LP Dual Ray Heuristic

• LP relaxation
• dual LP

max cTx
s.t. Ax ? b
0 ? x ? ? ? u
local bounds
min bTy ?Tr
s.t. ATy r ? c
y, r ? 0
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Infeasible LP Dual Ray Heuristic

• LP relaxation
• dual LP
• dual ray

max cTx
s.t. Ax ? b
0 ? x ? ? ? u
local bounds
min bTy ?Tr
s.t. ATy r ? c
y, r ? 0
(y, r) ? 0, ATy r ? 0, bTy ?Tr lt 0
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Infeasible LP Dual Ray Heuristic

• LP relaxation
• dual LP
• dual ray

max cTx
s.t. Ax ? b
0 ? x ? ? ? u
local bounds
min bTy ?Tr
s.t. ATy r ? c
y, r ? 0
(y, r) ? 0, ATy r ? 0, bTy ?Tr lt 0
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Infeasible LP Dual Ray Heuristic

• LP relaxation
• dual LP
• dual ray
• relax bounds

max cTx
s.t. Ax ? b
0 ? x ? ? ? u
local bounds
min bTy ?Tr
s.t. ATy r ? c
y, r ? 0
(y, r) ? 0, ATy r ? 0, bTy ?Tr lt 0
?i ui , s.t. still bTy ?Tr lt 0
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Outline

• Conflict Analysis in SAT
• deductions lead to conflict ? conflict graph
• cut in conflict graph ? conflict clause
• Conflict Analysis in MIP
• deductions lead to infeasible LP
• LP analysis links conflict vertex
• cut in conflict graph ? conflict constraint
• Implementation
• Computational Results

conflict graph
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Outline

• Conflict Analysis in SAT
• deductions lead to conflict ? conflict graph
• cut in conflict graph ? conflict clause
• Conflict Analysis in MIP
• deductions lead to infeasible LP
• LP analysis links conflict vertex
• cut in conflict graph ? conflict constraint
• Implementation
• Computational Results

conflict graph
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Implementation

• Which pruned subproblems should be analyzed?
• propagation conflicts yes
• infeasible LP conflicts yes
• bound-exceeding LP conflicts no
• strong branching conflicts no
• How many conflict constraints per conflict?
• all FUIP constraints 1-FUIP, 2-FUIP, ...
• How should conflict constraints be used?
• only for propagation, not as cutting planes
• How can useless conflict constraints be
  identified?
• "easy" for SAT due to depth-first-search
• open topic for best-first-search MIP, needs more
  ideas!
• currently aging mechanism

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Outline

• Conflict Analysis in SAT
• deductions lead to conflict ? conflict graph
• cut in conflict graph ? conflict clause
• Conflict Analysis in MIP
• deductions lead to infeasible LP
• LP analysis links conflict vertex
• cut in conflict graph ? conflict constraint
• Implementation
• Computational Results

conflict graph
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Outline

• Conflict Analysis in SAT
• deductions lead to conflict ? conflict graph
• cut in conflict graph ? conflict clause
• Conflict Analysis in MIP
• deductions lead to infeasible LP
• LP analysis links conflict vertex
• cut in conflict graph ? conflict constraint
• Implementation
• Computational Results

conflict graph
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Computational Results

• Instances of MIPLIB 2003 and Mittelmann
• that could be solved by one of the three in 1
  hour
• for which one of the three needed at least 1000
  nodes

47 instances CPLEX 10.0 SCIP SCIP ConfA
Fails 5 3 3
Nodes 13 429 18 192 14 507
Node Winner (24) 14 (7) 27 (17)
Time 65.1 134.7 116.5
Time Winner (33) 16 (4) 25 (11)
nodes -20 time -14
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Computational Results

• ALU instances up to 8 bits
• for which one of the three needed at least 1000
  nodes
• infeasible MIP instances

21 instances CPLEX 10.0 SCIP SCIP ConfA
Fails 0 1 1
Nodes 35 657 6 272 910
Node Winner (7) 0 (3) 16 (14)
Time 23.4 29.9 12.7
Time Winner (12) 1 (4) 16 (9)
nodes -85 time -58
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Computational Results

• "Enlight" instances up to board size 10 ? 10
• for which one of the three needed at least 1000
  nodes
• 2 infeasible and 4 feasible MIP instances

6 instances CPLEX 10.0 SCIP SCIP ConfA
Fails 1 0 0
Nodes 50 176 70 216 15 774
Node Winner 0 0 6
Time 22.2 32.3 13.1
Time Winner (4) 0 (0) 5 (2)
nodes -78 time -59
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Computational Results

• instance "arki001" from MIPLIB 3.0 / MIPLIB 2003
• first solved by Balas and Saxena (Dec 2005)
• "Optimizing over the Split Closure"
• 54 hours rank-1 split cut generation 11 hours
  CPLEX 9.0
• Bob-tuned CPLEX
• "using a lot of strong branching"
• 628 hours, 103 million nodes
• SCIP
• using an insane amount of strong branching

• full conflict analysis (bound-exceeding LPs and
  strong branchings)

SCIP ConfA
1 857 512
5.2 hours
SCIP
Nodes 5 054 427
Time 11.4 hours
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Computational Results
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Computational Results
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Computational Results

• instance roll3000" from MIPLIB 2003
• SCIP
• using an insane amount of strong branching
• moderate conflict analysis
• 3.2GHz Pentium-IV

SCIP SCIP ConfA
Nodes 9 104 084 4 139 860
Time 20.0 hours 10.3 hours

• Unfortunately...
• XPressMP 2006B with tuned settings 1 hour (3 GHz
  Athlon-64)
• Reported yesterday by Alkis Vazacopoulos
• As far as I understood lots of local cuts