Transcript Advances in Mixed Integer Programming

Branching
SCIP Workshop at ZIB
October 2007
Tobias Achterberg
Konrad-Zuse-Zentrum für Informationstechnik Berlin
[email protected]
2
Branching
 current solution is infeasible
3
Branching on Variables
 split problems into sub problems to cut off
current solution
4
Branching
 current solution is infeasible
5
Branching on Constraints
 split problems into subproblems to cut off
current solution
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Branching in SCIP
 in constraint handlers and branching rules
 „last resort“ for dealing with infeasible node
solutions
 no domain propagation or cuts available/desired
 split current problem into any number of
subproblems (children) such that
 each child is „more restricted“ than current problem
(„children become smaller“)
 at least one child has the same optimum value as the
current problem
(„optimal solution is not lost“)
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Implementing Branching Rules in SCIP
1. create child node
SCIPcreateChild(scip, &node, prio);
2. modify child node
SCIPaddConsNode(scip, node, cons, NULL);
SCIPchgVarLbNode(scip, node, var, newlb);
SCIPchgVarUbNode(scip, node, var, newub);
3. if more children needed, goto 1.
4. set result code
*result = SCIP_BRANCHED;
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Branching on Variables in SCIP
 Calling
SCIPbranchVar(scip, var, ...)
is shortcut for:
SCIP_NODE* node;
SCIP_Real x = SCIPvarGetLPSol(var);
SCIPcreateChild(scip, &node,
SCIPchgVarUbNode(scip, node,
SCIPcreateChild(scip, &node,
SCIPchgVarLbNode(scip, node,
downprio);
var, floor(x));
upprio);
var, ceil(x));
 node selection priorities are automatically calculated by
child selection rule
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Example: Random Branching
SCIP_DECL_BRANCHEXECLP(branchExeclpRandom)
{
SCIP_BRANCHRULEDATA* branchruledata;
SCIP_VAR** lpcands;
int nlpcands;
int k;
branchruledata = SCIPbranchruleGetData(branchrule);
SCIP_CALL(SCIPgetLPBranchCands(scip, &lpcands,
NULL, NULL, NULL, &nlpcands));
k = SCIPgetRandomInt(0, nlpcands-1,
&branchruledata->randseed);
SCIP_CALL(SCIPbranchVar(scip, lpcands[k],
NULL, NULL, NULL));
*result = SCIP_BRANCHED;
return SCIP_OKAY;
}
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Branching Rules for MIP
 most common MIP branching rules branch on
variables:
 two children
 split domain of single variable into two parts
 choose variable with fractional LP value such that LP
solution changes in both children
 remaining choices:
 which fractional variable to branch on?
 which of the two children to process next
 related to node selection strategy
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Branching Variable Selection
 most fractional branching
 choose variable with fractional value closest to 0.5
 full strong branching
 solve the LP relaxations for all possible branchings
 choose the variable that yields largest LP objectives
 strong branching
 only apply strong branching on some candidates
 only perform a limited number of simplex iterations
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Pseudo Costs
c=2
 LP relaxation yields lower bound
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Pseudo Costs
x3 = 7.3
c=2
 LP relaxation yields lower bound
 integer variable has fractional LP value
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Pseudo Costs
x3 ≤ 7
x3 = 7.3
c=2
x3  8
 LP relaxation yields lower bound
 integer variable has fractional LP value
 branching decomposes problem into subproblems
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Pseudo Costs
c=5
x3 ≤ 7
x3 = 7.3
c=2
x3  8
 LP relaxation yields lower bound
 integer variable has fractional LP value
 branching decomposes problem into subproblems
 LP relaxation is solved for subproblems
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Pseudo Costs
c=5
x3 ≤ 7
x3 = 7.3
c=2
x3  8
 history of objective changes caused by branching
on specific variable
 objective gain per unit:
52
3
 

 10
7.3  7 0.3

3
 down/upwards pseudo costs j-, j+:
average of all objective gains per unit
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Pseudo Cost Branching
 choose variable with largest estimated LP
objective gain:
j  f j  j , j  1  f j  j
 What to do if pseudo costs are uninitialized?
 pure pseudo cost branching
 use average pseudo costs over all variables, or
j  1
 pseudo cost with strong branching initialization
 apply strong branching to initialize pseudo costs
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Reliability Branching
 choose variable with largest estimated LP
objective gain:
j  f j  j , j  1  f j  j
 pseudo costs are unreliable, if number of
updates is small: min  j , j  rel
 apply strong branching on unreliable candidates
 psc with strong branching initialization:  rel  1
 (full) strong branching:
 reasonable value:
rel  
rel  8
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Branching in SAT
 „Strong Branching“ equivalent:
 apply domain propagation on all potential
subproblems
 choose variable which leads to largest number of
inferences
 Conflict Activity
 choose variable that is contained in many recently
generated conflict clauses
 „recently“: exponentially decreasing importance of
older conflict clauses
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Hybrid Reliability/Inference Branching
 Reliability Value
 pseudo costs
 strong branching on unreliable candidates
 Inference History
 like pseudo costs, but for number of inferences due to
branching on a variable
 Conflict Score
 number of conflicts for which branching on this
variable was part of the conflict reason
 exponentially decreasing weight for older conflicts
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Computational Results: nodes
1200
 244 instances
 shifted geometric nodes
 ratio to „hybrid“ in percent
100
317
127
198
0
35
200
25
400
528
600
643
800
109
1076
1000
random
most inf
least inf
full strong
strong
pseudo cost
psc str init
reliability
inference
hybrid
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Computational Results: time
441
450
400
250
319
300
326
350
 244 instances
 shifted geometric time
 ratio to „hybrid“ in percent
100
106
195
0
108
50
133
100
144
150
195
200
random
most inf
least inf
full strong
strong
pseudo cost
psc str init
reliability
inference
hybrid
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Branching Score Functions
 pseudo costs yield LP objective gain estimates
for both branching directions
 how to combine the two values into a single
score?
 current approach: weighted sum



s j   min j , j  1    max j , j
 new approach: product



s j  max j ,   max j , 


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Computational Results
180
175
160
60
40
20
0
137
100
80
129
100
114
120
129
140
min (1)
weighted (1/6)
weighted (1/3)
average (1/2)
max (0)
product
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Comparison to CPLEX and CBC
387.1
400
350
300
250
CPLEX
SCIP/CPX
SCIP/SPX
CBC
62.5
41.7
170.1
150.4
50
26.3
100
27.0
150
40.3
200
0
nodes (x100)
ratios to CPLEX 10.1
time (s)
nodes
time
SCIP/CPX
0.67
1.50
1.5x slower
SCIP/Soplex
0.65
3.61
3.6x slower
CBC/CLP
4.22
9.28
9.3x slower