Download The AIMMS Presolver

The AIMMS Presolver
Marcel Hunting
AIMMS Optimization Specialist
Webinar, July 15, 2015
Overview
> Introduction
> Preprocessing techniques
• Basic
• Advanced
> Activating the AIMMS presolver
> Analyse infeasible problems
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Why Preprocessing?
> Speed up the solving process
> Get better solutions
> Improve numerical stability
> Detect infeasibilities
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Why should AIMMS do Preprocessing?
> Not needed for linear problems because the linear solvers (CPLEX,
Gurobi, CBC) do very advanced preprocessing
> But useful for nonlinear problems because many nonlinear solvers
do no or very basic preprocessing
• Exceptions: KNITRO, BARON
> Optimization algorithms implemented in AIMMS can benefit from it
• Outer Approximation (MINLP)
• Multi-Start (NLP)
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Basic Preprocessing Techniques
> Delete simple constraints
• Constraint x ≤ 100 becomes a bound
> Delete fixed variables
> Remove doubletons
• Constraint x = 2 y
> Delete duplicate constraints
> Delete redundant constraints
• Constraint sqrt(x) + y ≤ 20 can be removed if x ≤ 100 and y ≤ 5
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Advanced Preprocessing Techniques
> Bounds tightening (feasibility based)
• Variable x: range [0,inf)
range [10,55]
• Linear & nonlinear constraints
> Linearize quadratic constraints
• Multiplication of a binary variable with a bounded variable
> Improve coefficients
• For models with binary variables
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Feasibility Based Bound Tightening
Constraint:
L ≤ x + f(y1,…,yn) ≤ U
Assume for some lf and uf:
lf ≤ f(y1,…,yn) ≤ uf
Then:
x ≥ L - uf
x ≤ U - lf
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FBBT: Example
x: range [10,100]
y: range [5,8]
z: range [1,2]
Constraint: x + 4 y - 2 z3 = 60
x = 60 – 4 y + 2 z3 ≤ 60 – 20 + 16 = 56
x = 60 – 4 y + 2 z3 ≥ 60 – 32 + 12 = 30
x: range [30,56]
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FBBT: Expression Tree
𝑒𝑧
tighten
bounds
variables
3
∈ [0,8]
𝑒 𝑧 ∈ [0,2]
𝑧 ∈ (−inf,ln(2)]
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FBBT: Expression Tree
𝑒𝑧
3
∈ [8,64]
𝑒 𝑧 ∈ [2,4]
tighten
bounds
expression
𝑧 ∈ [ln(2),ln(4)]
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FBBT: Binary Operator
𝑥𝑦 + 𝑒 𝑧
3
∈ [2,6]
𝑥𝑦 ∈ [−2,2]
𝑥 ∈ [−2,2]
𝑒𝑧
𝑦 ∈ [0,1]
𝑧 ∈ (−inf,ln(2)]
𝑧
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3
∈ [0,8]
Linearize Constraints
Constraint
zt ≥ xt yt
with
xt binary, yt ∈ [20,100], zt free
xt = 1
xt = 0
zt ≥ yt
zt ≥ 0
zt ≥ yt + 100 (xt – 1)
zt ≥ 20 xt
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Linearize Constraints (cont’d)
Constraint
(1 – xt) yt ≤ yt-1
with
xt binary, yt ∈ [0,100]
xt = 1
xt = 0
yt-1 ≥ 0
yt ≤ yt-1
yt ≤ yt-1 + 100 xt
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Improving Coefficients
y + 6 x ≤ 14
y + 12 x ≤ 20
x binary
0 ≤ y ≤ 14
x=0
x=1
x binary
0 ≤ y ≤ 14
y ≤ 20 loose
y ≤ 8 tight
x=0
x=1
y ≤ 14 tight
y ≤ 8 tight
(8,1)
(0,1)
x
y + 12 x ≤ 20
(14,0)
(0,0)
14
y
y + 6 x ≤ 14
Improving Coefficients: Nonlinear
g(y1,…,yk) + M x ≤ b
x binary
If we have an upper bound gu on g such that gu < b then reformulate:
g(y1,…,yk) + (M – b + gu ) x ≤ gu
x binary
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Improving Coefficients: Probing
g(y1,…,yk) + M x ≤ b
x binary
y≥0
yi ≤ mi x i = 1,…,k
x=0
y1 = … = yk = 0
If g(0,…,0) < b
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tighten
(implication)
AIMMS Presolver Algorithm
RemoveDuplicateConstraints;
RemoveDoubletonVariables;
LinearizeConstraints;
while ( iter <= iterLimit and someBoundTightened ) do
TightenVariableBounds;
RemoveRedundantConstraints;
endwhile;
RemoveDoubletonVariables;
RemoveFixedVariables;
LinearizeConstraints;
ImproveCoefficients;
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Activate Presolver: Normal Solve
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Activate Presolver: GMP Solve
myGMP := GMP::Instance::Generate(MathProgram);
GMP::Instance::Solve(myGMP);
> Option 1: Set ‘Nonlinear presolve’ option (same as before)
> Option 2:
myGMP := GMP::Instance::Generate(MathProgram);
myPresolvedGMP := GMP::Instance::CreatePresolved(myGMP);
GMP::Instance::Solve(myPresolvedGMP);
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Activate Presolver: Standard Algorithms
> Outer Approximation (MINLP):
myGMP := GMP::Instance::Generate(MathProgram);
GMPOuterApprox::UseNonlinearPresolver := 1;
GMPOuterApprox::DoOuterApproximation(myGMP);
> Multi-Start (NLP):
myGMP := GMP::Instance::Generate(MathProgram);
MulStart::UsePresolver := 1;
MulStart::DoMultiStart(myGMP,…);
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Infeasibility Analysis
> Activated by setting options ‘Display Infeasibility Analysis’ and
‘Nonlinear Presolve’
> Warnings :
• Local nonlinear solver (CONOPT, KNITRO, SNOPT, IPOPT, AOA) might
return Locally Infeasible while problem is feasible
• For an infeasible problem the AIMMS presolver often cannot detect that
the problem is indeed infeasible
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References
> Language Reference
• The AIMMS Presolver: Chapter 17.1 (AIMMS 4.8)
> Gondzio, J., Presolve analysis of linear programs prior to applying
the interior-point method, INFORMS Journal on Computing 9, 1997,
pp. 73-91.
> Savelsbergh, M.W.P., Preprocessing and Probing Techniques for
Mixed Integer Programming Problems, ORSA Journal on Computing
6, 1994, pp. 445-454.
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