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1 Introduction to ILOG CPLEX Copyright © 2004 ILOG, Inc.
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Introduction to ILOG CPLEX
Copyright © 2004 ILOG, Inc.
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Overview
• ILOG Optimization Suite
• Mathematical programming Problems
• CPLEX Algorithms
• Building CPLEX applications
• Parallel CPLEX
• Conclusion
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ILOG Optimization Suite
C API VB6 API
ILOGILOGCPLEXCPLEX ILOG SolverILOG Solver
ILOG Concert Technology (C++, Java, .NET)
ILOGOPL
Studio
ILOGOPL
StudioILOGILOG
SchedulerSchedulerILOGILOG
DispatcherDispatcherILOGILOG
ConfiguratorConfigurator
AMPL
A Common API for CPLEX and Solver
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Mathematical Programming
Problem Types
• Linear Programs• Mixed Integer Linear Programs• Quadratic Programs• Mixed Integer Quadratic Programs• Quadratic Constrained Programs• Mixed Integer Quadratic Constrained
Programs
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Linear Programming
Minimize cTxSubject to Ax = b
l ≤ x ≤ u
Objective Function
Constraints
Decision Variables
Lower Bounds
Upper Bounds
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Linear Programming
Minimize cTxSubject to Ax = b
l ≤ x ≤ u
(LP)
Maximizex1 + 2 x2 + 3 x3
Subject To- x1 + x2 + x3 ≤ 20x1 - 3 x2 + x3 ≤ 30
0 ≤ x1 ≤ 40x2, x3 ≥ 0
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Maximize x1 + 2 x2 + 3 x3 + x4Subject To
- x1 + x2 + x3 + 10 x4 ≤ 20x1 - 3 x2 + x3 ≤ 30
x2 - 3.5 x4 = 0
0 ≤ x1 ≤ 40 x2, x3 ≥ 02 ≤ x4 ≤ 3x4 integer
(MIP)
Mixed Integer Programming
Minimize cTxSubject to Ax = b
l ≤ x ≤ uSome x are integer
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Quadratic Programming
Minimize cTx + ½ xTQxSubject to Ax = b
l ≤ x ≤ u
(QP)
Maximizex1 + 2 x2 + 3 x3
- 0.5 ( 33*x1*x1 + 22*x2*x2 +
11*x3*x3 - 12*x1*x2 - 23*x2*x3 )Subject To
- x1 + x2 + x3 ≤ 20x1 - 3 x2 + x3 ≤ 30
0 ≤ x1 ≤ 40x2, x3 ≥ 0
Quadratic Function
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Mixed Integer Quadratic Programming
Minimize cTx + ½ xTQxSubject to Ax = b
l ≤ x ≤ u(MIQP)
Maximizex1 + 2 x2 + 3 x3
- 0.5 ( 33*x1*x1 + 22*x2*x2 + 11*x3*x3 - 12*x1*x2 - 23*x2*x3 )
Subject To- x1 + x2 + x3 ≤ 20
x1 - 3 x2 + x3 ≤ 30
0 ≤ x1 ≤ 40 x1 integerx2, x3 ≥ 0
Some x are integer
Quadratic Function
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Quadratic Constrained Programs
Minimize cTx + ½ xTQxSubject to Ax = b
½ xTQkx + akx ≤ bkl ≤ x ≤ u
(QCP)
Maximizex1 + 2 x2 + 3 x3
- 0.5 ( 33*x1*x1 + 22*x2*x2 + 11*x3*x3 - 12*x1*x2 - 23*x2*x3 )
Subject To- x1 + x2 + x3 ≤ 20
x1 - 3 x2 + x3 ≤ 30x1*x1 + x2*x2 + x3*x3 ≤ 1
0 ≤ x1 ≤ 40x2, x3 ≥ 0
Quadratic Constraint
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Mixed Integer Quadratic Constrained Programs
Minimize cTx + ½ xTQxSubject to Ax = b
½ xTQkx + akx ≤ bkl ≤ x ≤ u
(MIQCP)
Maximizex1 + 2 x2 + 3 x3
- 0.5 ( 33*x1*x1 + 22*x2*x2 + 11*x3*x3 - 12*x1*x2 - 23*x2*x3 )
Subject To- x1 + x2 + x3 ≤ 20
x1 - 3 x2 + x3 ≤ 30x1*x1 + x2*x2 + x3*x3 ≤ 1
0 ≤ x1 ≤ 40 x1 integerx2, x3 ≥ 0
Some x are integer
Quadratic Constraint
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What’s inside ILOG CPLEX?
• ILOG CPLEX Suite contains:• Simplex Optimizers (Primal, Dual, Network) for LP
and QP• Barrier Optimizer for LP, QP, and QCP• Mixed Integer Optimizer for MIP, MIQP and MIQCP• CPLEX Interactive Optimizer• CPLEX Component Libraries
• CPLEX Callable Library (C API and VB6 API)• ILOG Concert Technology (C++, Java, Microsoft .NET APIs)
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CPLEX algorithms
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Linear Programming Algorithms
• Primal simplex• Dual simplex• Network simplex• Primal/dual log barrier
• Includes crossover to simplex solutions
All use automatic CPLEX presolve algorithms
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Quadratic Programming Algorithms
Note: Minimizing (maximizing) a quadratic function requires convexity (concavity)
• Primal dual log barrier• Primal Simplex• Dual Simplex
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Quadratic Constraint Programming Algorithms
• Primal dual log barrier• Based on second-order cone programming
Note: Inequalities require convex regions
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Choosing LP Algorithm
Trends
• Which is Best Algorithm?• Primal versus Dual: 2.3X• Primal versus Barrier: 4.2X• Primal versus Best: 7.5X
• Eliminate the need to choose• Use all algorithms at one time• Concurrent optimization• Requires multiple CPUs
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Concurrent Optimization
• CPU Utilization• # threads = 1: dual• # threads = 2: best of (dual, barrier)• # threads = 3: best of (dual, primal, barrier)• # threads >= 4: best of (dual, primal, parallel
barrier)
• User can customize• Requires a Parallel CPLEX license
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Mixed Integer Programming
Branch-and-Bound Fundamentals
Choose an integer variable: V in [0 ..40]with a non integer current value: 3.7
A new integer feasiblesolution has been foundyesyesdo all integer variables
have integer values?
no
Create two new problems to solve
V ≤ 3V in [0..3]
V ≥ 4V in [4..40]Or
Solve a linear program relaxation
Is it a better solution?
yesyes
Have new incumbent: Fathom
no
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Branch and Bound Tree
Root
Integer
v ≤ 3 v ≥ 4
x ≤2 x ≥ 3
y ≤0 y ≥ 1
z ≤0 z ≥ 1
Integer
Infeas
z ≤0 z ≥ 1
GAP
Lower Bound
Upper Bound
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Branch and Bound Tree
Root
Integer
v ≤ 3 v ≥ 4
x ≤2 x ∆ 3
y ≤0 y ≥ 1
z ≤0 z ≥ 1
Integer
Infeas
z ≤0 z ≥ 1
In the CPLEX MIP Algorithm:
• You pick a variable selection strategy to decide which variable to branch on, and what value to try next
• You pick a node strategy to select which node to work on next
• Often, the default built-in strategies work quite well!
• You can also write your own strategies based upon your own problem knowledge
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CPLEX Presolve Algorithms
Minimize cTxSubject to Ax = b
l ≤ x ≤ uPresolve
Minimize c'Tx' Subject to A'x' = b'
l' ≤ x' ≤ u'
Solve
Solution x'Basis
Dual ValuesUnpresolve
Solution xBasis
Dual Values
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Building a CPLEX application
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ILOG Optimization Suite
ILOGILOGCPLEXCPLEX
ILOGOPL
Studio
ILOGOPL
Studio
CPLEX Callable LibraryCPLEX Callable LibraryILOG Concert TechnologyILOG Concert Technology
CPLEX Callable LibraryILOG Concert Technology
OPL Studio GUIOPL Studio GUIOPL Component LibrariesOPL Component Libraries
CPLEX Component libraries
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Component Libraries Features
• Problem Creation Routines and Methods• Optimization Result Routines and Methods• Utility Routines and Methods• Problem Modification Routines and Methods• Problem Query Routines and Methods• File Reading and Writing Routines and Methods• Parameter Setting Routines and Methods
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Simple LP Application
Raw Data(Database,
Spreadsheet,GUI)
Minimize cTxSubject to Ax = b
l ≤ x ≤ u
CPLEX
Output(Database,
Spreadsheet,GUI)
Transform (c, A, b
, l, u)
(x, c Tx)
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LP Application with Modifications
Raw Data(Database,
Spreadsheet,GUI)
Minimize cTxSubject to Ax = b
l ≤ x ≤ u
CPLEX
Output(Database,
Spreadsheet,GUI)
Transform (c, A, b
, l, u)
(x, c Tx)
Modify
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What are Hot Starts?
Basis
Minimize cTxSubject to Ax = b
l ≤ x ≤ u
Minimize c'TxSubject to A'x = b'
l' ≤ x ≤ u'
Solve
Use
Modify
Solve One ProblemObtain Solution
Modify The ProblemSolution Not Optimal
Solve the Modified ProblemUsing the Prior Solution to Start
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It’s all about how you model…..
• CPLEX Callable Library uses matrices to represent a problem
• Users map their model to a linear ordering
• ILOG Concert Technology uses objects and methods to represent a problem
• Users write out the mathematical representation in C++, Java, VB .NET, C# (or any .NET language)
• C++ features operator overloading for ease of expression
• A modeling language in C++, Java and .NET
• ILOG OPL Component Libraries allow OPL models to be embedded inside an application
• Provided inside ILOG OPLStudio (Requires an ILOG OPL Studio license)
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A Simple MIP Example
Maximize
obj: x1 + 2 x2 + 3 x3 + x4
Subject To
c1: - x1 + x2 + x3 + 10 x4 ≤ 20
c2: x1 - 3 x2 + x3 ≤ 30
c3: x2 - 3.5 x4 = 0
Bounds
0 ≤ x1 ≤ 40 x2, x3 ≥ 0
2 ≤ x4 ≤ 3
Integers
x4
End
-1 1 1 101 -3 1 00 1 0 -3.5
Matrix representation
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Simple MIP with C
strcpy (probname, "example");*numcols_p = 4;*numrows_p = 3;*objsen_p = CPX_MAX; /* The problem is maximization */
/* The code is formatted to make a visual correspondencebetween the mathematical linear program and the specific dataitems. */
obj[0] = 1.0; obj[1] = 2.0; obj[2] = 3.0; obj[3] = 1.0;
matbeg[0] = 0; matbeg[1] = 2; matbeg[2] = 5; matbeg[3] = 7;matcnt[0] = 2; matcnt[1] = 3; matcnt[2] = 2; matcnt[3] = 2;
matind[0] = 0; matind[2] = 0; matind[5] = 0; matind[7] = 0;matval[0] = -1.0; matval[2] = 1.0; matval[5] = 1.0; matval[7] = 10.0;
matind[1] = 1; matind[3] = 1; matind[6] = 1;matval[1] = 1.0; matval[3] = -3.0; matval[6] = 1.0;
matind[4] = 2; matind[8] = 2;matval[4] = 1.0; matval[8] = -3.5;
lb[0] = 0.0; lb[1] = 0.0; lb[2] = 0.0; lb[3] = 2.0;ub[0] = 40.0; ub[1] = CPX_INFBOUND; ub[2] = CPX_INFBOUND; ub[3] = 3.0;
ctype[0] = 'C'; ctype[1] = 'C'; ctype[2] = 'C'; ctype[3] = 'I';
/* The right-hand-side values don't fit nicely on a line above. So putthem here. */
sense[0] = 'L';rhs[0] = 20.0;
sense[1] = 'L';rhs[1] = 30.0;
sense[2] = 'E';rhs[2] = 0.0;
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Data Structures for C Callable Library
1.0E200.0
1.0 1.0
1.0E20
1.0
matbeg 2 5 7matcnt 3 2 3matind 1 0 1 2 0 1 0 2matval -1.0 -3.0 1.0 1.0 10.0 -3.5obj 1.0 2.0 3.0 4.0lb 0.0 0.0 2.0ub 40.0 3.0ctype C C C
sense L L E -1 1 1 101 -3 1 00 1 0 -3.5
Non zero elements
0
Irhs 20.0 30.0 0.0
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Same MIP with Concert Technology (C++)
IloEnv env;IloModel mip(env);
// Declaring the variables; x4 is integerIloNumVar x1(env, 0.0, 40.0, "x1");IloNumVar x2(env, 0.0, IloInfinity, "x2");IloNumVar x3(env, 0.0, IloInfinity, "x3");IloNumVar x4(env, 2, 3, ILOINT, "x4");
// Adding the constraints :mip.add(-x1 + x2 + x3 + 10*x4 <= 20);mip.add( x1 - 3*x2 + x3 <= 30);mip.add( x2 - 3.5*x4 == 0);
// Setting the objectivemip.add(IloMaximize(env, x1 + 2*x2 + 3*x3 + x4));
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Solving the MIP
IloCplex cplex(mip);cplex.solve();
env.out() << "Solution status = " << cplex.getStatus() << endl;
env.out() << "Solution value = " << cplex.getObjValue() << endl;
env.out() << cplex.getValue(x1) << endl;env.out() << cplex.getValue(x2) << endl;env.out() << cplex.getValue(x3) << endl;env.out() << cplex.getValue(x4) << endl;
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Simple Arrays
/* Read in an array of numbers */IloNumArray mydata(env);ifstream file("mydata.dat");file >> mydata;IloInt n = mydata.getSize();
/* Create an array x of variables */IloNumVarArray x(env, n, 0.0, IloInfinity);
/* Add a constraint */model.add (IloScalProd(mydata, x) <= 15);
[5, 4, 9, 8, 7,6, 4, 3, 10, 2]
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Complex Arrays
var float make[Prod,Time];// ...forall(t in Time)
sum(p in Prod) (1/rate[p])*make[p,t] <= avail[t];
OPL
typedef IloArray<IloNumVarArray> IloNumVarArray2;// ...IloNumVarArray2 Make(env);
for (p = 0; p < nProd; p++)Make.add(IloNumVarArray(env, nTime, 0.0,IloInfinity));
// ...for (t = 0; t < nTime; t++) {
IloExpr availExpr(env); for (p = 0; p < nProd; p++)
availExpr += (1/rate[p]) * Make[p][t]; mod.add(availExpr <= avail[t]);
}
Concert(C++)
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CPLEX Parameters
• CPLEX offers a variety of parameters to control various algorithmic and display choices.
• All parameters have a default value which is sufficient in most of the cases.
• Parameters are accessible via all theinterfaces
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CPLEX Parameters
• CPLEX parameters can be subdivided in several categories according to their aims. There are:
• General parameters (example : set clocktype)• Starting and preprocessing parameters• Simplex parameters• Barrier parameters• Network parameters• MIP parameters
• CPLEX parameters are too numerous for a complete enumeration here (see the CPLEX 9.0 Reference Manual).
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CPLEX Parameters Settings Hints
• Default works well to prove optimality or to find good feasible solutions for most models
• Try it first• CPLEX has an emphasis setting.
• Using it to specify a goal may help for some models• Several features are off by default
• Turning them on or changing parameter settings can help solving hard models
• CPLEX provides advanced routines for exploiting user knowledge
• They can be helpful for hard models, e.g. their use for extending Gomory cuts for handling side constraints
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Parallel CPLEX
• Provides scalability on symmetric multi-processor (SMP) shared memory systems
• For extremely high-performance applications• Very large LPs or QPs• Very hard MIPs
• Speed-ups vary by problem type and instance• Barrier: sub-linear, but better than simplex• MIP: variable, may be super-linear!
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Conclusion
• CPLEX Algorithms• Simplex Optimizers (Primal, Dual, Network) (LP / QP)• Barrier Optimizer (LP / QP/ QCP)• Mixed Integer Optimizer (MILP / MIQP/ MIQCP)
• CPLEX Component Libraries• CPLEX Callable Library (C API)• ILOG Concert Technology (C++/Java/.NET API)
• Parallel CPLEX as an option
