IBM ILOG CPLEX V12.1

User's Manual for CPLEX

Copyright International Business Machines Corporation 1987, 2009US Government Users Restricted Rights - Use, duplication or disclosure restricted by GSA ADP Schedule Contract with IBM Corp.

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Copyright notice Copyright International Business Machines Corporation 1987, 2009.US Government Users Restricted Rights - Use, duplication or disclosure restricted byGSA ADP Schedule Contract with IBM Corp.

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Additional registered trademarksPython is a registered trademark of the Python Software Foundation.MATLAB is a registered trademark of The MathWorks, Inc.

Acknowledgement of use: dtoa routine of the gdtoa packageILOG acknowledges use of the dtoa routine of the gdtoa package, available athttp://www.netlib.org/fp/.

The author of this software is David M. Gay.All Rights Reserved.

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Table of contents

For technical support...............................................................................................23Contacting IBM Support...........................................................................................................24

Meet CPLEX..............................................................................................................27What is CPLEX?........................................................................................................................28What does CPLEX do?..............................................................................................................29What you need to know............................................................................................................31Examples online........................................................................................................................32Notation in this manual.............................................................................................................35Related documentation.............................................................................................................36Online services..........................................................................................................................38Further reading..........................................................................................................................39

Languages and APIs................................................................................................41Concert Technology for C++ users..........................................................................................43Overview.......................................................................................................................................................45Architecture of a CPLEX C++ application....................................................................................................46Licenses.......................................................................................................................................................47Compiling and linking...................................................................................................................................48Creating a C++ application with Concert Technology...................................................................................49Modeling an optimization problem with Concert Technology........................................................................51

Copyright IBM Corp. 1987, 2009 5

C O N T E N T S

Overview...........................................................................................................................................52Creating the environment: IloEnv......................................................................................................53Defining variables and expressions: IloNumVar................................................................................54Declaring the objective: IloObjective.................................................................................................56Adding constraints: IloConstraint and IloRange................................................................................57Formulating a problem: IloModel.......................................................................................................58Managing data..................................................................................................................................59

Solving the model.........................................................................................................................................61Overview...........................................................................................................................................62Extracting a model............................................................................................................................64Invoking a solver...............................................................................................................................65Choosing an optimizer......................................................................................................................66Controlling the optimizers..................................................................................................................69

Accessing solution information.....................................................................................................................71Accessing solution status..................................................................................................................72Querying solution data......................................................................................................................73Accessing basis information..............................................................................................................74Performing sensitivity analysis..........................................................................................................75Analyzing infeasible problems...........................................................................................................76Assessing solution quality.................................................................................................................77

Modifying a model........................................................................................................................................79Overview...........................................................................................................................................80Deleting and removing modeling objects..........................................................................................81Changing variable type.....................................................................................................................82

Handling errors.............................................................................................................................................83Example: optimizing the diet problem in C++...............................................................................................85

Overview...........................................................................................................................................86Problem representation.....................................................................................................................87Application description......................................................................................................................89Creating multi-dimensional arrays with IloArray................................................................................90Using arrays for input or output.........................................................................................................91Solving the model with IloCplex........................................................................................................93Complete program............................................................................................................................94

Concert Technology for Java users.........................................................................................95Architecture of a CPLEX Java application....................................................................................................97

Overview...........................................................................................................................................98Licenses in a Java application.........................................................................................................100Compiling and linking a Java application........................................................................................101

Creating a Java application with Concert Technology................................................................................102Modeling an optimization problem with Concert Technology in the Java API.............................................103

Overview.........................................................................................................................................104Using IloModeler.............................................................................................................................107The active model.............................................................................................................................110

U S E R ' S M A N U A L F O R C P L E X6

Building the model...........................................................................................................................111Solving the model.......................................................................................................................................113Accessing solution information...................................................................................................................115Choosing an optimizer................................................................................................................................117

Overview.........................................................................................................................................118What does CPLEX solve?...............................................................................................................119Solving a single continuous model..................................................................................................121Solving subsequent continuous relaxations in a MIP......................................................................122

Controlling CPLEX optimizers....................................................................................................................123Overview.........................................................................................................................................124Parameters......................................................................................................................................125Priority orders and branching directions.........................................................................................127

More solution information...........................................................................................................................129Overview.........................................................................................................................................130Writing solution files........................................................................................................................131Dual solution information.................................................................................................................132Basis information.............................................................................................................................133Infeasible solution information.........................................................................................................134Solution quality................................................................................................................................135

Advanced modeling with IloLPMatrix..........................................................................................................136Modeling by column....................................................................................................................................137

What is modeling by column?.........................................................................................................138Procedure for Modeling by Column.................................................................................................139

Example: optimizing the diet problem in Java.............................................................................................140Modifying the model...................................................................................................................................142Concert Technology for .NET users.......................................................................................143Prerequisites..............................................................................................................................................146Describe.....................................................................................................................................................147Step 1: Describe the problem.....................................................................................................................148Step 2: Open the file...................................................................................................................................149Model..........................................................................................................................................................150Step 3: Create the model............................................................................................................................151Step 4: Create an array to store the variables............................................................................................152Step 5: Specify by row or by column..........................................................................................................153Build by Rows.............................................................................................................................................154Step 6: Set up rows....................................................................................................................................155Step 7: Create the variables: build and populate by rows...........................................................................156Step 8: Add objective..................................................................................................................................157Step 9: Add nutritional constraints..............................................................................................................158Build by Columns........................................................................................................................................159Step 10: Set up columns............................................................................................................................160Step 11: Add empty objective function and constraints..............................................................................161Step 12: Create variables...........................................................................................................................162

U S E R ' S M A N U A L F O R C P L E X 7

Solve...........................................................................................................................................................163Step 13: Solve............................................................................................................................................164Step 14: Display the solution......................................................................................................................165Step 15: End application and free license..................................................................................................166Good programming practices.....................................................................................................................167Step 16: Read the command line (data from user).....................................................................................168Step 17: Show correct use of command line..............................................................................................169Step 18: Enclose the application in try catch statements...........................................................................170Example: optimizing the diet problem in C#.NET.......................................................................................171Callable Library.......................................................................................................................173Architecture of the Callable Library............................................................................................................175

Overview.........................................................................................................................................176Licenses..........................................................................................................................................177Compiling and linking......................................................................................................................178

Using the Callable Library in an application...............................................................................................179Overview.........................................................................................................................................180Initialize the CPLEX environment....................................................................................................181Instantiate the problem as an object...............................................................................................182Put data in the problem object........................................................................................................183Optimize the problem......................................................................................................................184Change the problem object.............................................................................................................185Destroy the problem object.............................................................................................................186Release the CPLEX environment...................................................................................................187

CPLEX programming practices..................................................................................................................189Overview.........................................................................................................................................191Variable names and calling conventions.........................................................................................192Data types.......................................................................................................................................194Ownership of problem data.............................................................................................................195Problem size and memory allocation issues...................................................................................196Status and return values.................................................................................................................197Symbolic constants.........................................................................................................................198Parameter routines..........................................................................................................................199Null arguments................................................................................................................................200Row and column references...........................................................................................................201Character strings.............................................................................................................................202Problem data...................................................................................................................................203Callbacks.........................................................................................................................................205Portability........................................................................................................................................206FORTRAN interface........................................................................................................................207C++ interface...................................................................................................................................209

Managing parameters from the Callable Library........................................................................................210Example: optimizing the diet problem in the Callable Library.....................................................................213

Overview.........................................................................................................................................214

U S E R ' S M A N U A L F O R C P L E X8

Problem representation...................................................................................................................215Program description........................................................................................................................217Solving the model with CPXlpopt....................................................................................................218Complete program..........................................................................................................................219

Using surplus arguments for array allocations...........................................................................................220Example: using query routines lpex7.c.......................................................................................................222

Programming considerations................................................................................223Developing CPLEX applications............................................................................................225Tips for successful application development..............................................................................................227

Prototype the model........................................................................................................................229Identify routines to use....................................................................................................................230Test interactively..............................................................................................................................231Assemble data efficiently................................................................................................................232Test data..........................................................................................................................................233Test and debug the model...............................................................................................................234Choose an optimizer.......................................................................................................................235Program with a view toward maintenance and modifications..........................................................236

Using the Interactive Optimizer for debugging............................................................................................239Eliminating common programming errors...................................................................................................241

Check your include files..................................................................................................................243Clean house and try again..............................................................................................................244Read your messages......................................................................................................................245Check return values........................................................................................................................246Beware of numbering conventions..................................................................................................247Make local variables temporarily global..........................................................................................248Solve the problem you intended......................................................................................................249Special considerations for FORTRAN.............................................................................................250Tell us..............................................................................................................................................251

Managing input and output....................................................................................................253Overview.....................................................................................................................................................254Understanding file formats.........................................................................................................................255

Overview.........................................................................................................................................256Working with LP files.......................................................................................................................257Working with MPS files...................................................................................................................259Converting file formats....................................................................................................................261

Using Concert XML extensions..................................................................................................................262Using Concert csvReader..........................................................................................................................263Managing log files......................................................................................................................................265

Overview.........................................................................................................................................266Creating, renaming, relocating log files...........................................................................................267Closing log files...............................................................................................................................268

Controlling message channels...................................................................................................................269

U S E R ' S M A N U A L F O R C P L E X 9

Overview.........................................................................................................................................270Parameter for output channels........................................................................................................271Callable Library routines for message channels.............................................................................272Example: Callable Library message channels................................................................................274Concert Technology message channels.........................................................................................275

Timing interface.......................................................................................................................277Using the timing interface...........................................................................................................................278Licensing an application........................................................................................................281Overview.....................................................................................................................................................282Types of ILM runtime licenses....................................................................................................................283

File-based RTNODE, RTSTOKEN or TOKEN keys........................................................................284Memory-based RUNTIME keys.......................................................................................................285

Routines and methods for licensing...........................................................................................................286Examples....................................................................................................................................................287

CPXputenv routine for C and C++ users.........................................................................................288The putenv method for Java users..................................................................................................289The Putenv method for .NET users.................................................................................................290CPXRegisterLicense routine for C and C++ users..........................................................................291The registerLicense method for Java users....................................................................................292The RegisterLicense method for .NET users..................................................................................293

Summary....................................................................................................................................................294Tuning tool...............................................................................................................................295Meet the tuning tool....................................................................................................................................297

Overview.........................................................................................................................................298If CPLEX solves your problem to optimality....................................................................................299If CPLEX finds solutions but does not prove optimality...................................................................300Tuning and time limits......................................................................................................................301Tuning results..................................................................................................................................302

Invoking the tuning tool...............................................................................................................................303Example: time limits on tuning in the Interactive Optimizer........................................................................304Fixing parameters and tuning multiple models in the Interactive Optimizer...............................................305

Invoking the tuning tool in the Interactive Optimizer........................................................................306Fixed parameters to respect...........................................................................................................307Files of models to tune....................................................................................................................308

Tuning models in the Callable Library (C API)............................................................................................309Callbacks for tuning....................................................................................................................................310Terminating a tuning session......................................................................................................................311

Continuous optimization.......................................................................................313Solving LPs: simplex optimizers...........................................................................................315Introducing the primal and dual optimizers.................................................................................................316Choosing an optimizer for your LP problem...............................................................................................317

Overview of LP optimizers..............................................................................................................319

U S E R ' S M A N U A L F O R C P L E X10

Automatic selection of an optimizer................................................................................................321Dual simplex optimizer....................................................................................................................322Primal simplex optimizer.................................................................................................................323Network optimizer...........................................................................................................................324Barrier optimizer..............................................................................................................................325Sifting optimizer...............................................................................................................................326Concurrent optimizer.......................................................................................................................327Parameter settings and optimizer choice........................................................................................328

Tuning LP performance..............................................................................................................................329Introducing performance tuning for LP models...............................................................................330Preprocessing.................................................................................................................................331Starting from an advanced basis....................................................................................................334Simplex parameters........................................................................................................................336

Diagnosing performance problems.............................................................................................................341Lack of memory..............................................................................................................................342Numeric difficulties..........................................................................................................................344

Diagnosing LP infeasibility..........................................................................................................................349Infeasibility reported by LP optimizers............................................................................................350Coping with an ill-conditioned problem or handling unscaled infeasibilities....................................351Interpreting solution quality.............................................................................................................352Finding a conflict.............................................................................................................................355Repairing infeasibility: FeasOpt.......................................................................................................356

Examples: using a starting basis in LP optimization...................................................................................357Overview.........................................................................................................................................358Example ilolpex6.cpp......................................................................................................................359Example lpex6.c..............................................................................................................................360

Solving LPs: barrier optimizer...............................................................................................361Introducing the barrier optimizer.................................................................................................................363Barrier simplex crossover...........................................................................................................................365Differences between barrier and simplex optimizers..................................................................................366Using the barrier optimizer.........................................................................................................................367Special options in the Interactive Optimizer...............................................................................................368Controlling crossover..................................................................................................................................369Using SOL file format.................................................................................................................................370Interpreting the barrier log file....................................................................................................................371

Accessing and managing the log file of the barrier optimizer..........................................................372Sample log file from the barrier optimizer.......................................................................................373Preprocessing in the log file............................................................................................................374Nonzeros in lower triangle of A*A' in the log file..............................................................................375Ordering-algorithm time in the log file.............................................................................................376Cholesky factor in the log file..........................................................................................................377Iteration progress in the log file.......................................................................................................378Infeasibility ratio in the log file.........................................................................................................379

U S E R ' S M A N U A L F O R C P L E X 11

Understanding solution quality from the barrier LP optimizer.....................................................................380Tuning barrier optimizer performance.........................................................................................................383

Overview of parameters for tuning the barrier optimizer.................................................................384Memory emphasis: letting the optimizer use disk for storage.........................................................386Preprocessing.................................................................................................................................387Detecting and eliminating dense columns.......................................................................................388Choosing an ordering algorithm......................................................................................................389Using a starting-point heuristic........................................................................................................390

Overcoming numeric difficulties..................................................................................................................391Default behavior of the barrier optimizer with respect to numeric difficulty.....................................392Numerical emphasis settings..........................................................................................................393Difficulties in the quality of solution.................................................................................................394Difficulties during optimization........................................................................................................396Difficulties with unbounded problems..............................................................................................398

Diagnosing infeasibility reported by barrier optimizer.................................................................................399Solving network-flow problems.............................................................................................401Choosing an optimizer: network considerations.........................................................................................402Formulating a network problem..................................................................................................................403Example: network optimizer in the Interactive Optimizer............................................................................405

Network flow problem description...................................................................................................406Understanding the network log file..................................................................................................407Tuning performance of the network optimizer.................................................................................408

Solving problems with the network optimizer.............................................................................................409Invoking the network optimizer........................................................................................................410Network extraction..........................................................................................................................411Preprocessing and the network optimizer.......................................................................................412

Example: using the network optimizer with the Callable Library netex1.c..................................................413Solving network-flow problems as LP problems.........................................................................................415Example: network to LP transformation netex2.c.......................................................................................417Solving problems with a quadratic objective (QP)...............................................................419Identifying convex QPs...............................................................................................................................421Entering QPs..............................................................................................................................................423

Matrix view......................................................................................................................................424Algebraic view.................................................................................................................................425Examples for entering QPs.............................................................................................................426Reformulating QPs to save memory...............................................................................................428

Saving QP problems...................................................................................................................................429Changing problem type in QPs...................................................................................................................430Changing quadratic terms..........................................................................................................................431Optimizing QPs..........................................................................................................................................433Diagnosing QP infeasibility.........................................................................................................................434Examples: creating a QP, optimizing, finding a solution.............................................................................435

Problem description of a quadratic program...................................................................................436

U S E R ' S M A N U A L F O R C P L E X12

Example: iloqpex1.cpp....................................................................................................................437Example: QPex1.java......................................................................................................................438Example: qpex1.c............................................................................................................................439

Example: reading a QP from a file qpex2.c................................................................................................440Solving problems with quadratic constraints (QCP)...........................................................441Identifying a quadratically constrained program (QCP)..............................................................................443

Characteristics of a quadratically constrained program..................................................................444Convexity.........................................................................................................................................445Semi-definiteness............................................................................................................................447Second order cone programming (SOCP)......................................................................................448

Detecting the problem type of a QCP or SOCP.........................................................................................449Overview.........................................................................................................................................450Concert Technology and QCP problem type...................................................................................451Callable Library and QCP problem type..........................................................................................452Interactive Optimizer and QCP problem type..................................................................................453File formats and QCP problem type................................................................................................454

Changing problem type..............................................................................................................................459Changing quadratic constraints..................................................................................................................460Solving with quadratic constraints..............................................................................................................461Numeric difficulties and quadratic constraints............................................................................................462Examples: QCP..........................................................................................................................................463

Discrete optimization.............................................................................................465Solving mixed integer programming problems (MIP)..........................................................467Stating a MIP problem................................................................................................................................469Preliminary issues......................................................................................................................................471

Entering MIP problems....................................................................................................................472Displaying MIP problems.................................................................................................................473Changing problem type in MIPs......................................................................................................474Changing variable type...................................................................................................................476

Using the mixed integer optimizer..............................................................................................................477Invoking the optimizer for a MIP model...........................................................................................478Emphasizing feasibility and optimality.............................................................................................479Terminating MIP optimization..........................................................................................................481

Tuning performance features of the mixed integer optimizer......................................................................483Branch & cut or dynamic search?...................................................................................................485Introducing performance features of the MIP optimizer..................................................................486Applying cutoff values.....................................................................................................................487Applying tolerance parameters.......................................................................................................488Applying heuristics..........................................................................................................................489When an integer solution is found: the incumbent..........................................................................490Controlling strategies: diving and backtracking...............................................................................491Selecting nodes...............................................................................................................................492

U S E R ' S M A N U A L F O R C P L E X 13

Selecting variables..........................................................................................................................493Changing branching direction.........................................................................................................494Solving subproblems.......................................................................................................................495Using node files...............................................................................................................................496Probing............................................................................................................................................497

Cuts............................................................................................................................................................499What are cuts?................................................................................................................................501Clique cuts......................................................................................................................................502Cover cuts.......................................................................................................................................503Disjunctive cuts...............................................................................................................................504Flow cover cuts...............................................................................................................................505Flow path cuts.................................................................................................................................506Gomory fractional cuts....................................................................................................................507Generalized upper bound (GUB) cover cuts...................................................................................508Implied bound cuts..........................................................................................................................509Mixed integer rounding (MIR) cuts..................................................................................................510Multi-commodity flow (MCF) cuts....................................................................................................511Zero-half cuts..................................................................................................................................512Adding cuts and re-optimizing.........................................................................................................513Counting cuts..................................................................................................................................514Parameters affecting cuts................................................................................................................515

Heuristics....................................................................................................................................................517What are heuristics?.......................................................................................................................518Node heuristic.................................................................................................................................519Relaxation induced neighborhood search (RINS) heuristic............................................................520Solution polishing............................................................................................................................521Feasibility pump..............................................................................................................................527

Preprocessing: presolver and aggregator...................................................................................................528Starting from a solution: MIP starts............................................................................................................531Issuing priority orders.................................................................................................................................535Using the MIP solution...............................................................................................................................537Progress reports: interpreting the node log................................................................................................539Troubleshooting MIP performance problems..............................................................................................543

Introducing trouble shooting for MIP performance..........................................................................545Too much time at node 0.................................................................................................................546Trouble finding more than one feasible solution..............................................................................547Large number of unhelpful cuts.......................................................................................................548Lack of movement in the best node................................................................................................549Time wasted on overly tight optimality criteria................................................................................550Slightly infeasible integer variables.................................................................................................551Running out of memory...................................................................................................................552Difficulty solving subproblems: overcoming degeneracy.................................................................556Unsatisfactory optimization of subproblems....................................................................................557

U S E R ' S M A N U A L F O R C P L E X14

Examples: optimizing a simple MIP problem..............................................................................................561ilomipex1.cpp..................................................................................................................................562MIPex1.java.....................................................................................................................................563MIPex1.cs and MIPex1.vb...............................................................................................................564mipex1.c..........................................................................................................................................565

Example: reading a MIP problem from a file...............................................................................................567ilomipex2.cpp..................................................................................................................................568mipex2.c..........................................................................................................................................569

Solution pool: generating and keeping multiple solutions.................................................571What is the solution pool?..........................................................................................................................573Example: simple facility location problem...................................................................................................574Filling the solution pool...............................................................................................................................576Accumulating incumbents in the solution pool............................................................................................577Populating the solution pool.......................................................................................................................579

What is populating the solution pool?.............................................................................................580Invoking the populate procedure.....................................................................................................581Algorithm of the populate procedure...............................................................................................582Example: calling populate...............................................................................................................583Stopping criteria for the populate procedure...................................................................................586Stored solutions, populate limit, and pool capacity.........................................................................588

Choosing whether to accumulate or populate............................................................................................589Whats the difference between accumulating and populating?.......................................................590Advanced use: interaction of MIP optimization and populate..........................................................591Example: using populate after MIP optimization.............................................................................592

Enumerating all solutions...........................................................................................................................593How to enumerate all solutions.......................................................................................................594Limitations due to continuous variables and finite precision...........................................................595Limitations due to unbounded MIP models.....................................................................................596Limitations due to numeric difficulties.............................................................................................597

Impact of change on the solution pool........................................................................................................599Changes between MIP optimization and populate..........................................................................600Persistence of solutions in the solution pool...................................................................................601Model changes and the solution pool..............................................................................................602

Examining the solution pool.......................................................................................................................603Accessing a solution in the solution pool....................................................................................................604Using solutions from the solution pool........................................................................................................606Deleting solutions from the solution pool....................................................................................................608The incumbent and the solution pool.........................................................................................................609Parameters of the solution pool..................................................................................................................611

Which parameters control the solution pool?..................................................................................612Example: quality control through the solution pool gap parameter.................................................613Example: few or many solutions through intensity parameter.........................................................614Example: diverse solutions through replacement parameter..........................................................615

U S E R ' S M A N U A L F O R C P L E X 15

Filtering the solution pool...........................................................................................................................617What are filters of the solution pool?...............................................................................................618Diversity filters.................................................................................................................................619Range filters....................................................................................................................................620Filter files.........................................................................................................................................621Example: controlling properties of solutions with filters...................................................................622Incumbent callback as a filter..........................................................................................................623

Using special ordered sets (SOS)..........................................................................................625What is a special ordered set (SOS)?........................................................................................................626Example: SOS Type 1 for sizing a warehouse............................................................................................627Declaring SOS members............................................................................................................................628Example: using SOS and priority................................................................................................................629

ilomipex3.cpp..................................................................................................................................630mipex3.c..........................................................................................................................................631

Using semi-continuous variables: a rates example.............................................................633What are semi-continuous variables?........................................................................................................634Describing the problem..............................................................................................................................635Representing the problem..........................................................................................................................636Building a model.........................................................................................................................................637Solving the problem....................................................................................................................................638Ending the application................................................................................................................................639Complete program......................................................................................................................................640Using piecewise linear functions in optimization: a transport example............................641What is a piecewise linear function?..........................................................................................................643Syntax of piecewise linear functions...........................................................................................................644Discontinuous piecewise linear functions...................................................................................................646Isolated points in piecewise linear functions...............................................................................................649Using IloPiecewiseLinear in expressions...................................................................................................650Describing the problem..............................................................................................................................651

Problem statement..........................................................................................................................652Variable shipping costs...................................................................................................................653Model with varying costs.................................................................................................................655

Developing a model....................................................................................................................................657Creating the environment and model..............................................................................................658Representing the data.....................................................................................................................659Adding constraints...........................................................................................................................660Checking convexity and concavity...................................................................................................661Adding an objective.........................................................................................................................662

Solving the problem....................................................................................................................................663Displaying a solution...................................................................................................................................664Ending the application................................................................................................................................665Complete program: transport.cpp...............................................................................................................666

U S E R ' S M A N U A L F O R C P L E X16

Logical constraints in optimization.......................................................................................667What are logical constraints?.....................................................................................................................668What can be extracted from a model with logical constraints?...................................................................669

Overview.........................................................................................................................................670Logical constraints in the C++ API..................................................................................................671Logical constraints in the Java API.................................................................................................672Logical constraints in the .NET API.................................................................................................673

Which nonlinear expressions can be extracted?........................................................................................674Logical constraints for counting..................................................................................................................675Logical constraints as binary variables.......................................................................................................676How are logical constraints extracted?.......................................................................................................677Indicator constraints in optimization....................................................................................679What is an indicator constraint?.................................................................................................................680Example: fixnet.c........................................................................................................................................681Indicator constraints in the Interactive Optimizer........................................................................................682What are indicator variables?.....................................................................................................................683Restrictions on indicator constraints...........................................................................................................684Best practices with indicator constraints.....................................................................................................685Using logical constraints: Food Manufacture 2...................................................................687Introducing the example.............................................................................................................................688Describing the problem..............................................................................................................................689Representing the data................................................................................................................................690Developing the model.................................................................................................................................693Formulating logical constraints...................................................................................................................694Solving the problem....................................................................................................................................695Early tardy scheduling............................................................................................................697Describing the problem..............................................................................................................................699Understanding the data file........................................................................................................................700Reading the data........................................................................................................................................701Creating variables.......................................................................................................................................702Stating precedence constraints..................................................................................................................703Stating resource constraints.......................................................................................................................704Representing the piecewise linear cost function........................................................................................705Transforming the problem...........................................................................................................................706Solving the problem....................................................................................................................................707Using column generation: a cutting stock example.............................................................709What is column generation?.......................................................................................................................711Column-wise models in Concert Technology..............................................................................................712Describing the problem..............................................................................................................................713Representing the data................................................................................................................................715Developing the model: building and modifying...........................................................................................717

The master model and column generator in this application...........................................................718

U S E R ' S M A N U A L F O R C P L E X 17

Adding extractable objects: both ways............................................................................................719Adding columns to a model.............................................................................................................720Changing the type of a variable......................................................................................................721Cut optimization model....................................................................................................................722Pattern generator model.................................................................................................................723

Changing the objective function.................................................................................................................724Solving the problem: using more than one algorithm.................................................................................725Ending the program....................................................................................................................................727Complete program......................................................................................................................................728

Infeasibility and unboundedness..........................................................................729Preprocessing and feasibility.................................................................................................731Issues of infeasibility and unboundedness.................................................................................................732Early reports of infeasibility based on preprocessing reductions...............................................................733Managing unboundedness.....................................................................................................735What is unboundedness?...........................................................................................................................736Avoiding unboundedness in a model..........................................................................................................737Diagnosing unboundedness.......................................................................................................................738Diagnosing infeasibility by refining conflicts.......................................................................739What is a conflict?......................................................................................................................................741What a conflict is not..................................................................................................................................742How to invoke the conflict refiner................................................................................................................743How a conflict differs from an IIS................................................................................................................744Meet the conflict refiner in the Interactive Optimizer..................................................................................745

Limits of the conflict refiner in the Interactive Optimizer..................................................................746A model for the conflict refiner........................................................................................................747Optimizing the example...................................................................................................................748Interpreting the results and detecting conflict.................................................................................749Displaying a conflict in the Interactive Optimizer.............................................................................750

Interpreting conflict.....................................................................................................................................751Understanding the conflict in the model..........................................................................................752Deleting a constraint.......................................................................................................................753Understanding a conflict report.......................................................................................................755Summing equality constraints.........................................................................................................756Changing a bound...........................................................................................................................757Adding a constraint.........................................................................................................................758Changing bounds on cost...............................................................................................................759Relaxing a constraint.......................................................................................................................760

More about the conflict refiner....................................................................................................................761Refining a conflict in a MIP start.................................................................................................................763Using the conflict refiner in an application..................................................................................................765

Example: modifying ilomipex2.cpp..................................................................................................766What belongs in an application to refine conflict.............................................................................768

U S E R ' S M A N U A L F O R C P L E X18

Comparing a conflict application to Interactive Optimizer...........................................................................769Preferences in the conflict refiner....................................................................................................770Groups in the conflict refiner...........................................................................................................771

Repairing infeasibilities with FeasOpt...................................................................................773What is FeasOpt?.......................................................................................................................................774Invoking FeasOpt........................................................................................................................................775Specifying preferences...............................................................................................................................776Example: FeasOpt in Concert Technology.................................................................................................777

Advanced programming techniques....................................................................783User-cut and lazy-constraint pools........................................................................................785What are user cuts and lazy constraints?...................................................................................................786What are pools of user cuts or lazy constraints?........................................................................................787Differences between user cuts and lazy constraints..................................................................................788Limitations on pools in the C API...............................................................................................................789Adding user cuts and lazy constraints........................................................................................................791

Using the Component Libraries.......................................................................................................792Using the Interactive Optimizer.......................................................................................................793Reading and writing LP files...........................................................................................................794Reading and writing SAV files.........................................................................................................796Reading and writing MPS files........................................................................................................797

Deleting user cuts and lazy constraints......................................................................................................799Using goals..............................................................................................................................801Branch & cut with goals..............................................................................................................................803

What is a goal?...............................................................................................................................804Overview of goals in the search......................................................................................................805How goals are implemented in branch & cut...................................................................................806About the method execute in a goal................................................................................................807

Special goals in branch & cut.....................................................................................................................809Or goal............................................................................................................................................810And goal..........................................................................................................................................811Fail goal...........................................................................................................................................812Local cut goal..................................................................................................................................813Null goal..........................................................................................................................................814Branch as CPLEX goal...................................................................................................................815Solution goal...................................................................................................................................816

Aggregating goals......................................................................................................................................817Example: goals in branch & cut...................................................