R FICO Xpress Optimization Last update 02 October 2020 37.01 REFERENCE MANUAL FICO R Xpress Optimizer ©1983–2020 Fair Isaac Corporation. All rights reserved. This documentation is the property of Fair Isaac Corporation ("FICO"). Receipt or possession of this documentation does not convey rights to disclose, reproduce, make derivative works, use, or allow others to use it except solely for internal evaluation purposes to determine whether to purchase a license to the software described in this documentation, or as otherwise set forth in a written software license agreement between you and FICO (or a FICO affiliate). Use of this documentation and the software described in it must conform strictly to the foregoing permitted uses, and no other use is permitted. The information in this documentation is subject to change without notice. If you find any problems in this documentation, please report them to us in writing. 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Xpress Optimizer Deliverable Version: A Last Revised: 02 October 2020 Version 37.01 Contents 1 Introduction 1 1.1 The FICO Xpress Optimizer....................................1 1.2 Starting the First Time.......................................2 1.2.1 Licensing..........................................2 1.2.2 Starting Console Optimizer................................2 1.2.3 Scripting Console Optimizer...............................3 1.2.4 Interrupting Console Optimizer..............................5 1.3 Manual Layout...........................................5 2 Basic Usage 6 2.1 Initialization............................................6 2.2 The Problem Pointer........................................7 2.3 Logging...............................................7 2.4 Problem Loading..........................................8 2.5 Problem Solving..........................................8 2.6 Interrupting the Solve.......................................9 2.7 Results Processing........................................ 10 2.8 Function Quick Reference..................................... 11 2.8.1 Administration....................................... 11 2.8.2 Problem Loading..................................... 11 2.8.3 Problem Solving...................................... 11 2.8.4 Results Processing.................................... 12 2.9 Summary.............................................. 12 3 Problem Types 13 3.1 Linear Programs (LPs)...................................... 13 3.2 Mixed Integer Programs (MIPs)................................. 13 3.3 Quadratic Programs (QPs).................................... 14 3.4 Quadratically Constrained Quadratic Programs (QCQPs)................... 15 3.4.1 Algebraic and matrix form................................ 15 3.4.2 Convexity.......................................... 15 3.4.3 Characterizing Convexity in Quadratic Constraints................... 16 3.5 Second Order Cone problems (SOCPs)............................. 16 4 Solution Methods 18 4.1 Simplex Method.......................................... 18 4.1.1 Output........................................... 19 4.2 Newton Barrier Method...................................... 19 4.2.1 Crossover......................................... 19 4.2.2 Output........................................... 20 4.3 Branch and Bound......................................... 20 4.3.1 Theory........................................... 20 4.3.2 Variable Selection and Cutting.............................. 22 4.3.3 Variable Selection for Branching............................. 22 4.3.4 Cutting Planes....................................... 23 4.3.5 Node Selection....................................... 23 Fair Isaac Corporation Confidential and Proprietary Information i Contents 4.3.6 Adjusting the Cutoff Value................................ 24 4.3.7 Stopping Criteria...................................... 24 4.3.8 Integer Preprocessing................................... 24 4.4 QCQP and SOCP Methods.................................... 25 4.4.1 Convexity Checking.................................... 25 4.4.2 Quadratically Constrained and Second Order Cone Problems............ 26 5 Advanced Usage 27 5.1 Problem Names.......................................... 27 5.2 Manipulating the Matrix...................................... 27 5.2.1 Reading the Matrix.................................... 28 5.2.2 Modifying the Matrix................................... 28 5.3 Working with Presolve....................................... 29 5.3.1 (Mixed) Integer Programming Problems........................ 29 5.4 Working with LP Folding..................................... 30 5.5 Working with Heuristics...................................... 30 5.6 Analyzing and Handling Numerical Issues........................... 31 5.6.1 Analyzing Models for Numerical Issues......................... 32 5.6.2 Scaling........................................... 32 5.6.3 Solution Refinement.................................... 32 5.6.4 Other Ways to Handle Numerical Issues........................ 33 5.7 Common Causes of Confusion.................................. 33 5.8 Using the Callbacks........................................ 33 5.8.1 Output Callbacks..................................... 34 5.8.2 LP Callbacks........................................ 34 5.8.3 Global Search Callbacks................................. 34 5.9 Working with the Cut Manager.................................. 35 5.9.1 Cuts and the Cut Pool................................... 35 5.9.2 Cut Management Routines................................ 35 5.9.3 User Cut Manager Routines............................... 36 5.10 Solving Problems Using Multiple Threads............................ 36 5.10.1 The concurrent solver................................... 37 5.11 Solving Large Models (the 64 bit Functions).......................... 38 5.12 Using the Tuner.......................................... 38 5.12.1 Basic Usage........................................ 38 5.12.2 The Tuner Method..................................... 39 5.12.3 The Tuner Output..................................... 40 5.12.4 The Tuner Target..................................... 40 5.12.5 Restarting the Tuner.................................... 40 5.12.6 Tuner with Multiple Threads............................... 40 5.12.7 Tuner with Problem Permutations............................ 41 5.12.8 Tuning a Set of Problems................................. 41 5.12.9 Advanced Topics..................................... 42 5.13 Remote Solving with Xpress Insight Compute Interface.................... 42 5.13.1 Authentication....................................... 43 5.13.2 Callbacks.......................................... 43 5.13.3 Licensing.......................................... 43 5.13.4 Advanced Configuration................................. 43 6 Infeasibility, Unboundedness and Instability 44 6.1 Infeasibility............................................. 44 6.1.1 Diagnosis in Presolve................................... 44 6.1.2 Diagnosis using Primal Simplex............................. 45 6.1.3 Irreducible Infeasible Sets................................ 45 6.1.4 The Infeasibility Repair Utility............................... 46 Fair Isaac Corporation Confidential and Proprietary Information ii Contents 6.1.5 Integer Infeasibility.................................... 47 6.2 Unboundedness.......................................... 48 6.3 Instability.............................................. 48 6.3.1 Scaling........................................... 48 6.3.2 Accuracy.......................................... 49 7 Goal Programming 51 7.1 Overview.............................................. 51 7.2 Pre–emptive Goal Programming Using Constraints...................... 51 7.3 Archimedean Goal Programming Using Constraints...................... 52 7.4 Pre–emptive Goal Programming Using Objective Functions................. 52 7.5 Archimedean Goal Programming Using Objective Functions................. 53 8 Console and Library Functions 55 8.1 Console Mode Functions..................................... 55 8.2 Layout for Function Descriptions................................. 57 Function Name........................................... 57 Purpose..............................................