DOCSLIB.ORG

Xpress Nonlinear Manual This Material Is the Confidential, Proprietary, and Unpublished Property of Fair Isaac Corporation

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

R FICO Xpress Optimization Last update 01 May, 2018 version 33.01 FICO R Xpress Nonlinear Manual This material is the confidential, proprietary, and unpublished property of Fair Isaac Corporation. Receipt or possession of this material does not convey rights to divulge, reproduce, use, or allow others to use it without the specific written authorization of Fair Isaac Corporation and use must conform strictly to the license agreement. The information in this document is subject to change without notice. If you find any problems in this documentation, please report them to us in writing. Neither Fair Isaac Corporation nor its affiliates warrant that this documentation is error-free, nor are there any other warranties with respect to the documentation except as may be provided in the license agreement. ©1983–2018 Fair Isaac Corporation. All rights reserved. Permission to use this software and its documentation is governed by the software license agreement between the licensee and Fair Isaac Corporation (or its affiliate). Portions of the program may contain copyright of various authors and may be licensed under certain third-party licenses identified in the software, documentation, or both. In no event shall Fair Isaac Corporation or its affiliates be liable to any person for direct, indirect, special, incidental, or consequential damages, including lost profits, arising out of the use of this software and its documentation, even if Fair Isaac Corporation or its affiliates have been advised of the possibility of such damage. The rights and allocation of risk between the licensee and Fair Isaac Corporation (or its affiliates) are governed by the respective identified licenses in the software, documentation, or both. Fair Isaac Corporation and its affiliates specifically disclaim any warranties, including, but not limited to, the implied warranties of merchantability and fitness for a particular purpose. The software and accompanying documentation, if any, provided hereunder is provided solely to users licensed under the Fair Isaac Software License Agreement. Fair Isaac Corporation and its affiliates have no obligation to provide maintenance, support, updates, enhancements, or modifications except as required to licensed users under the Fair Isaac Software License Agreement. FICO and Fair Isaac are trademarks or registered trademarks of Fair Isaac Corporation in the United States and may be trademarks or registered trademarks of Fair Isaac Corporation in other countries. Other product and company names herein may be trademarks of their respective owners. Xpress Nonlinear Deliverable Version: A Last Revised: 01 May, 2018 Version version 33.01 Contents I Overview1 1 Introduction 2 1.1 Mathematical programs......................................2 1.1.1 Linear programs......................................3 1.1.2 Convex quadratic programs...............................3 1.1.3 Convex quadratically constrained quadratic programs................3 1.1.4 Second order conic problems..............................3 1.1.5 General nonlinear optimization problems........................4 1.1.6 Mixed integer programs..................................4 1.2 Technology Overview.......................................4 1.2.1 The Simplex Method...................................4 1.2.2 The Logarithmic Barrier Method.............................4 1.2.3 Outer approximation schemes..............................5 1.2.4 Successive Linear Programming.............................5 1.2.5 Second Order Methods..................................5 1.2.6 Mixed Integer Solvers...................................5 1.3 API naming convention......................................5 2 The Problem 7 2.1 Problem Definition.........................................7 2.2 Problem Formulation.......................................7 3 Modeling in Mosel 9 3.1 Basic formulation.........................................9 3.2 Setting up and solving the problem............................... 10 3.3 Looking at the results....................................... 11 3.4 Parallel evaluation of Mosel user functions........................... 11 4 Modeling in Extended MPS Format 13 4.1 Basic formulation......................................... 13 4.2 Using the XSLP console-based interface............................ 17 4.3 Coefficients and terms...................................... 18 4.4 User functions........................................... 18 4.4.1 A user function in an Excel macro............................ 19 4.4.2 Extending the polygon model.............................. 19 5 The Xpress NonLinear API Functions 21 5.1 Header files............................................. 21 5.2 Initialization............................................ 21 5.3 Callbacks.............................................. 21 5.4 Creating the linear part of the problem............................. 22 5.5 Adding the non-linear part of the problem............................ 25 5.6 Adding the non-linear part of the problem using character formulae............. 27 5.7 Checking the data......................................... 28 Fair Isaac Corporation Confidential and Proprietary Information i Contents 5.8 Solving and printing the solution................................. 28 5.9 Closing the program........................................ 29 5.10 Adding initial values........................................ 29 5.11 User functions........................................... 30 5.11.1 A user function in C.................................... 30 5.11.2 Extending the polygon model.............................. 31 5.11.3 Internal user functions.................................. 32 6 The XSLP Console Program 33 6.1 The Console XSLP........................................ 33 6.1.1 The XSLP console extensions.............................. 33 6.1.2 Common features of the Xpress Optimizer and the Xpress XSLP console..... 34 II Advanced 36 7 Nonlinear Problems 37 7.1 Coefficients and terms...................................... 37 7.2 SLP variables............................................ 38 7.3 Local and global optimality.................................... 38 7.4 Convexity.............................................. 38 7.5 Converged and practical solutions................................ 39 7.6 The duals of general, nonlinear program............................ 39 8 Extended MPS file format 41 8.1 Formulae.............................................. 41 8.2 COLUMNS............................................. 42 8.3 BOUNDS.............................................. 43 8.4 SLPDATA.............................................. 43 8.4.1 CV (Character variable).................................. 43 8.4.2 DR (Determining row)................................... 44 8.4.3 EC (Enforced constraint)................................. 44 8.4.4 FR (Free variable)..................................... 44 8.4.5 FX (Fixed variable)..................................... 45 8.4.6 IV (Initial value)...................................... 45 8.4.7 LO (Lower bounded variable)............................... 45 8.4.8 Rx, Tx (Relative and absolute convergence tolerances)................ 45 8.4.9 SB (Initial step bound).................................. 46 8.4.10 UF (User function)..................................... 46 8.4.11 UP (Free variable)..................................... 47 8.4.12 WT (Explicit row weight)................................. 47 8.4.13 DL (variable specific Determining row cascade iteration Limit)............ 47 9 Xpress-SLP Solution Process 48 9.1 Analyzing the solution process.................................. 49 9.2 The initial point........................................... 49 9.3 Derivatives............................................. 49 9.3.1 Finite Differences..................................... 50 9.3.2 Symbolic Differentiation................................. 50 9.3.3 Automatic Differentiation................................. 50 9.4 Points of inflection......................................... 50 9.5 Trust regions............................................ 51 10 Handling Infeasibilities 52 10.1 Infeasibility Analysis in the Xpress Optimizer.......................... 52 10.2 Managing Infeasibility with Xpress Knitro............................ 52 Fair Isaac Corporation Confidential and Proprietary Information ii Contents 10.3 Managing Infeasibility with Xpress-SLP............................. 53 10.4 Penalty Infeasibility Breakers in XSLP.............................. 53 11 Cascading 55 11.1 Determining rows and determining columns.......................... 56 12 Convergence criteria 57 12.1 Convergence criteria........................................ 57 12.2 Convergence overview...................................... 57 12.2.1 Strict Convergence.................................... 57 12.2.2 Extended Convergence.................................. 57 12.2.3 Stopping Criterion..................................... 58 12.2.4 Step Bounding....................................... 59 12.3 Convergence: technical details.................................. 59 12.3.1 Closure tolerance (CTOL)................................. 62 12.3.2 Delta tolerance (ATOL).................................. 62 12.3.3 Matrix tolerance (MTOL)................................. 62 12.3.4 Impact tolerance (ITOL).................................. 63 12.3.5 Slack impact tolerance (STOL).............................. 64 12.3.6 Fixed variables due to determining columns smaller than threshold (FX)...... 64 12.3.7 User-defined convergence...............................
Recommended publications
  • Solving Mixed Integer Linear and Nonlinear Problems Using the SCIP Optimization Suite

    Solving Mixed Integer Linear and Nonlinear Problems Using the SCIP Optimization Suite

    Takustraße 7 Konrad-Zuse-Zentrum D-14195 Berlin-Dahlem fur¨ Informationstechnik Berlin Germany TIMO BERTHOLD GERALD GAMRATH AMBROS M. GLEIXNER STEFAN HEINZ THORSTEN KOCH YUJI SHINANO Solving mixed integer linear and nonlinear problems using the SCIP Optimization Suite Supported by the DFG Research Center MATHEON Mathematics for key technologies in Berlin. ZIB-Report 12-27 (July 2012) Herausgegeben vom Konrad-Zuse-Zentrum f¨urInformationstechnik Berlin Takustraße 7 D-14195 Berlin-Dahlem Telefon: 030-84185-0 Telefax: 030-84185-125 e-mail: [email protected] URL: http://www.zib.de ZIB-Report (Print) ISSN 1438-0064 ZIB-Report (Internet) ISSN 2192-7782 Solving mixed integer linear and nonlinear problems using the SCIP Optimization Suite∗ Timo Berthold Gerald Gamrath Ambros M. Gleixner Stefan Heinz Thorsten Koch Yuji Shinano Zuse Institute Berlin, Takustr. 7, 14195 Berlin, Germany, fberthold,gamrath,gleixner,heinz,koch,[email protected] July 31, 2012 Abstract This paper introduces the SCIP Optimization Suite and discusses the ca- pabilities of its three components: the modeling language Zimpl, the linear programming solver SoPlex, and the constraint integer programming frame- work SCIP. We explain how these can be used in concert to model and solve challenging mixed integer linear and nonlinear optimization problems. SCIP is currently one of the fastest non-commercial MIP and MINLP solvers. We demonstrate the usage of Zimpl, SCIP, and SoPlex by selected examples, give an overview of available interfaces, and outline plans for future development. ∗A Japanese translation of this paper will be published in the Proceedings of the 24th RAMP Symposium held at Tohoku University, Miyagi, Japan, 27{28 September 2012, see http://orsj.or.
  • Hydraulic Optimization Demonstration for Groundwater Pump

    Hydraulic Optimization Demonstration for Groundwater Pump

    Figure 5-18: Shallow Particles, Contain Shallow 20-ppb Plume, & 500 gpm for Deep 20-ppb plume (1573 gpm, 3 new wells, 1 existing well) Injection Well Well Layer 1 Well Layer 2 30000 25000 20000 15000 10000 5000 0 0 5000 10000 15000 A "+" symbol indicates that a particle starting at that location is captured by one of the remediation wells, based on particle tracking with MODPATH. Shallow particles originate half-way down in layer 1. Figure 5-19: Deep Particles, Contain Shallow 20-ppb Plume, & 500 gpm for Deep 20-ppb plume (1573 gpm, 3 new wells, 1 existing well) Injection Well Well Layer 1 Well Layer 2 30000 25000 20000 15000 10000 5000 0 0 5000 10000 15000 A "+" symbol indicates that a particle starting at that location is captured by one of the remediation wells, based on particle tracking with MODPATH. Deep particles originate half-way down in layer 2. Figure 5-20: Shallow Particles, Contain Shallow 20-ppb & 50-ppb Plumes, & 500 gpm for Deep 20-ppb plume (2620 gpm, 6 new wells, 0 existing wells) Injection Well Well Layer 1 Well Layer 2 30000 25000 20000 15000 10000 5000 0 0 5000 10000 15000 A "+" symbol indicates that a particle starting at that location is captured by one of the remediation wells, based on particle tracking with MODPATH. Shallow particles originate half-way down in layer 1. Figure 5-21: Deep Particles, Contain Shallow 20-ppb & 50-ppb Plumes, & 500 gpm for Deep 20-ppb plume (2620 gpm, 6 new wells, 0 existing wells) Injection Well Well Layer 1 Well Layer 2 30000 25000 20000 15000 10000 5000 0 0 5000 10000 15000 A "+" symbol indicates that a particle starting at that location is captured by one of the remediation wells, based on particle tracking with MODPATH.
  • Numericaloptimization

    Numericaloptimization

    Numerical Optimization Alberto Bemporad http://cse.lab.imtlucca.it/~bemporad/teaching/numopt Academic year 2020-2021 Course objectives Solve complex decision problems by using numerical optimization Application domains: • Finance, management science, economics (portfolio optimization, business analytics, investment plans, resource allocation, logistics, ...) • Engineering (engineering design, process optimization, embedded control, ...) • Artificial intelligence (machine learning, data science, autonomous driving, ...) • Myriads of other applications (transportation, smart grids, water networks, sports scheduling, health-care, oil & gas, space, ...) ©2021 A. Bemporad - Numerical Optimization 2/102 Course objectives What this course is about: • How to formulate a decision problem as a numerical optimization problem? (modeling) • Which numerical algorithm is most appropriate to solve the problem? (algorithms) • What’s the theory behind the algorithm? (theory) ©2021 A. Bemporad - Numerical Optimization 3/102 Course contents • Optimization modeling – Linear models – Convex models • Optimization theory – Optimality conditions, sensitivity analysis – Duality • Optimization algorithms – Basics of numerical linear algebra – Convex programming – Nonlinear programming ©2021 A. Bemporad - Numerical Optimization 4/102 References i ©2021 A. Bemporad - Numerical Optimization 5/102 Other references • Stephen Boyd’s “Convex Optimization” courses at Stanford: http://ee364a.stanford.edu http://ee364b.stanford.edu • Lieven Vandenberghe’s courses at UCLA: http://www.seas.ucla.edu/~vandenbe/ • For more tutorials/books see http://plato.asu.edu/sub/tutorials.html ©2021 A. Bemporad - Numerical Optimization 6/102 Optimization modeling What is optimization? • Optimization = assign values to a set of decision variables so to optimize a certain objective function • Example: Which is the best velocity to minimize fuel consumption ? fuel [ℓ/km] velocity [km/h] 0 30 60 90 120 160 ©2021 A.
  • Using the COIN-OR Server

    Using the COIN-OR Server

    Using the COIN-OR Server Your CoinEasy Team November 16, 2009 1 1 Overview This document is part of the CoinEasy project. See projects.coin-or.org/CoinEasy. In this document we describe the options available to users of COIN-OR who are interested in solving opti- mization problems but do not wish to compile source code in order to build the COIN-OR projects. In particular, we show how the user can send optimization problems to a COIN-OR server and get the solution result back. The COIN-OR server, webdss.ise.ufl.edu, is 2x Intel(R) Xeon(TM) CPU 3.06GHz 512MiB L2 1024MiB L3, 2GiB DRAM, 4x73GiB scsi disk 2xGigE machine. This server allows the user to directly access the following COIN-OR optimization solvers: • Bonmin { a solver for mixed-integer nonlinear optimization • Cbc { a solver for mixed-integer linear programs • Clp { a linear programming solver • Couenne { a solver for mixed-integer nonlinear optimization problems and is capable of global optiomization • DyLP { a linear programming solver • Ipopt { an interior point nonlinear optimization solver • SYMPHONY { mixed integer linear solver that can be executed in either parallel (dis- tributed or shared memory) or sequential modes • Vol { a linear programming solver All of these solvers on the COIN-OR server may be accessed through either the GAMS or AMPL modeling languages. In Section 2.1 we describe how to use the solvers using the GAMS modeling language. In Section 2.2 we describe how to call the solvers using the AMPL modeling language. In Section 3 we describe how to call the solvers using a command line executable pro- gram OSSolverService.exe (or OSSolverService for Linux/Mac OS X users { in the rest of the document we refer to this executable using a .exe extension).
  • SDDP.Jl: a Julia Package for Stochastic Dual Dynamic Programming

    SDDP.Jl: a Julia Package for Stochastic Dual Dynamic Programming

    Optimization Online manuscript No. (will be inserted by the editor) SDDP.jl: a Julia package for Stochastic Dual Dynamic Programming Oscar Dowson · Lea Kapelevich Received: date / Accepted: date Abstract In this paper we present SDDP.jl, an open-source library for solving multistage stochastic optimization problems using the Stochastic Dual Dynamic Programming algorithm. SDDP.jl is built upon JuMP, an algebraic modelling lan- guage in Julia. This enables a high-level interface for the user, while simultaneously providing performance that is similar to implementations in low-level languages. We benchmark the performance of SDDP.jl against a C++ implementation of SDDP for the New Zealand Hydro-Thermal Scheduling Problem. On the bench- mark problem, SDDP.jl is approximately 30% slower than the C++ implementa- tion. However, this performance penalty is small when viewed in context of the generic nature of the SDDP.jl library compared to the single purpose C++ imple- mentation. Keywords julia · stochastic dual dynamic programming 1 Introduction Solving any mathematical optimization problem requires four steps: the formula- tion of the problem by the user; the communication of the problem to the com- puter; the efficient computational solution of the problem; and the communication of the computational solution back to the user. Over time, considerable effort has been made to improve each of these four steps for a variety of problem classes such linear, quadratic, mixed-integer, conic, and non-linear. For example, con- sider the evolution from early file-formats such as MPS [25] to modern algebraic modelling languages embedded in high-level languages such as JuMP [11], or the 73-fold speed-up in solving difficult mixed-integer linear programs in seven years by Gurobi [16].
  • ILOG AMPL CPLEX System Version 10.0 User's Guide

    ILOG AMPL CPLEX System Version 10.0 User's Guide

    ILOG AMPL CPLEX System Version 10.0 User’s Guide Standard (Command-line) Version Including CPLEX Directives January 2006 COPYRIGHT NOTICE Copyright © 1987-2006, by ILOG S.A., 9 Rue de Verdun, 94253 Gentilly Cedex, France, and ILOG, Inc., 1080 Linda Vista Ave., Mountain View, California 94043, USA. All rights reserved. General Use Restrictions This document and the software described in this document are the property of ILOG and are protected as ILOG trade secrets. They are furnished under a license or nondisclosure agreement, and may be used or copied only within the terms of such license or nondisclosure agreement. No part of this work may be reproduced or disseminated in any form or by any means, without the prior written permission of ILOG S.A, or ILOG, Inc. Trademarks ILOG, the ILOG design, CPLEX, and all other logos and product and service names of ILOG are registered trademarks or trademarks of ILOG in France, the U.S. and/or other countries. All other company and product names are trademarks or registered trademarks of their respective holders. Java and all Java-based marks are either trademarks or registered trademarks of Sun Microsystems, Inc. in the United States and other countries. Microsoft, Windows, and Windows NT are either trademarks or registered trademarks of Microsoft Corporation in the United States and other countries. document version 10.0 CO N T E N T S Table of Contents Chapter 1 Welcome to AMPL . 9 Using this Guide. .9 Installing AMPL . .10 Requirements. .10 Unix Installation . .10 Windows Installation . .11 AMPL and Parallel CPLEX.
  • Notes 1: Introduction to Optimization Models

    Notes 1: Introduction to Optimization Models

    Notes 1: Introduction to Optimization Models IND E 599 September 29, 2010 IND E 599 Notes 1 Slide 1 Course Objectives I Survey of optimization models and formulations, with focus on modeling, not on algorithms I Include a variety of applications, such as, industrial, mechanical, civil and electrical engineering, financial optimization models, health care systems, environmental ecology, and forestry I Include many types of optimization models, such as, linear programming, integer programming, quadratic assignment problem, nonlinear convex problems and black-box models I Include many common formulations, such as, facility location, vehicle routing, job shop scheduling, flow shop scheduling, production scheduling (min make span, min max lateness), knapsack/multi-knapsack, traveling salesman, capacitated assignment problem, set covering/packing, network flow, shortest path, and max flow. IND E 599 Notes 1 Slide 2 Tentative Topics Each topic is an introduction to what could be a complete course: 1. basic linear models (LP) with sensitivity analysis 2. integer models (IP), such as the assignment problem, knapsack problem and the traveling salesman problem 3. mixed integer formulations 4. quadratic assignment problems 5. include uncertainty with chance-constraints, stochastic programming scenario-based formulations, and robust optimization 6. multi-objective formulations 7. nonlinear formulations, as often found in engineering design 8. brief introduction to constraint logic programming 9. brief introduction to dynamic programming IND E 599 Notes 1 Slide 3 Computer Software I Catalyst Tools (https://catalyst.uw.edu/) I AIMMS - optimization software (http://www.aimms.com/) Ming Fang - AIMMS software consultant IND E 599 Notes 1 Slide 4 What is Mathematical Programming? Mathematical programming refers to \programming" as a \planning" activity: as in I linear programming (LP) I integer programming (IP) I mixed integer linear programming (MILP) I non-linear programming (NLP) \Optimization" is becoming more common, e.g.
  • ILOG AMPL CPLEX System Version 9.0 User's Guide

    ILOG AMPL CPLEX System Version 9.0 User's Guide

    ILOG AMPL CPLEX System Version 9.0 User’s Guide Standard (Command-line) Version Including CPLEX Directives September 2003 Copyright © 1987-2003, by ILOG S.A. All rights reserved. ILOG, the ILOG design, CPLEX, and all other logos and product and service names of ILOG are registered trademarks or trademarks of ILOG in France, the U.S. and/or other countries. JavaTM and all Java-based marks are either trademarks or registered trademarks of Sun Microsystems, Inc. in the United States and other countries. Microsoft, Windows, and Windows NT are either trademarks or registered trademarks of Microsoft Corporation in the U.S. and other countries. All other brand, product and company names are trademarks or registered trademarks of their respective holders. AMPL is a registered trademark of AMPL Optimization LLC and is distributed under license by ILOG.. CPLEX is a registered trademark of ILOG.. Printed in France. CO N T E N T S Table of Contents Chapter 1 Introduction . 1 Welcome to AMPL . .1 Using this Guide. .1 Installing AMPL . .2 Requirements. .2 Unix Installation . .3 Windows Installation . .3 AMPL and Parallel CPLEX. 4 Licensing . .4 Usage Notes . .4 Installed Files . .5 Chapter 2 Using AMPL. 7 Running AMPL . .7 Using a Text Editor . .7 Running AMPL in Batch Mode . .8 Chapter 3 AMPL-Solver Interaction . 11 Choosing a Solver . .11 Specifying Solver Options . .12 Initial Variable Values and Solvers. .13 ILOG AMPL CPLEX SYSTEM 9.0 — USER’ S GUIDE v Problem and Solution Files. .13 Saving temporary files . .14 Creating Auxiliary Files . .15 Running Solvers Outside AMPL. .16 Using MPS File Format .
  • Open Source Tools for Optimization in Python

    Open Source Tools for Optimization in Python

    Open Source Tools for Optimization in Python Ted Ralphs Sage Days Workshop IMA, Minneapolis, MN, 21 August 2017 T.K. Ralphs (Lehigh University) Open Source Optimization August 21, 2017 Outline 1 Introduction 2 COIN-OR 3 Modeling Software 4 Python-based Modeling Tools PuLP/DipPy CyLP yaposib Pyomo T.K. Ralphs (Lehigh University) Open Source Optimization August 21, 2017 Outline 1 Introduction 2 COIN-OR 3 Modeling Software 4 Python-based Modeling Tools PuLP/DipPy CyLP yaposib Pyomo T.K. Ralphs (Lehigh University) Open Source Optimization August 21, 2017 Caveats and Motivation Caveats I have no idea about the background of the audience. The talk may be either too basic or too advanced. Why am I here? I’m not a Sage developer or user (yet!). I’m hoping this will be a chance to get more involved in Sage development. Please ask lots of questions so as to guide me in what to dive into! T.K. Ralphs (Lehigh University) Open Source Optimization August 21, 2017 Mathematical Optimization Mathematical optimization provides a formal language for describing and analyzing optimization problems. Elements of the model: Decision variables Constraints Objective Function Parameters and Data The general form of a mathematical optimization problem is: min or max f (x) (1) 8 9 < ≤ = s.t. gi(x) = bi (2) : ≥ ; x 2 X (3) where X ⊆ Rn might be a discrete set. T.K. Ralphs (Lehigh University) Open Source Optimization August 21, 2017 Types of Mathematical Optimization Problems The type of a mathematical optimization problem is determined primarily by The form of the objective and the constraints.
  • Benchmarks for Current Linear and Mixed Integer Optimization Solvers

    Benchmarks for Current Linear and Mixed Integer Optimization Solvers

    ACTA UNIVERSITATIS AGRICULTURAE ET SILVICULTURAE MENDELIANAE BRUNENSIS Volume 63 207 Number 6, 2015 http://dx.doi.org/10.11118/actaun201563061923 BENCHMARKS FOR CURRENT LINEAR AND MIXED INTEGER OPTIMIZATION SOLVERS Josef Jablonský1 1 Department of Econometrics, Faculty of Informatics and Statistics, University of Economics, Prague, nám. W. Churchilla 4, 130 67 Praha 3, Czech Republic Abstract JABLONSKÝ JOSEF. 2015. Benchmarks for Current Linear and Mixed Integer Optimization Solvers. Acta Universitatis Agriculturae et Silviculturae Mendelianae Brunensis, 63(6): 1923–1928. Linear programming (LP) and mixed integer linear programming (MILP) problems belong among very important class of problems that fi nd their applications in various managerial consequences. The aim of the paper is to discuss computational performance of current optimization packages for solving large scale LP and MILP optimization problems. Current market with LP and MILP solvers is quite extensive. Probably among the most powerful solvers GUROBI 6.0, IBM ILOG CPLEX 12.6.1, and XPRESS Optimizer 27.01 belong. Their attractiveness for academic research is given, except their computational performance, by their free availability for academic purposes. The solvers are tested on the set of selected problems from MIPLIB 2010 library that contains 361 test instances of diff erent hardness (easy, hard, and not solved). Keywords: benchmark, linear programming, mixed integer linear programming, optimization, solver INTRODUCTION with integer variables need not be solved even Solving linear and mixed integer linear in case of a very small size of the given problem. optimization problems (LP and MILP) that Real-world optimization problems have usually belong to one of the most o en modelling tools, many thousands of variables and/or constraints.
  • Computational Aspects of Infeasibility Analysis in Mixed Integer Programming

    Computational Aspects of Infeasibility Analysis in Mixed Integer Programming

    Takustr. 7 Zuse Institute Berlin 14195 Berlin Germany JAKOB WITZIG TIMO BERTHOLD STEFAN HEINZ Computational Aspects of Infeasibility Analysis in Mixed Integer Programming ZIB Report 19-54 (November 2019) Zuse Institute Berlin Takustr. 7 14195 Berlin Germany Telephone: +49 30-84185-0 Telefax: +49 30-84185-125 E-mail: [email protected] URL: http://www.zib.de ZIB-Report (Print) ISSN 1438-0064 ZIB-Report (Internet) ISSN 2192-7782 Computational Aspects of Infeasibility Analysis in Mixed Integer Programming Jakob Witzig,1 Timo Berthold,2 and Stefan Heinz3 1Zuse Institute Berlin, Takustr. 7, 14195 Berlin, Germany [email protected] 2Fair Isaac Germany GmbH, Stubenwald-Allee 19, 64625 Bensheim, Germany [email protected] 3Gurobi GmbH, Ulmenstr. 37–39, 60325 Frankfurt am Main, Germany [email protected] November 6, 2019 Abstract The analysis of infeasible subproblems plays an important role in solv- ing mixed integer programs (MIPs) and is implemented in most major MIP solvers. There are two fundamentally different concepts to gener- ate valid global constraints from infeasible subproblems. The first is to analyze the sequence of implications, obtained by domain propagation, that led to infeasibility. The result of this analysis is one or more sets of contradicting variable bounds from which so-called conflict constraints can be generated. This concept is called conflict graph analysis and has its origin in solving satisfiability problems and is similarly used in con- straint programming. The second concept is to analyze infeasible linear programming (LP) relaxations. Every ray of the dual LP provides a set of multipliers that can be used to generate a single new globally valid linear constraint.
  • Computing in Operations Research Using Julia

    Computing in Operations Research Using Julia

    Computing in Operations Research using Julia Miles Lubin and Iain Dunning MIT Operations Research Center INFORMS 2013 { October 7, 2013 1 / 25 High-level, high-performance, open-source dynamic language for technical computing. Keep productivity of dynamic languages without giving up speed. Familiar syntax Python+PyPy+SciPy+NumPy integrated completely. Latest concepts in programming languages. 2 / 25 Claim: \close-to-C" speeds Within a factor of 2 Performs well on microbenchmarks, but how about real computational problems in OR? Can we stop writing solvers in C++? 3 / 25 Technical advancements in Julia: Fast code generation (JIT via LLVM). Excellent connections to C libraries - BLAS/LAPACK/... Metaprogramming. Optional typing, multiple dispatch. 4 / 25 Write generic code, compile efficient type-specific code C: (fast) int f() { int x = 1, y = 2; return x+y; } Julia: (No type annotations) Python: (slow) function f() def f(): x = 1; y = 2 x = 1; y = 2 return x + y return x+y end 5 / 25 Requires type inference by compiler Difficult to add onto exiting languages Available in MATLAB { limited scope PyPy for Python { incompatible with many libraries Julia designed from the ground up to support type inference efficiently 6 / 25 Simplex algorithm \Bread and butter" of operations research Computationally very challenging to implement efficiently1 Matlab implementations too slow to be used in practice High-quality open-source codes exist in C/C++ Can Julia compete? 1Bixby, Robert E. "Solving Real-World Linear Programs: A Decade and More of Progress", Operations Research, Vol. 50, pp. 3{15, 2002. 7 / 25 Implemented benchmark operations in Julia, C++, MATLAB, Python.