Decomposition Methods for Mathematical Programming Problems. GAMS Software GmbH / GAMS Development Corp.

Transcription

1 Decomposition Methods for Mathematical Programming Problems Michael R. Bussieck Stefan Vigerske GAMS Software GmbH / GAMS Development Corp. Aachen, June 11, 2013

2 Agenda Motivation GAMS Simple Example 2

3 Mathematical Programming/Optimization 3 en.wikipedia.org/wiki/mathematical_optimization

4 Major Subfields of Math. Optimization

5 5 Linear Programming

6 Typical Application Areas * Agricultural Economics Chemical Engineering Econometrics Environmental Economics Finance International Trade Macro Economics Management Science/OR Micro Economics Applied General Equilibrium Economic Development Energy Engineering Forestry Logistics Military Mathematics Physics 6 * Illustrative examples in the GAMS Model Library

7 Modeling in GAMS A mathematical model is a description of a system using mathematical language (from: Wikipedia)

8 INFORMS Lanchester Prize Frederick W. Lanchester Prize: Winner [-hide] 8 Citation: Multi-Level Planning: Case Studies in Mexico, edited by Louis M. Goreux and Alan S. Manne. This book is a bench mark quantitative study of policy oriented issues in a growing economy. In the modern tradition of Professor W. Leontief's inputoutput analysis, a team of researchers from several institutions employed advanced mathematical programming approaches to study in depth the problems of interdependency among national economic choices. This monograph on multi-level planning is impressive in its dedication to developing and testing large-scale models based on available statistical data. The team's decision a half-decade ago to give special emphasis to the agricultural and energy sectors was prophetic in anticipating many of today's critical world-wide problems. Beyond its substantial contribution to empirical analysis, the book also enhances conceptual understanding of multi-level national planning as well as demonstrates the benefits to strategic policy analysis of continuing technical innovations in operations research.

9 9 Model Structure

10 10 Model Data

11 11 Matrix Generator

12 GAMS (General) Algebraic Modeling System What s that? Formulation mathematical programming problems Notation similar to algebraic notation Ready-for-use links to state-of-the-art algorithms Simplified model building Efficient solution process

13 What is a Model in GAMS? A mathematical model is a description of a system using mathematical language (from: Wikipedia) Mathematical Programming (MP) Model: List of Constraints defining relationships of Variables Collection of several intertwined MP Models Data Preparation Data Calibration Solution Module (e.g. sequential, parallel, loop) Report Module 13

14 Agenda Motivation GAMS Simple Example 14

15 General Algebraic Modeling System Roots: World Bank, 1976 Went commercial in 1987 GAMS Development Corporation (Washington, Houston) GAMS Software GmbH (Köln, Braunschweig) Broad academic & commercial user community and network

16 GAMS Fundamental concepts Platform independence Hassle-free switch of solution methods 10+ Supported Platforms Solaris Open architecture and interfaces to other systems Balanced mix of declarative and procedural elements AXU Windows 64bit Linux Linux 64bit Windows HP 16

17 GAMS Fundamental concepts Platform independence Hassle-free switch of solution methods Open architecture and interfaces to other systems Balanced mix of declarative and procedural elements 25+ Integrated Solvers XPRESS XA COIN-OR BARON LINDOGLOBAL CPLEX BDMLP GUROBI 17

18 GAMS Fundamental concepts Platform independence Binary Data Exchange Hassle-free switch of solution methods Application GDX GAMS SOLVER Open architecture and interfaces to other systems Balanced mix of declarative and procedural elements Fast exchange of data Syntactical check on data before model starts Data Exchange at any stage (Compile and Run-time) Platform Independent Direct GDX interfaces and general API Scenario Management Support Full Support of Batch Runs 18

19 GAMS Fundamental concepts Platform independence Hassle-free switch of solution methods Open architecture and interfaces to other systems Balanced mix of declarative and procedural elements Declaration of.. - Sets - Parameters - Variables - Equations - Models - Procedural Elements like - loops - if-then-else - 19

20 GAMS Fundamental concepts cont d Different layers with separation of model and data model and solution methods model and operating system model and interface Interface Data Model Solver Interface Models benefit from advancing hardware enhanced / new solver technology improved / upcoming interfaces to other systems 20

21 GAMS at a Glance The GAMS/BASE Module Compiler and Execution System GAMS IDE (Windows) Documentation + Model libraries GDX Utilities Free Solvers/Solver Links 21

22 GAMS at a Glance The GAMS/BASE Module Compiler and Execution System GAMS IDE (Windows) Documentation + Model libraries GDX Utilities Free Solvers/Solver Links 22

23 Integrated Development Environment Project management Editor / Syntax coloring / Spell checking Launching and monitoring of (multiple) GAMS processes Listing file / Tree view / Syntax-error navigation Solver selection / Option selection GDX viewer Data cube Data export (e.g. to MS Excel) Charting facilities Model libraries Documentation 23

24 GAMS at a Glance The GAMS/BASE Module Compiler and Execution System GAMS IDE (Windows) Documentation + Model libraries GDX Utilities Free Solvers/Solver Links 24

25 Documentation Distributed Documentation GAMS Users Guide Expanded GAMS Users Guide (McCarl) Solver Manuals GAMS Utility Manuals Wikis Support Wiki Interfaces Wiki 25

26 Documentation Groups User Group Google Group Newsletter McCarl s News Release List Search all GAMS Websites 26

27 Distributed Model Libraries GAMS Model Library Example and user-contributed models Very often used as templates Tests for Solver robustness and correctness Backward compatibility GAMS Data Utilities Library Demonstration of the various utilities interfacing GAMS with other applications E.g. gdxxrw, mdb2gms, sql2gms 27

28 GAMS at a Glance The GAMS/BASE Module Compiler and Execution System GAMS IDE (Windows) Documentation + Model libraries GDX Utilities Free Solvers/Solver Links 28

29 GAMS at a Glance The GAMS/BASE Module Compiler and Execution System GAMS IDE (Windows) Documentation + Model libraries GDX Utilities Free Solvers/Solver Links 29

30 GAMS at a Glance The GAMS/BASE Module Free Solvers Convert (convert model to different formats) EMP, LOGMIP, NLPEC BENCH, EXAMINER, GAMSCHK BDMLP, LS, and MILES COIN-OR Cbc, IpOpt, BonMin, Couenne Glpk, Scip (academic only) 30

31 GAMS at a Glance Commercial Solvers and Links GUROBI CONOPT CPLEX DICOPT KNITRO MINOS MOSEK XPRESS 31

32 Agenda Motivation GAMS Simple Example 32

33 A Transportation Model Seattle (350) Chicago (300) New York (325) San Diego (600) Topeka (275) 33

34 A Transportation Model San Diego New York Topeka Seattle Chicago Minimize subject to Transportation cost Demand satisfaction at markets Supply constraints 34

35 35 Mathematical Algebra

36 36 GAMS Algebra

37 A few Word about GAMS Syntax Symbols: Sets Parameters Variables Equations Models ASCII Output Files 37 Statements Declarations Data Assignments Equation Definition Programming Flow Control Option statement

38 38 Demo! Transportation Model

39 Organization of the Lecture 13:00 14:30 Introduction Theoretical Background 39 15:00 16:30 Examples Benders Decomposition Outer Approximation Column Generation Outlook and Conclusions

40 40 Outlook and Conclusions

41 Solver Prototypes in GAMS DICOPT (OA, Grossmann) BARON (Spatial B&B, Sahinidis) SBB (B&B, Drud) NLPEC (Reformulations, Ferris) EMP 41 What makes GAMS a good environment for decomposition? Procedural elements, difficult to make programming mistakes (addressing) Concise language Access to state-of-the art solvers

42 New Modeling and Solution Concepts Breakouts of traditional MP classes Extended Nonlinear Programs Chance Constraints CVaR Constraints Robust Programming Bilevel Programs Generalized Disjunctive Programs Multi Agent Equilibrium 42 Limited support with common model representation No conventional syntax Incomplete/experimental solution approaches Lack of reliable/any software

43 What now? Do not: overload existing GAMS notation right away! attempt to build new solvers right away! But: Use existing language features to specify additional model features, structure, and semantics Express extended model in symbolic (source) form and apply existing modeling/solution technology Package new tools with the production system Extended Mathematical Programming (EMP) 43

44 JAMS: a GAMS EMP Solver EMP Information Original Model Viewable Translation Reformulated Model Solving using established Algorithms Mapping Solution Into original space 44 Solution

45 EMP Library Distributed with GAMS 45 Available on website

46 Contacting GAMS USA GAMS Development Corp Potomac Street, NW Washington, DC USA Phone: Fax: Europe GAMS Software GmbH P.O. Box Frechen Germany Phone: Fax: