decomposition methods for mathematical programming problems webseite... · 2020. 3. 20. · cplex...

Decomposition Methods forMathematical Programming Problems

Michael R. Bussieck [email protected]

Stefan Vigerske [email protected]

GAMS Software GmbH / GAMS Development Corp.

www.gams.com

Aachen, June 11, 2013

Agenda

2

Motivation

GAMS

Simple Example

Mathematical Programming/Optimization

3en.wikipedia.org/wiki/Mathematical_optimization

Major Subfields of Math. Optimization

http://en.wikipedia.org/wiki/Mathematical_optimizati on#Major_subfields

Linear Programming http://en.wikipedia.org/wiki/Linear_programming

5

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

* Illustrative examples in the GAMS Model Library6

Modeling in GAMS

A mathematical model is a description of a system using mathematical language (from: Wikipedia)

1973 Frederick W. Lanchester Prize: Winner [-hide]

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 input-output 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.

INFORMS Lanchester Prize 1973

8

Model Structure

9

Model Data

10

Matrix Generator

11

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

…

GAMS (General) Algebraic Modeling System

• 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

What is a Model in GAMS?

13

Agenda

14

Motivation

GAMS

Simple Example

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

• Platform independence

• Hassle-free switch of solution methods

• Open architecture and interfaces to other systems

• Balanced mix of declarative and procedural elements

GAMS’ Fundamental concepts

16

10+ Supported Platforms

AXU

Solaris

Windows 64bit

Windows

Linu

x

Linu

x 64

bit

HP

25+ Integrated Solvers

XAXPRESS

COIN-OR

BDMLP CPLEX

BA

RO

N

LIN

DO

GLO

BA

L

GUROBI


17





Binary Data Exchange

• 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

Application GDX

GAMS

SOLVER

Declaration of..- Sets- Parameters- Variables- Equations- Models- …

Procedural Elements like…- loops- if-then-else- …






19

Interface Data Model Solver Interface

GAMS’ Fundamental concepts cont’d

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• Different layers with separation of– model and data– model and solution methods– model and operating system– model and interface

� Models benefit from– advancing hardware– enhanced / new solver technology– improved / upcoming interfaces to other systems

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

GAMS at a Glance





• GDX Utilities


22

• 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

Integrated Development Environment

23

GAMS at a Glance





• GDX Utilities


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• Distributed Documentation– GAMS Users Guide– Expanded GAMS Users Guide (McCarl)– Solver Manuals– GAMS Utility Manuals

• Wikis– Support Wiki http://support.gams-software.com– Interfaces Wiki http://interfaces.gams-software.com

Documentation

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• Groups– User Group http://www.gams.com/maillist/gams_l.htm– Google Group http://groups.google.de/group/gamsworld

• Newsletter– McCarl‘s News

http://www.gams.com/maillist/newsletter.htm– Release List

• Search all GAMS Websiteshttp://www.gams.com/search.htm

Documentation

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• 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

Distributed Model Libraries

27

GAMS at a Glance





• GDX Utilities


28

GAMS at a Glance





• GDX Utilities


29

GAMS at a Glance


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)

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GAMS at a Glance

Commercial Solvers and Links

• GUROBI

• CONOPT

• CPLEX

• DICOPT

• KNITRO

• MINOS

• MOSEK

• XPRESS

• …

31

Agenda

32

Motivation

GAMS

Simple Example

A Transportation Model

33

Seattle(350)

San Diego(600)

Chicago(300)

New York(325)

Topeka(275)

A Transportation Model

34

Minimize Transportation costsubject to Demand satisfaction at markets

Supply constraints

600San Diego

350

325

275

300

Seattle

New York

Topeka

Chicago

2500

17001800

2500

14001800

Mathematical Algebra

35

GAMS Algebra

36

• Symbols:– Sets– Parameters– Variables– Equations– Models– ASCII Output Files

• Statements– Declarations– Data Assignments– Equation Definition– Programming Flow Control – Option statement

A few Word about GAMS Syntax

37

Demo! Transportation Model

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• 13:00 – 14:30– Introduction– Theoretical Background

• 15:00 – 16:30– Examples

• Benders Decomposition• Outer Approximation• Column Generation

– Outlook and Conclusions

Organization of the Lecture

39

40

Outlook and Conclusions

• DICOPT (OA, Grossmann)• BARON (Spatial B&B, Sahinidis)• SBB (B&B, Drud)• NLPEC (Reformulations, Ferris)• …• EMP

• 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

Solver Prototypes in GAMS

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• Breakouts of traditional MP classes– Extended Nonlinear Programs– Chance Constraints– CVaR Constraints– Robust Programming– Bilevel Programs– Generalized Disjunctive Programs– Multi Agent Equilibrium

• Limited support with common model representation• No conventional syntax • Incomplete/experimental solution approaches• Lack of reliable/any software

New Modeling and Solution Concepts

42

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)

What now?

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JAMS: a GAMS EMP Solver

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EMP Information

ReformulatedModel

Solution

Map

ping

Sol

utio

nIn

to o

rigin

al s

pace

Original Model

Translation

Solving usingestablished Algorithms

Viewable

• Distributed with GAMS

• Available on websitehttp://www.gams.com/emplib/emplib.htm

EMP Library

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Contacting GAMS

USA

GAMS Development Corp. 1217 Potomac Street, NW Washington, DC 20007USA Phone: +1 202 342 0180 Fax: +1 202 342 0181http://www.gams.com

[email protected]@gams.com

Europe

GAMS Software GmbHP.O. Box 40 5950216 FrechenGermanyPhone: +49 221 949 9170Fax: +49 221 949 9171http://www.gams.de

[email protected]

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decomposition methods for mathematical programming problems webseite... · 2020. 3. 20. · cplex...

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