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AMPL (A Mathematical Programming Language) is an algebraic modeling language to describe and solve high-complexity problems for large-scale mathematical computing (i.e., large-scale optimization and scheduling-type problems).[1] It was developed by Robert Fourer, David Gay, and Brian Kernighan at Bell Laboratories. AMPL supports dozens of solvers, both open source and commercial software, including CBC, CPLEX, FortMP, Gurobi, MINOS, IPOPT, SNOPT, KNITRO, and LGO. Problems are passed to solvers as nl files. AMPL is used by more than 100 corporate clients, and by government agencies and academic institutions.[2]

One advantage of AMPL is the similarity of its syntax to the mathematical notation of optimization problems. This allows for a very concise and readable definition of problems in the domain of optimization. Many modern solvers available on the NEOS Server (formerly hosted at the Argonne National Laboratory, currently hosted at the University of Wisconsin, Madison[3]) accept AMPL input. According to the NEOS statistics AMPL is the most popular format for representing mathematical programming problems.


AMPL features a mix of declarative and imperative programming styles. Formulating optimization models occurs via declarative language elements such as sets, scalar and multidimensional parameters, decision variables, objectives and constraints, which allow for concise description of most problems in the domain of mathematical optimization.

Procedures and control flow statements are available in AMPL for

the exchange of data with external data sources such as spreadsheets, databases, XML and text files
data pre- and post-processing tasks around optimization models
the construction of hybrid algorithms for problem types for which no direct efficient solvers are available.

To support re-use and simplify construction of large-scale optimization problems, AMPL allows separation of model and data.

AMPL supports a wide range of problem types, among them:

Linear programming
Quadratic programming
Nonlinear programming
Mixed-integer programming
Mixed-integer quadratic programming with or without convex quadratic constraints
Mixed-integer nonlinear programming
Second-order cone programming
Global optimization
Semidefinite programming problems with bilinear matrix inequalities
Complementarity theory problems (MPECs) in discrete or continuous variables
Constraint programming[4]

AMPL invokes a solver in a separate process which has these advantages:

User can interrupt the solution process at any time
Solver errors do not affect the interpreter
32-bit version of AMPL can be used with a 64-bit solver and vice versa

Interaction with the solver is done through a well-defined nl interface.
NEOS input statistics for January 2011.

AMPL is available for many popular 32- and 64-bit operating systems including Linux, macOS, Solaris, AIX, and Windows.[5] The translator is proprietary software maintained by AMPL Optimization LLC. However, several online services exist, providing free modeling and solving facilities using AMPL.[6][7] A free student version with limited functionality and a free full-featured version for academic courses are also available.[8]

AMPL can be used from within Microsoft Excel via the SolverStudio Excel add-in.

The AMPL Solver Library (ASL), which allows reading nl files and provides the automatic differentiation, is open-source. It is used in many solvers to implement AMPL connection.
Status history

This table present significant steps in AMPL history.

Year Highlights
1985 AMPL was designed and implemented[1]
1990 Paper describing the AMPL modeling language was published in Management Science[9]
1991 AMPL supports nonlinear programming and automatic differentiation
1993 Robert Fourer, David Gay and Brian Kernighan were awarded ORSA/CSTS Prize[10] by the Operations Research Society of America, for writings on the design of mathematical programming systems and the AMPL modeling language
1995 Extensions for representing piecewise-linear and network structures
1995 Scripting constructs
1997 Enhanced support for nonlinear solvers
1998 AMPL supports complementarity theory problems
2000 Relational database and spreadsheet access
2002 Support for constraint programming[4]
2003 AMPL Optimization LLC was founded by the inventors of AMPL, Robert Fourer, David Gay, and Brian Kernighan. The new company took over the development and support of the AMPL modeling language from Lucent Technologies, Inc.
2005 AMPL Modeling Language Google group opened[11]
2008 Kestrel: An AMPL Interface to the NEOS Server introduced
2012 Robert Fourer, David Gay, and Brian Kernighan were awarded the 2012 INFORMS Impact Prize as the originators of one of the most important algebraic modeling languages.[12]
2012 AMPL book becomes freely available online
2013 A new cross-platform integrated development environment (IDE) for AMPL becomes available[13]

A sample model

A transportation problem from George Dantzig is used to provide a sample AMPL model. This problem finds the least cost shipping schedule that meets requirements at markets and supplies at factories.[14]

set Plants;
set Markets;

# Capacity of plant p in cases
param Capacity{p in Plants};

# Demand at market m in cases
param Demand{m in Markets};

# Distance in thousands of miles
param Distance{Plants, Markets};

# Freight in dollars per case per thousand miles
param Freight;

# Transport cost in thousands of dollars per case
param TransportCost{p in Plants, m in Markets} :=
Freight * Distance[p, m] / 1000;

# Shipment quantities in cases
var shipment{Plants, Markets} >= 0;

# Total transportation costs in thousands of dollars
minimize cost:
sum{p in Plants, m in Markets} TransportCost[p, m] * shipment[p, m];

# Observe supply limit at plant p
s.t. supply{p in Plants}: sum{m in Markets} shipment[p, m] <= Capacity[p];

# Satisfy demand at market m
s.t. demand{m in Markets}: sum{p in Plants} shipment[p, m] >= Demand[m];


set Plants := seattle san-diego;
set Markets := new-york chicago topeka;

param Capacity :=
seattle 350
san-diego 600;

param Demand :=
new-york 325
chicago 300
topeka 275;

param Distance : new-york chicago topeka :=
seattle 2.5 1.7 1.8
san-diego 2.5 1.8 1.4;

param Freight := 90;


Here is a partial list of solvers supported by AMPL:[15]

Solver Supported problem types
APOPT mixed integer nonlinear programming
Artelys Knitro linear, quadratic and nonlinear programming
Bonmin mixed integer nonlinear programming
BPMPD linear and quadratic programming
COIN-OR CBC mixed integer programming
COIN-OR CLP linear programming
CONOPT nonlinear programming
Couenne[16] mixed integer nonlinear programming (MINLP)
CPLEX linear, quadratic, second-order cone and mixed integer programming
CPLEX CP Optimizer[17] constraint programming
FILTER nonlinear programming
FortMP linear, quadratic and mixed integer programming
Gecode[18] constraint programming
Gurobi linear, quadratic, second-order cone and mixed integer programming
IPOPT nonlinear programming
JaCoP[19] constraint programming
LGO[20] global and local nonlinear optimization
lp_solve[21] linear and mixed integer programming
MINOS linear and nonlinear programming
MINTO mixed integer programming
MOSEK linear, mixed integer linear, quadratic, mixed integer quadratic, quadratically constrained, conic and convex nonlinear programming
Octeract Engine All types of optimisation problems without differential or integral terms, including discontinuous problems with min and max elementary functions.
SCIP mixed integer programming
SNOPT nonlinear programming
Sulum[22] linear and mixed integer programming
WORHP nonlinear programming
XA linear and mixed integer programming
Xpress linear and convex quadratic optimization and their mixed integer counterparts

See also

sol (format)


Fourer, Robert; Gay, David M; Kernighan, Brian W (2003). AMPL: a modeling language for mathematical programming. USA: Duxbury Press/Brooks/Cole Publishing Company. ISBN 978-0-534-38809-6.
"Position Available". Retrieved 29 July 2011.
"About". Retrieved 11 August 2015.
Fourer, Robert; Gay, David M. (2002). "Extending an Algebraic Modeling Language to Support Constraint Programming". INFORMS Journal on Computing. 14 (4): 322–344. CiteSeerX doi:10.1287/ijoc.14.4.322.2825.
"Platforms". AMPL Optimizations Inc.
"NEOS Server for Optimization". Retrieved 11 August 2015.
"Try AMPL!". Retrieved 11 August 2015.
"AMPL Downloads". Archived from the original on 26 May 2015. Retrieved 11 August 2015.
Fourer, Robert; Gay, David M.; Kernighan, Brian W. (1990). "A Modeling Language for Mathematical Programming" (PDF). Management Science. 36 (5): 519–554–83. doi:10.1287/mnsc.36.5.519.
INFORMS. "ICS - INFORMS" (PDF). Retrieved 11 August 2015.
"Google Groups".
INFORMS. "INFORMS Impact Prize". Archived from the original on 22 October 2013. Retrieved 11 August 2015.
"Google Groups". Retrieved 11 August 2015.
Dantzig, George (2016) [1963]. "3. Formulating a Linear Programming Model". Linear Programming and Extensions. Princeton University Press. pp. 32–62. ISBN 978-1-4008-8417-9.
"Solvers - AMPL". Retrieved 21 January 2018.
"Couenne". Archived from the original on 29 October 2013. Retrieved 27 October 2013.
"mp/solvers/ilogcp at master · ampl/mp · GitHub". GitHub. Retrieved 11 August 2015.
"mp/solvers/gecode at master · ampl/mp · GitHub". GitHub. Retrieved 11 August 2015.
"mp/solvers/jacop at master · ampl/mp · GitHub". GitHub. Retrieved 11 August 2015.
"LGO - AMPL". Retrieved 11 August 2015.
"Using lpsolve from AMPL". Retrieved 11 August 2015.

"mp/solvers/sulum at master · ampl/mp · GitHub". GitHub. Retrieved 11 August 2015.

External links

Official website
Prof. Fourer's home page at Northwestern University

Undergraduate Texts in Mathematics

Graduate Texts in Mathematics

Graduate Studies in Mathematics

Mathematics Encyclopedia



Hellenica World - Scientific Library

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