# AMPL

AMPL (A Mathematical Programming Language) is an algebraic modeling language to describe and solve high-complexity problems for large-scale mathematical computing (e.g. large-scale optimization and scheduling-type problems). 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, MOSEK, 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.
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) accept AMPL input. According to the NEOS statistics AMPL is the most popular format for representing mathematical programming problems.

## Features

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
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.

## Availability

AMPL is available for many popular 32 & 64-bit operating systems including Linux, macOS, Solaris, AIX, and Windows. The translator is proprietary software maintained by AMPL Optimization LLC. However, several online services exist, providing free modeling and solving facilities using AMPL. A free student version with limited functionality and a free full-featured version for academic courses are also available.
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 |
| 1990 | Paper describing the AMPL modeling language was published in Management Science |
| 1991 | AMPL supports nonlinear programming and automatic differentiation |
| 1993 | Robert Fourer, David Gay and Brian Kernighan were awarded ORSA/CSTS Prize 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 |
| 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 |
| 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. |
| 2012 | AMPL book becomes freely available online |
| 2013 | A new cross-platform integrated development environment (IDE) for AMPL becomes available |

## 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.

## Solvers

Here is a partial list of solvers supported by AMPL:
| 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 | mixed integer nonlinear programming (MINLP) |
| CPLEX | linear, quadratic, second-order cone and mixed integer programming |
| CPLEX CP Optimizer | constraint programming |
| FILTER | nonlinear programming |
| FortMP | linear, quadratic and mixed integer programming |
| Gecode | constraint programming |
| IPOPT | nonlinear programming |
| JaCoP | constraint programming |
| LGO | global and local nonlinear optimization |
| lp_solve | 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 | 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 |