Algebraic modeling language

Algebraic Modeling Languages (AML) are high-level computer programming languages for describing and solving high complexity problems for large scale mathematical computation (i.e. large scale optimization type problems). One particular advantage of some algebraic modeling languages like AIMMS,AMPL,GAMS , OPL or Mosel is the similarity of their syntax to the mathematical notation of optimization problems. This allows for a very concise and readable definition of problems in the domain of optimization, which is supported by certain language elements like sets, indices, algebraic expressions, powerful sparse index and data handling variables, constraints with arbitrary names. The algebraic formulation of a model does not contain any hints how to process it.

An AML does not solve those problems directly; instead, it calls appropriate external algorithms to obtain a solution. These algorithms are called solvers and can handle certain kind of mathematical problems like:

The core elements of an AML are:

Most AML follow certain design principles:

Algebraic modelling languages find their roots in matrix-generator and report-writer programs (MGRW), developed in the late seventies. Some of these are MAGEN, MGRW (IBM), GAMMA.3, DATAFORM and MGG/RWG. These systems simplified the communication of problem instances to the solution algorithms and the generation of a readable report of the results.

An early matrix-generator for LP was developed around 1969 at the Mathematisch Centrum (now CWI), Amsterdam. Its syntax was very close to the usual mathematical notation, using subscripts en sigmas. Input for the generator consisted of separate sections for the model and the data. It found users at universities and in industry. The main industrial user was the steel maker Hoogovens (now Tata Steel) where it was used for nearly 25 years.

A big step towards the modern modelling languages is found in UIMP , where the structure of the Mathematical Programming models taken from real life is analysed for the first time, to highlight the natural grouping of variables and constraints arising from such models. This led to data-structure features, which supported structured modelling; in this paradigm, all the input and output tables, together with the decision variables, are defined in terms of these structures, in a way comparable to the use of subscripts and sets. This is probably the single most notable feature common to all modern AMLs and enabled, in time, a separation between the model structure and its data, and a correspondence between the entities in a MP model and data in relational databases. So, a model could be finally instantiated and solved over different datasets, just by modifying its datasets.

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