What are the transformations that AMPL applies to optimization problems?

AMPL’s presolve phase attempts to transform your problem to an equivalent one that is smaller and easier to solve. Presolve first removes trivial model components, such as variables fixed at constant values and constraints that express simple lower or upper bounds. Then it applies an iterative procedure to tighten certain bounds on variables and constraints, with the result that additional variables may be fixed and constraints may be dropped. Many of these transformations are based on ideas first presented by A.L. Brearly, G. Mitra and H.P. Williams, in “Analysis of Mathematical Programming Problems Prior to Applying the Simplex Algorithm,” Mathematical Programming 8 (1975) 54-83. See also Experience with a Primal Presolve Algorithm for a detailed discussion of the implementation in AMPL.

If your model uses AMPL’s notation for piecewise-linear terms in individual variables, then AMPL transforms your problem’s piecewise-linear expressions to equivalent linear expressions, through the addition of auxiliary variables and constraints. Chapter 17 of the AMPL book describes the piecewise-linear notation in more detail. See also Expressing Special Structures in an Algebraic Modeling Language for Mathematical Programming for further discussion of how the transformation is carried out.

You can cause AMPL to eliminate certain variables from your problem, by substituting expressions specified in var declarations or in certain constraints. If the substitution comes from a constraint, then the constraint is also eliminated. See Sections 18.2 and A.8.1 of the AMPL book for more information.