Numerical Issues and Influences in the Design of Algebraic Modeling Languages for Optimization

The idea of a modeling language is to describe mathematical problems symbolically in a way that is familiar to people but that allows for processing by computer systems. In particular the concept of an algebraic modeling language, based on objective and constraint expressions in terms of decision variables, has proved to be valuable for a broad range of optimization and related problems. One modeling language can work with numerous solvers, each of which implements one or more optimization algorithms. The separation of model specification from solver execution is thus a key tenet of modeling language design. Nevertheless, several issues in numerical analysis that are critical to solvers are also important in implementations of modeling languages. So-called presolve procedures, which tighten bounds with the aim of eliminating some variables and constraints, are numerical algorithms that require carefully chosen tolerances and can benefit from directed roundings. Correctly rounded binary-decimal conversion is valuable in portably conveying problem instances and in debugging. Further rounding options offer tradeoffs between accuracy, convenience, and readability in displaying numerical results. Modeling languages can also strongly influence the development of solvers. Most notably, for smooth nonlinear optimization, the ability to provide numerically computed, exact first and second derivatives has made modeling languages a valuable tool in solver development. The generality of modeling languages has also encouraged the development of more general solvers, such as for optimization problems with equilibrium constraints. This paper draws from our experience in developing the AMPL modeling language [14, 15] to provide examples in all of the above areas. An associated set of slides [13] shows more completely the contexts from which our examples are taken.