Numerica: A Modeling Language for Global Optimization

Many science and engineering applications require the user to find solutions to systems of nonlinear constraints over real numbers or to optimize a nonlinear function subject to nonlinear constraints. This includes applications such the modeling of chemical engineering processes and of electrical circuits, robot kinematics, chemical equil ibrium problems, and design problems (e.g., nuclear reactor design). The field of global optimization is the study of methods to find all solutions to systems of nonlinear constraints and all global opt ima to optimization problems. Nonlinear problems raise many issues from a computation standpoint. On the one hand, deciding if a set of polynomial constraints has a solution is NP-hard. In fact, Canny [Canny, 1988] and Renegar [Renegar, 1988] have shown that the problem is in PSPACE and it is not known whether the problem lies in NP. Nonlinear programming problems can be so hard that some methods are designed only to solve problems up to, say, 20 variables. On the other hand, computing over real numbers raises numerical problems because of the finite nature of computers. N U M E R I C A [Van Hentenryck et a/., 1997c] is a modeling language for global optimization which makes it possible to solve nonlinear problems written in a form close to the statements tradit ionally found in textbooks and scientific papers. In addit ion, and contrary to most nonlinear programming tools, N U M E R I C A provides many guarantees on its results (modulo implementation errors):