Environments for Global Optimization Using Interval Arithmetic and Computational (Automatic) Differentiation

Interval arithmetic and computational (automatic) differentiation are powerful methods for use in optimization. Interval arithmetic operates on interval values rather than points and can be used to examine large areas of a space, often eliminating portions shown not to contain the global optimum. Interval derivative information may be used similarly. Computational differentiation computes derivative information in a fashion that requires very little user input. Three packages that integrate these methods are considered. Leading authorities on interval arithmetic and computational differentiation have developed GLOBSOL, a Fortran90 based public domain program and a very efficient global optimization package. The user provides a functional form of the objective along with any constraints. This form is parsed and expressions for the derivatives are computed. These are fed to the main program, which performs computations using interval variables. Choices for matrix preconditioners are provided along with (interval) Gaussian elimination and Gauss-Seidel solvers. The Newton method is also used. The Fritz-John conditions are checked on the constraints and routines are provided for ill-conditioned LS problems. While this package is in the public domain, it does require a Fortran90 compiler. Numerica is a commercially vended product for Windows based machines. It provides much of the capability of GLOBSOL and, due to its interface, is easier to use. It is not easy to incorporate complex user defined functions into Numerica. C-XSC is a commercially vended C ++ class library that includes an interval class the methods needed for interval arithmetic. It does not include a computational differentiation class, but this is available with the mathematics package (including global optimization) available on the Web. The computational differentiation method used is not as efficient as GLOBSOL's, and the optimization routines are not as sophisticated and cannot handling constraints. They work as well as GLOBSOL for unconstrained problems, but may be slower. All three of these packages are applied to optimization problems, including parameter estimation problems.