EAGOParametricInterval.jl

A library for bounding functions via parametric interval methods

Authors

Matthew Wilhelm, Department of Chemical and Biomolecular Engineering, University of Connecticut (UCONN)

Installation

julia> Pkg.add("EAGOParametricInterval")

Capabilities

EAGOParametricInterval.jl provides methods for performing parametric interval calculations such as (Parametric Interval Newton/Krawczyk) as well as a series of tests to verify the (non)existence of unique enclosed functions. This routine are used extensively in the EAGO.jl package solver

Currently, it exports four types of contractor functions for use. This contractors include embedded test for the guaranteed existence or non-existence. Please see the example.jl file for usage cases.

Future Work

• A parametric bisection routine will be updated that can divide the (X,P) space into a a series of boxes that all contain unique branches of the implicit function p->y(p).
• Minor modifications to the contractors are planned to improve computational performance. Namely, row handling for the sparse LU factorization for interval midpoint computation.

Related Packages

• EAGO.jl: A package containing global and robust solvers based mainly on McCormick relaxations. This package supports a JuMP and MathProgBase interface.
• IntervalRootFinding.jl: Provides root finding routines using Interval Newton and Krawczyk methods but don't include parametric method, methods with embedded test, or handling of large sparse matrices for preconditioner calculation.

References

• E. R. Hansen and G. W. Walster. Global Optimization Using Interval Analysis. Marcel Dekker, New York, second edition, 2004.
• R. Krawczyk. Newton-algorithmen zur bestimmung con nullstellen mit fehler-schranken. Computing, 4:187–201, 1969.
• R. Krawczyk. Interval iterations for including a set of solutions. Computing, 32:13–31, 1984.
• C. Miranda. Un’osservatione su un teorema di brower. Boll. Un. Mat. Ital., 3:5–7, 1940.
• A. Neumaier. Interval Methods for Systems of Equations. Cambridge University Press, Cambridge, 1990.
• R. E. Moore. A test for existence of solutions to nonlinear systems. SIAM Journal on Numerical Analysis, 14(4):611–615, 1977.