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Treating Non-Smooth Functions as Smooth Functions in Global Optimization and Nonlinear Systems Solvers

Techniques of interval extensions and interval Newton methods have been developed for verified solution of nonlinear systems of equations and for global optimization. In most of the literature to date, such interval extensions and interval Newton methods are applicable when the functions are given by smooth expressions, without conditional branches. In fact, however, many practical problems, in particular those containing expressions such as |E(X)| and max{E(X), F (X)}, E,F : R → R, or expressions defined by IF-THEN-ELSE branches, result in functions whose derivatives have jump discontinuities. However, in [4], continuous, order-1 interval extensions were proposed for such continuous but non-smooth functions.