Long-term stability of the Tevatron by verified global optimization

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The tools used to compute high-order transfer maps based on differential algebraic (DA) methods have recently been augmented by methods that also allow a rigorous computation of an interval bound for the remainder. In this paper we will show how such methods can also be used to determine rigorous bounds for the global extrema of functions in an efficient way. The method is used for the bounding of normal form defect functions, which allows rigorous stability estimates for repetitive particle accelerator. However, the method is also applicable to general lattice design problems and can enhance the commonly used local optimization with heuristic successive starting point modification. The global optimization approach studied rests on the ability of the method to suppress the so-called dependency problem common to validated computations, as well as effective polynomial bounding techniques. We review the linear dominated bounder (LDB) and the quadratic fast bounder (QFB) and study their performance for various example problems in global optimization. We observe that the method is superior to other global optimization approaches and can prove stability times similar to what is desired, without any need for expensive long-term tracking and in a fully rigorous way.