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Geometry and combinatorics of the cutting angle method. (English) Zbl 1055.65071

Lower approximation of Lipschitz functions plays an important role in deterministic global optimization. This paper examines in detail the lower piecewise linear approximation which arises in the cutting angle method. All its local minima can be explicitly enumerated, and a special data structure is designed to process them very efficiently, improving previous results by several orders of magnitude. Also, some geometrical properties of the lower approximation are studied. Connection to a spectral distance function and Voronoi diagrams is established. An application of these results is a black-box multivariate random number generator, based on an acceptance-rejection approach.

MSC:

65K05 Numerical mathematical programming methods
90C59 Approximation methods and heuristics in mathematical programming
65C10 Random number generation in numerical analysis
90C30 Nonlinear programming

Software:

LGO
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