The objective of global optimization is to find the
globally best solution of a model. Nonlinear models are ubiquitous
in many applications and their solution often requires a global
search approach; i.e. for a function f from a set A ⊂ Rn to
the real numbers, an element x0 ∈ A is sought-after, such that
∀ x ∈ A : f(x0) ≤ f(x). Depending on the field of application,
the question whether a found solution x0 is not only a local minimum
but a global one is very important.
This article presents a probabilistic approach to determine the
probability of a solution being a global minimum. The approach is
independent of the used global search method and only requires a
limited, convex parameter domain A as well as a Lipschitz continuous
function f whose Lipschitz constant is not needed to be known.
A. Conn, N. I. M. Gould, and P. L. Toint, "Large-Scale Nonlinear
Constrained Optimization: A Current Survey," Algorithms for continuous
optimization: the state of the art, vol. 434, pp. 287-332, 1994.
 N. Gould, D. Orban, and P. Toint, "Numerical methods for large-scale
nonlinear optimization," Acta Numerica, vol. 14, pp. 299-361, 2005.
 H.-G. Beyer and B. Sendhoff, "Robust optimization - A comprehensive
survey," Computer Methods in Applied Mechanics and Engineering, vol.
196, pp. 3190-3218, 2007.
 M. Kieffer, M. C. Mark'ot, H. Schichl, and E. Walter, "Verified global
optimization for estimating the parameters of nonlinear models," Modeling,
Design, and Simulation of Systems with Uncertainties, vol. 1, pp.
 R. Hammer, M. Hocks, U. Kulisch, and D. Ratz, C++ Toolbox for
Verified Computing, R. Hammer, M. Hocks, U. Kulisch, and D. Ratz,
Eds. Springer, 1997.
 M. C. Mark'ot and H. Schichl, "Comparison and Automated Selection of
Local Optimization Solvers for Interval Global Optimization Methods,"
SIAM Journal on Optimization, vol. 21, pp. 1371-1391, 2011.
 N. Henze, Stochastik - Einf┬¿uhrung in die Wahrscheinlichkeitstheorie
und Statistik (english: Stochastics - Introduction to probability calculus
and statistics), N. Henze, Ed. Technische Universit┬¿at Karlsruhe, 1995.
 ÔÇöÔÇö, Stochastik f┬¿ur Einsteiger (english: Stochastics for Beginners),
N. Henze, Ed. Vieweg, 1997.
 H. Bandemer and A. Bellmann, Statistische Versuchsplanung (english:
Statistical Test Planning), H. Bandemer and A. Bellmann, Eds. Verlag
H. Deutsch / BSB B. G. Teubner, 1979.
 E. Weisstein, MathWorld - A Wolfram Web Resource, E. Weisstein, Ed.
Wolfram Research, 2009.
 M. P. McLaughlin, A Compendium of Common Probability Distributions,
M. P. McLaughlin, Ed. Regress+, 1999.
 P. K. Agarwal, S. Har-Peled, and K. R. Varadarajan, "Geometric Approximations
via Coresets," Combinatorial and Computational Geometry -
MSRI Publications, vol. 52, pp. 1-30, 2005.
 G. Zachmann, "Rapid Collision Detection by Dynamically Aligned
DOP-Trees," Proceedings of the Virtual Reality Annual International
Symposium, vol. 1, pp. 90-97, 1998.
 J. D. Pinter, "Globally Optimized Spherical Point Arrangements: Model
Variants and Illustrative Results," Annals of Operations Research, vol.
104, pp. 213-230, 2001.
 A. Katanforoush and M. Shahshahani, "Distributing Points on a Sphere,"
Experimental Mathematics, vol. 12, pp. 199-208, 2003.
 C. Schinko, T. Ullrich, and D. W. Fellner, "Simple and Efficient Normal
Encoding with Error Bounds," Proceedings of Theory and Practice of
Computer Graphics, vol. 29, pp. 63-66, 2011.
 R. Storn and K. Price, "Differential Evolution: A simple and efficient
heuristic for global optimization over continuous spaces," Journal of
Global Optimization, vol. 11, pp. 341-359, 1997.
 Z. Michalewicz and M. Schoenauer, "Evolutionary Algorithms for Constrained
Parameter Optimization Problems," Evolutionary Computation,
vol. 4, pp. 1-32, 1996.
J. C. Nash, Compact Numerical Methods for Computers: Linear Algebra
and Function Minimisation, second edition ed., J. C. Nash, Ed. Adam
J. D. Pinter, "Global Optimization: Software, Test Problems, and Applications,"
Handbook of Global Optimization, P.M. Pardalos and H.E.
Romeijn (eds), vol. 2, pp. 515-569, 2002.
J. Nocedal and S. J. Wright, Numerical Optimization, J. Nocedal and
S. J. Wright, Eds. Springer, 1999.
K. H┬¿ollig, J. H┬¿oner, and M. Pfeil, Numerische Methoden der Analysis,
K. H┬¿ollig, J. H┬¿oner, and M. Pfeil, Eds. Mathematik-Online, 2010.
P. E. Gill, W. Murray, and M. H. Wright, Practical Optimization, P. E.
Gill, W. Murray, and M. H. Wright, Eds. Academic Press, 1982.
R. Fletcher, Practical Methods of Optimization, R. Fletcher, Ed. Wiley,
U. Diwekar, Introduction to Applied Optimization, ser. Applied Optimization,
U. Diwekar, Ed. Springer, 2003, vol. 80.
W. Boehm and H. Prautzsch, Numerical Methods, W. Boehm and
H. Prautzsch, Eds. Vieweg, 1993.