Journal article Open Access

Probability of Globality

Eva Eggeling; Dieter W. Fellner; Torsten Ullrich

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.

Files (219.0 kB)
Name Size
219.0 kB Download
  • 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. [10] N. Gould, D. Orban, and P. Toint, "Numerical methods for large-scale nonlinear optimization," Acta Numerica, vol. 14, pp. 299-361, 2005. [11] H.-G. Beyer and B. Sendhoff, "Robust optimization - A comprehensive survey," Computer Methods in Applied Mechanics and Engineering, vol. 196, pp. 3190-3218, 2007. [12] 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. 129-151, 2011. [13] 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. [14] 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. [15] 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. [16] ÔÇöÔÇö, Stochastik f┬¿ur Einsteiger (english: Stochastics for Beginners), N. Henze, Ed. Vieweg, 1997. [17] 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. [18] E. Weisstein, MathWorld - A Wolfram Web Resource, E. Weisstein, Ed. Wolfram Research, 2009. [19] M. P. McLaughlin, A Compendium of Common Probability Distributions, M. P. McLaughlin, Ed. Regress+, 1999. [20] 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. [21] G. Zachmann, "Rapid Collision Detection by Dynamically Aligned DOP-Trees," Proceedings of the Virtual Reality Annual International Symposium, vol. 1, pp. 90-97, 1998. [22] J. D. Pinter, "Globally Optimized Spherical Point Arrangements: Model Variants and Illustrative Results," Annals of Operations Research, vol. 104, pp. 213-230, 2001. [23] A. Katanforoush and M. Shahshahani, "Distributing Points on a Sphere," Experimental Mathematics, vol. 12, pp. 199-208, 2003. [24] 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. [25] 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. [26] 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 Hilger, 1990.
  • 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, 2000.
  • 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.
All versions This version
Views 55
Downloads 11
Data volume 219.0 kB219.0 kB
Unique views 55
Unique downloads 11


Cite as