A Formally Verified Generic Branching Algorithm for Global Optimization
This paper presents a formalization in higher-order logic of a generic algorithm that is used in automated strategies for solving global optimization problems. It is a generalization of numerical branch and bound algorithms that compute the minimum of a function on a given domain by recursively dividing the domain and computing estimates for the range of the function on each sub-domain. The correctness statement of the algorithm has been proved in the Prototype Verification System (PVS) theorem prover. This algorithm can be instantiated with specific functions for performing particular global optimization methods. The correctness of the instantiated algorithms is guaranteed by simple properties that need to be verified on the specific input functions. The use of the generic algorithm is illustrated with an instantiation that yields an automated strategy in PVS for estimating the maximum and minimum values of real-valued functions.
KeywordsGlobal Optimization Interval Arithmetic Recursive Call Global Optimization Problem Bernstein Polynomial
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- 2.Crespo, L.G., Muñoz, C.A., Narkawicz, A.J., Kenny, S.P., Giesy, D.P.: Uncertainty analysis via failure domain characterization: Polynomial requirement functions. In: Proceedings of European Safety and Reliability Conference, Troyes, France (September 2011)Google Scholar
- 4.Harrison, J.: Metatheory and reflection in theorem proving: A survey and critique. Technical Report CRC-053, SRI Cambridge, Millers Yard, Cambridge, UK (1995), http://www.cl.cam.ac.uk/jrh13/papers/reflect.dvi.gz+
- 7.Moa, B.: Interval Methods for Global Optimization. PhD thesis, University of Victoria (2007)Google Scholar
- 8.Moore, R.E., Kearfott, R.B., Cloud, M.J.: Introduction to Interval Analysis. Cambridge University Press (2009)Google Scholar
- 11.Neumaier, A.: Complete search in continuous global optimization and constraint satisfaction. Acta Numerica 13, 271–369Google Scholar
- 12.Owre, S., Rushby, J., Shankar, N.: PVS: A prototype verification system. In: Kapur, D. (ed.) CADE 1992. LNCS, vol. 607, pp. 748–752. Springer, Heidelberg (1992)Google Scholar