Abstract
We describe how to deploy interval derivatives up to second order in the context of unconstrained global optimization with a branch and bound method. For computing these derivatives we combine the Boost interval library and the algorithmic differentiation tool dco/c++. The differentiation tool also computes the required floating-point derivatives for a local search algorithm that is embedded in our branch and bound implementation. First results are promising in terms of utility of interval adjoints in global optimization.
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References
Griewank, A., Walther, A.: Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation. 2nd edn. SIAM, Philadelphia, PA (2008)
Naumann, U.: The Art of Differentiating Computer Programs: An Introduction to Algorithmic Differentiation. SIAM, Philadelphia (2012)
Baydin, A.G., Pearlmutter, B.A., Radul, A.A., Siskind, J.M.: Automatic differentiation in machine learning: a survey. J. Mach. Learn. Res. 18(1), 5595–5637 (2017)
Giles, M., Glasserman, P.: Smoking adjoints: fast monte carlo greeks. Risk 19(1), 88–92 (2006)
Towara, M., Naumann, U.: SIMPLE adjoint message passing. Optim. Methods Softw. 33(4–6), 1232–1249 (2018)
Moore, R.E.: Methods and Applications of Interval Analysis, 2nd edn. SIAM, Philadelphia (1979)
Moore, R.E., Kearfott, R.B., Cloud, M.J.: Introduction to Interval Analysis. SIAM, Philadelphia (2009)
Hansen, E., Walster, G.W.: Global Optimization using Interval Analysis. Marcel Dekker, New York (2004)
Neumaier, A.: Complete search in continuous global optimization and constraint satisfaction. Acta Numer. 13, 271–369 (2004)
Floudas, C.A., Pardalos, P.M.: Encyclopedia of Optimization, 2nd edn. Springer, New York (2009)
Vassiliadis, V., Riehme, J., Deussen, J., Parasyris, K., Antonopoulos, C.D., Bellas, N., Lalis, S., Naumann, U.: Towards automatic significance analysis for approximate computing. In: Proceedings of CGO 2016, pp. 182–193. ACM, New York, (2016)
Hascoët, L., Naumann, U., Pascual, V.: “To be recorded” analysis in reverse-mode automatic differentiation. FGCS 21(8), 1401–1417 (2005)
Naumann, U., Lotz, J., Leppkes, K., Towara, M.: Algorithmic differentiation of numerical methods: tangent and adjoint solvers for parameterized systems of nonlinear equations. ACM Trans. Math. Softw. 41(4), 26:1–26:21 (2015)
Brönnimann, H., Melquiond, G., Pion, S.: The design of the Boost interval arithmetic library. Theor. Comput. Sci. 351(1), 111–118 (2006)
Dixon, L.C.W., Szegö, G.P.: The global optimization problem: an introduction. In: Towards Global Optimization, vol. 2, pp. 1–15. North-Holland, Amsterdam (1978)
Griewank, A.: Generalized decent for global optimization. J. Optim Theory Appl. 34(1), 11–39 (1981)
Rosenbrock, H.H.: An automatic method for finding the greatest or least value of a function. Comput. J. 3(3), 175–184 (1960)
Styblinski, M.A., Tang, T.S.: Experiments in nonconvex optimization: stochastic approximation with function smoothing and simulated annealing. Neural Netw. 3(4), 467–483 (1990)
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Deussen, J., Naumann, U. (2020). Discrete Interval Adjoints in Unconstrained Global Optimization. In: Le Thi, H., Le, H., Pham Dinh, T. (eds) Optimization of Complex Systems: Theory, Models, Algorithms and Applications. WCGO 2019. Advances in Intelligent Systems and Computing, vol 991. Springer, Cham. https://doi.org/10.1007/978-3-030-21803-4_8
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