In this work, we introduce a method of finding the global minimum of bounded from below functions with continuous partial derivatives in an Euclidean space
with existing finite points of the global minimum. We create a function that finds the zeros of the sign function of the sum of the squares of the partial derivatives of the function to minimize. We can solve with this method also systems of simultaneous continuous nonlinear equations. We create, in that case, a function that finds the zeros of the sign function of the sum of the squares of the equations of the system to solve.
References
[1]
Neumaier, A. (2004) Complete Search in Continuous Global Optimization and Constraint Satisfaction. Acta Numerica, 13, 271-369. https://doi.org/10.1017/s0962492904000194
[2]
Misener, R. and Floudas, C.A. (2014) ANTIGONE: Algorithms for Continuous/Integer Global Optimization of Nonlinear Equations. Journal of Global Optimization, 59, 503-526. https://doi.org/10.1007/s10898-014-0166-2
[3]
Horst, R., Pardalos, P.M. and Thoai, N.V. (2000) Introduction to Global Optimization. 2nd Edition, Kluwer Academic Publishers.
[4]
Jaulin, L., Kieffer, M., Didrit, O. and Walter, E. (2001) Applied Interval Analysis. Springer.
[5]
Hansen, E.R. (1992) Global Optimization Using Interval Analysis. Marcel Dekker.
[6]
Zhigljavsky, A. (1991) Theory of Global Random Search. Kluwer Academic Publishers.
[7]
Hamacher, K. (2006) Adaptation in Stochastic Tunneling Global Optimization of Complex Potential Energy Landscapes. Europhysics Letters (EPL), 74, 944-950. https://doi.org/10.1209/epl/i2006-10058-0
[8]
Dorigo, M. and Stützle, T. (2004) Ant Colony Optimization. The MIT Press. https://doi.org/10.7551/mitpress/1290.001.0001
[9]
Granville, V., Krivanek, M. and Rasson, J.-P. (1994) Simulated Annealing: A Proof of Convergence. IEEE Transactions on Pattern Analysis and Machine Intelligence, 16, 652-656. https://doi.org/10.1109/34.295910
[10]
Nolte, A. and Schrader, R. (1997) A Note on the Finite Time Behaviour of Simulated Annealing. In: Zimmermann, U., Derigs, U., Gaul, W., M?hring, R.H. and Schuster, K.-P., Eds., Operations Research Proceedings, Springer, 175-180. https://doi.org/10.1007/978-3-642-60744-8_32
[11]
Glover, F. and Laguna, M. (1997) Tabu Search. Kluwer Academic Publishers.
[12]
Slowik, A. and Kwasnicka, H. (2020) Evolutionary Algorithms and Their Applications to Engineering Problems. Neural Computing and Applications, 32, 12363-12379. https://doi.org/10.1007/s00521-020-04832-8
[13]
Aubin, J.P. and Ekeland, I. (1984) Applied Nonlinear Analysis. Wiley.
[14]
Bauschke, H.H. and Combetts, P.L. (2017) Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer.
[15]
Himmelblau, D. (1972) Applied Nonlinear Programming. McGraw-Hill.
[16]
Burden, R.L. and Faires, J.D. (2005) Numerical Solutions of Non-Linear Systems of Equations. In: Burden, R.L. and Faires, J.D., Eds., Numerical Analysis, Thomson Brooks/Cole, 597-640.