Abstract
We introduce a new regularization technique, using what we refer to as the Steklov regularization function, and apply this technique to devise an algorithm that computes a global minimizer of univariate coercive functions. First, we show that the Steklov regularization convexifies a given univariate coercive function. Then, by using the regularization parameter as the independent variable, a trajectory is constructed on the surface generated by the Steklov function. For monic quartic polynomials, we prove that this trajectory does generate a global minimizer. In the process, we derive some properties of quartic polynomials. Comparisons are made with a previous approach which uses a quadratic regularization function. We carry out numerical experiments to illustrate the working of the new method on polynomials of various degree as well as a non-polynomial function.
Similar content being viewed by others
References
Arıkan, O., Burachik, R.S., Kaya, C.Y.: “Backward differential flow” may not converge to a global minimizer of polynomials. J. Optim. Theory Appl. 167, 401–408 (2015)
Arnold, V.I.: Ordinary Differential Equations. The MIT Press, Cambridge (1978)
Attouch, H., Chbani, Z., Peypouquet, J., Redont, P.: Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity. Math. Program. 168(1–2), 123–175 (2018)
Borelli, R.L., Coleman, C.S.: Differential Equations. Wiley, New York (2004)
Bazaraa, M.S., Sherali, H.D., Shetti, C.M.: Nonlinear Programming: Theory and Algorithms, 3rd edn. Wiley, Hoboken (2006)
Boţ, R.I., Csetnek, E.R.: Convergence rates for forward-backward dynamical systems associated with strongly monotone inclusions. J. Math. Anal. Appl. 457(2), 1135–1152 (2018)
Chen, X.: Smoothing methods for nonsmooth, nonconvex minimization. Math. Program. Ser. B 134, 71–99 (2012)
Ermoliev, Y.M., Norkin, V.I., Wets, R.J.-B.: The minimization of semicontinuous functions: mollifier subgradients. SIAM J. Control Optim. 32, 149–167 (1995)
Garmanjani, R., Vicente, L.N.: Smoothing and worst-case complexity for direct-search methods in nonsmooth optimization. IMA J. Numer. Anal. 33, 1008–1028 (2013)
Gupal, A.M.: On a method for the minimization of almost-differentiable functions. Cybern. Syst. Anal. 13, 115–117 (1977)
Horst, R., Tuy, H.: Global Optimization: Deterministic Approaches. Springer, Berlin (1996)
Lera, D., Sergeyev, Y.D.: Acceleration of univariate global optimization algorithms working with Lipschitz functions and Lipschitz first derivatives. SIAM J. Optim. 23(1), 508–529 (2013)
Piyavskii, S.A.: An algorithm for finding the absolute extremum of a function. Comput. Math. Math. Phys. 12, 57–67 (1972)
Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (2004)
Scholz, D.: Deterministic Global Optimization: Geometric Branch-and-Bound Methods and Their Applications. Springer, New York (2012)
Snyman, J.A., Kok, S.: A reassessment of the Snyman–Fatti dynamic search trajectory method for unconstrained global optimization. J. Glob. Optim. 43, 67–82 (2009)
Stoer, J., Witzgall, C.: Convexity and Optimization in Finite Dimensions I. Springer, Berlin (1970)
Zhang, X., Xiong, Y.: Impulse noise removal using directional difference based noise detector and adaptive weighted mean filter. IEEE Signal Proc. Lett. 16, 295–298 (2009)
Zhu, J., Zhao, S., Liu, G.: Solution to global minimization of polynomials by backward differential flow. J. Optim. Theory Appl. 161, 828–836 (2014)
Acknowledgements
The authors offer their warm thanks to an anonymous referee whose comments and suggestions improved the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Arıkan, O., Burachik, R.S. & Kaya, C.Y. Steklov regularization and trajectory methods for univariate global optimization. J Glob Optim 76, 91–120 (2020). https://doi.org/10.1007/s10898-019-00837-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-019-00837-3