Browsing by Subject "Trajectory methods"

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    “Backward differential flow” may not converge to a global minimizer of polynomials
    (Springer New York LLC, 2015) Arıkan, Orhan; Burachik, R. S.; Kaya, C. Y.
    We provide a simple counter-example to prove and illustrate that the backward differential flow approach, proposed by Zhu, Zhao and Liu for finding a global minimizer of coercive even-degree polynomials, can converge to a local minimizer rather than a global minimizer. We provide additional counter-examples to stress that convergence to a local minimum via the backward differential flow method is not a rare occurence.
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    Steklov regularization and trajectory methods for univariate global optimization
    (Springer, 2020) Arıkan, Orhan; Burachik, R. S.; Kaya, C. Y.
    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.