Deformation Methods of Global Optimization in Chemistry and Physics

  • Lucjan Piela
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 62)


Deformation of the target function in global optimization has been a novel possibility for the last decade. The techniques based on the deformation turned out to be related to a variety of fundamental laws: diffusion equation, time-dependent Schrödinger equations, Smoluchowski dynamics, Bloch equation of canonical ensemble evolution with temperature, Gibbs free-energy principle. The progress indicator of global optimization in those methods takes different physical meanings: time, imaginary time or the inverse absolute temperature. Despite of the fact that the phenomena described are different, the resulting global optimization procedures have a remarkable similarity. In the case of the Gaussian Ansatz for the wave function or density distribution, the underlying differential equations of motion for the Gaussian position and width are of the same kind for all the phenomena. The original potential energy function is smoothed by a convolution with a Gaussian distribution, its center denoting the current position in space during the minimization. The Gaussian position moves according to the negative gradient of the smoothed potential energy function. The Gaussian width is position dependent through the curvature of the smoothed potential energy function, and evolves according to the following rule. For sufficiently positive curvatures (close to minima of the smoothed potential) the width decreases, thus leading to a smoothed potential approaching the original potential energy function, while for negative curvatures (close to maxima) the width increases leading eventually to disappearance of humps of the original potential energy function. This allows for crossing barriers separating the energy basins. Some methods result in an additional term, which increases the width, when the potential becomes flat. This may be described as a feature allowing hunting for distant minima. Some deformation methods that are of non-convolutional character are also discussed.


Global Optimization Target Function Potential Energy Function Bloch Equation Deformation Method 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer Science+Business Media Dordrecht 2002

Authors and Affiliations

  • Lucjan Piela
    • 1
  1. 1.Quantum Chemistry Laboratory Department of ChemistryUniversity of WarsawWarsawPoland

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