Canonical dual theory applied to a Lennard-Jones potential minimization problem

Abstract

The simplified Lennard-Jones (LJ) potential minimization problem is $f(x)=4{\sum }_{i=1}^N{\sum }_{j=1,j<i}^N(\frac{1}{{\tau }_{ij}^6}-\frac{1}{{\tau }_{ij}^3})subjecttox\in {\mathbb{R}}^n,$ where ${\tau }_{ij}=(x_{3i-2}-x_{3j-2}{)}^2+(x_{3i-1}-x_{3j-1}{)}^2+(x_{3i}-x_{3j}{)}^2$, $(x_{3i-2},x_{3i-1},x_{3i})$ is the coordinates of atom $i$ in ${\mathbb{R}}^3$, $i,j=1,2,...,N(\ge 2\text{integer})$, $n=3N$ and $N$ is the whole number of atoms. The nonconvexity of the objective function and the huge number of local minima, which is growing exponentially with $N$, interest many mathematical optimization experts. In this paper, the canonical dual theory elegantly tackles this problem illuminated by the amyloid fibril molecular model building. Keywords: Mathematical Canonical Duality Theory $\cdot$ Mathematical Optimization $\cdot$ Lennard-Jones Potential Minimization Problem $\cdot$ Global Optimization.

arXiv commentPlease cite this paper as a book-chapter: "Jiapu Zhang (2011), Canonical dual theory applied to a Lennard-Jones potential minimization problem, in Book [Jiapu Zhang, Practical Global Optimization Computing Methods in Molecular Modelling - for Atomic-resolution Structures of Amyloid Fibrils, ISBN 978-3-8465-2139-7, LAP Academic Publisher, 2011], Chapter 5, pages 94-114

References in corpus (1)

• On the Triality Theory in Global Optimization