正则对偶理论在Lennard-Jones势极小化问题中的应用
Canonical dual theory applied to a Lennard-Jones potential minimization problem
摘要：
简化的Lennard-Jones（LJ）势最小化问题是$f(x)=4\sum_{i=1}^N \sum_{j=1,j<i}^N (\frac{1}{τ_{ij}^6} -\frac{1}{τ_{ij}^3}) {subject to} x\in \mathbb{R}^n,$，其中$τ_{ij}=(x_{3i-2}-x_{3j-2})^2 +(x_{3i-1}-x_{3j-1})^2 +(x_{3i} -x_{3j})^2$，$(x_{3i-2},x_{3i-1},x_{3i})$是原子$i$在$\mathbb{R}^3$中的坐标，$i,j=1,2,...,N(\geq 2 \quad \text{integer})$，$n=3N$和$N$是原子的整数。目标函数的非凸性和大量的局部极小值，随着$N$呈指数增长，引起了许多数学优化专家的兴趣。在本文中，经典的对偶理论优雅地解决了这个问题，淀粉样蛋白纤维分子模型的建立。保留字：数学正则对偶理论$\cdot$数学优化$\cdot$ Lennard-Jones势最小化问题$\cdot$全局优化。
英文摘要：
The simplified Lennard-Jones (LJ) potential minimization problem is $f(x)=4\sum_{i=1}^N \sum_{j=1,j<i}^N (\frac{1}{τ_{ij}^6} -\frac{1}{τ_{ij}^3}) {subject to} x\in \mathbb{R}^n,$ where $τ_{ij}=(x_{3i-2}-x_{3j-2})^2 +(x_{3i-1}-x_{3j-1})^2 +(x_{3i} -x_{3j})^2$, $(x_{3i-2},x_{3i-1},x_{3i})$ is the coordinates of atom $i$ in $\mathbb{R}^3$, $i,j=1,2,...,N(\geq 2 \quad \text{integer})$, $n=3N$ 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.
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