The simplified Lennard-Jones (LJ) potential minimization problem is $f(x)=4\sum_{ i=1}^N \sum_{ j=1,j<i}^N (\frac{ 1}{ \tau_{ ij}^6} -\frac{ 1}{ \tau_{ ij}^3}) { subject to} x\in \mathbb{ R}^n,$ 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 $\mathbb{ R}^3$, $i,j=1,2,...,N(\geq 2 \quad \text{ integer})$, $n=3N$ 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.
Categories: math.OC physics.bio-ph q-bio.BM