What is the best way to minimize this constrained expression? Does it have a minimizer? Is it unique?

Question

Let $ N(x_0,r)$ be half a sphere in $\mathbb{R}^{3n}$ centered at $x_0$ and has radius $r$, and consider the function $U:\mathbb{R}^{3n}\to \mathbb{R}$ (This is an empirical potential for a cluster of $n$ particles, e.g Lennard Jones potential is defined via $$ U(x) =4 \epsilon \sum_{i<j}^n \left(\left[\frac{\sigma}{r_{ij}}\right]^{12}-\left[\frac{\sigma}{r_{ij}}\right]^{6}\right) $$ where $\epsilon$ and $\sigma$ are constants and $r_{ij}=$ distance between particles $i$ and $j$.) How one should think about this constrained minimization problem in $3n-1$ dimensions: $$ \text{Find }x\in N(x_0,r)\text{ such that }\quad U(x)+U(2x_0-x)=\min_{y\in N(x_0,r)} [U(y)+U(2x_0-y)] $$ What methods should I use to solve it?

Answer 1

With no loss in generality, we can set $x_0$ to $0$. Simply translate the system. Also with no loss in generality, we can imagine a sphere of $2n$ particles in a sphere centered at zero arranged in a pattern that has inversion symmetry (for every particle at $y$, there is another at $-y$).

This is a highly nonconvex minimization problem. It's not going to be easy to solve. Whether or not the solution is unique totally depends on the positions of these particles.

For each particle, there is an optimal distance $\rho$ you can be from it. However being at an optimal distance from one, may mean you're not at the ideal distance from another.

Example of uniqueness: Say that optimal distance is actually $\rho=r$! If all particles are arranged at the curved boundary of the sphere, the unique minimum is the origin.

Example of non-uniqueness: Say that we have $\rho$ much larger than $r$. You're going to want to be as far away from the particles are you can be. Say there are two particles: on at the "north pole" and one at the "south pole". Then the minimum is going to be any point on the "equator" of the sphere.

Existence: A minimum will always exist. Exclude a tiny radius around each particle where you know a minimum is never going to exist. If you remove all these spheres from your big sphere, then potential is a continuous function over a compact region, and therefore there will always be a minimizer. It can be proven that this minimizer can never be on the boundary of any of these little spheres. Therefore it is a minimizer over the entire sphere, and not just the restricted region. Therefore a minimizer must exist.