Thread: Finding a minimum for a "vector" function

Suppose I have a "scalar" function $\displaystyle H$ which takes a "vector" ($\displaystyle L_i$) for an argument:

$\displaystyle H(L_i) = \sum_{i=1}^D [\alpha (L_{i+1} - L_i)^2 + P \cdot L_i]$

The vector $\displaystyle L_i$ (D dimensional) takes only integer values.

Also suppose that $\displaystyle D$, $\displaystyle \alpha$ and $\displaystyle P$ are known.
To close the system we choose quasi-periodic boundaries $\displaystyle L_{D+1} = L_1$.

I wish to find a "vector" $\displaystyle L_i$ such that $\displaystyle H$ takes its minimum value, under an assumption that $\displaystyle L_i \geqslant X_i$, where the integer values of $\displaystyle X_i$ are known.

I'm looking for two kind's of solutions:
1) When I'm simply given a vector $\displaystyle X_i$ and asked to find $\displaystyle Min(H)$, with some arbitrary initial guess for $\displaystyle L_i$.
2) When I have found a solution to this problem, and then one of the $\displaystyle X_i$'s is changed by 1.

Any advice on how to economically solve this problem would be greatly appreciated!

What does it mean to square a vector? Do you mean the square of its magnitude?

I seem to have left out the thing asked by Prove It:

This creature $\displaystyle L_i$ is not really a proper vector, and all calculations are done "element by element".
Following this, $\displaystyle (L_{i+1} - L_i)^2$ would mean the subtraction of L's ith element from the (i+1)th element, squared.
This subtraction is just the discrete version of the gradient of L at point i, where now L can be viewed as a discrete function.

Same goes for $\displaystyle L_i \geqslant X_i$ which must be true element by element.


Now that I think about it, this problem is a discrete form of finding the minimum of a function H depending on the function f(x) like:
$\displaystyle H[f(x)] = \int_{0}^d [\alpha \nabla f(x)^2 + P \cdot f(x)]dx$
With the condition: $\displaystyle f(x) \geqslant g(x)$