Math Help - Finding a minimum for a "vector" function

Suppose I have a "scalar" function which takes a "vector" ( ) for an argument:



The vector (D dimensional) takes only integer values.

Also suppose that , and are known.
To close the system we choose quasi-periodic boundaries .

I wish to find a "vector" such that takes its minimum value, under an assumption that , where the integer values of are known.

I'm looking for two kind's of solutions:
1) When I'm simply given a vector and asked to find , with some arbitrary initial guess for .
2) When I have found a solution to this problem, and then one of the '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 is not really a proper vector, and all calculations are done "element by element".
Following this, 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 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:

With the condition: