Finding a minimum for a "vector" function

**Marril** · May 4th 2011, 04:55 AM

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

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

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

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

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

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

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

**Prove It** · May 4th 2011, 05:26 AM

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

**Marril** · May 4th 2011, 06:57 AM

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

This creature $L_i$ is not really a proper vector, and all calculations are done "element by element".
Following this, $(L_{i+1} - L_i)^2$ 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 $L_i \geqslant X_i$ 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:
$H[f(x)] = \int_{0}^d [\alpha \nabla f(x)^2 + P \cdot f(x)]dx$
With the condition: $f(x) \geqslant g(x)$

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