# Optimization of a matrix with an objective function (for ML)

Hi.

I need to do max. likelihood for an objective likelihood function L (minimize it), and the target is a matrix. ie:

$$min_KL(K)$$

For example: K is, let's say, of size 3x3 and with initial value of ones ($k_{i,j}=1∀i,j$). L is $L=∥\nabla(K)∥$ or $L=∥K∥^{1.1}$.

I know how to do gradient descend etc., but here I need to minimize the function L by iterating over K and I don't really know how to approach it. I'd expect something of this sort: $K:=K-f(\nabla(L))$, but I don't know what.

*note: It might have something to do with the Euler-Largange method ($L_x-L_t\left(L_{x'}\right)=0$) but I'd have to do it iteratively if any...

Appreciate any help.

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I think you will have to give significantly more information to get a useful answer. Optimization is a large field and you haven't restricted your problem very much so it is hard to say what methods would apply. Is $K$ constrained? If not, your second objective looks like it would be optimized at $K=0$. As for the first objective, it is not clear to me what $\nabla K$ means for a single matrix $K$. Finally, saying that you "need to" solve the problem in a certain way suggests homework help, which is well-received at math.stackexchange.com but generally not at MO as per the FAQ. – Noah Stein Jan 10 '12 at 13:47
Your notation is quite unclear. By $\| K \|^{1.1}$, do you mean the $p$ norm of $K$ with $p=1.1$?, or do you mean the $2$-norm of $K$ raised to the power 1.1? In the latter case, raising the norm to the power 1.1 has no effect on the minimum. By $\nabla K$, do you mean the kind of image gradient often used in image processing? It seems odd that you'd be applying that to a 3x3 matrix. Or, do you mean something else? – Brian Borchers Jan 11 '12 at 6:20
This is not a HW question, I didn't specify any other terms because I'm not sure about what I want to do yet. Obviously K=0 is a solution but since I do it iteratively I may not get there. The general form of L would be rather the sum of both L's I specified, so the exponents have a meaning. Also, I didn't decide yet whether I want the Frobenius or L2 norm. My question is more technical - Say I have a matrix K of size 3x3, how do I iteratively modify it so that I minimize L. Thanks. – id0 Jan 11 '12 at 7:11