# How to solve coupled optimization problem?

• May 27th 2011, 05:05 AM
ggyyree
How to solve coupled optimization problem?
Hi there,

I have got a problem of solving coupled optimization problem in real application. For example,

The objective function E(a,b) contains two sets of variables need to be optimized:

$\displaystyle E(x, a) = \frac{1}{2}\bigg(y - W(a)x\bigg)^2;$

Assume $\displaystyle W(a)$ is a transformation matrix describing the motion of the images, and $\displaystyle x$ is pixel intensity of the image. And $\displaystyle y$ is our observation. The purpose of this is to reconstruct the motion corrected image of several scene. Essentially, we would like to optimize w.r.t both $\displaystyle x$ and $\displaystyle a$.

For a general image reconstruction problem, the problem is linear; however, the motion introduce the nonlinearity. Therefore, the problem here is a coupled nonlinear optimization problem.

1. Is there a efficient way to solve this kind of coupled optimization problem?

2. Is that possible to linearize the problem?

Thanks for any suggestion or giving me some reference to read please.

Thanks a lot!
• May 27th 2011, 04:36 PM
ojones
Is this a discrete optimization problem? What's the size of the weight matrix $\displaystyle W(a)$?
• May 31st 2011, 03:12 AM
ggyyree
Quote:

Originally Posted by ojones
Is this a discrete optimization problem? What's the size of the weight matrix $\displaystyle W(a)$?

Thanks for your reply. Yes, it is.

The size of $\displaystyle W(a)$ is huge but $\displaystyle W(a)$ is sparse. Any ideas?

Thanks.
• May 31st 2011, 02:23 PM
ojones
And what's $\displaystyle a$, a scalar or a vector? What's complicating things is the dependence of the weight matrix on a parameter. Do you have any references for this type of problem?