1

I am new to optimization and I will try to state my question as clear as I can. I need to solve this constraint optimization problem.

I want to find real vectors $\mathbf{f}$ and $\mathbf{g}$ that minimize $e^2$ $$ e^2 = \mathbf{f}^TR_{yy}\mathbf{f}+\mathbf{g}^TR_{aa}\mathbf{g} -2\mathbf{f}^TR_{ya}\mathbf{g} $$ where $R_{yy}$,$R_{aa}$, and $R_{ya}$ are known. And

$$\mathbf{f}=[f_0 f_1 …f_n]^T$$ $$\mathbf{g}=[g_0 g_1 …g_m]^T$$

The constraint is on the last coefficient of $\mathbf{g}$ i-e $g_m=g_0g_1$.

My understanding is that $\mathbf{f}$ can be found by differentiating $e^2$ wrt $\mathbf{f}$ and equating it to zero.This leads to $\mathbf{f}_{opt}=R_{yy}^{-1}R_{ya}\mathbf{g}$. I thought of finding $g_i$ for $i=0,1,…,m-1$ in the same manner but this leads me to trivial solution $g_i=0 \forall i=0,1,…,m-1$. I understand I need to include constraints in my original equation to get to actual coefficients but I don't know how to do that. I will appreciate if someone can give me some useful lead here.

NAASI
  • 997
  • The solution to the unconstrained problem, $\mathbf f=\mathbf 0,\mathbf g=\mathbf 0$, also satisfies the constraint $g_m=g_0g_1$, so it is also a solution to the constrained problem. There may be some other conditions or constraints you have forgotten about. –  Jan 06 '15 at 07:33
  • @ Rahul…Last two lines of my question say the same thing. I need some help in modifying my objective function so that I get a non-trivial solution to this problem – NAASI Jan 06 '15 at 21:07

1 Answers1

0

It is possible to take the derivatives of $e^2=\mathbf{f}^TR_{yy}\mathbf{f}+\mathbf{g}^TR_{aa}\mathbf{g} -2\mathbf{f}^TR_{ya}\mathbf{g} $

You have to know the laws.

$\frac{\partial e^2}{\partial f}=\left( R_{yy}+R_{yy}^{T}\right)\mathbf{f}-2R_{ya}\mathbf{g}=0$

$\frac{\partial e^2}{\partial g}=\left( R_{aa}+R_{aa}^{T}\right)\mathbf{g}-2\left(\mathbf{f}^TR_{ya}\right)^T=\left( R_{aa}+R_{aa}^{T}\right)\mathbf{g}-2R_{ya}^T\mathbf{f}=0$

callculus42
  • 30,550
  • 1
    The constraint on the last entry of $\mathbf{g}=g_0g_1$ is not satisfied when $e^2$ is differentiated wrt $\mathbf{g}$. This approach leads to a general $\mathbf{g}$ which minimizes $e^2$ when we have no constraint on the entries of $\mathbf{g}$. – NAASI Dec 31 '14 at 17:26