# 2nd order approx. of barrier function

1. Sep 20, 2015

### FOIWATER

1. The problem statement, all variables and given/known data
write the 2nd order taylor series for the log barrier function $$-\sum_{i=1}^{m}(b_{i}-a_{i}^{T}x)$$

2. Relevant equations
See Above

3. The attempt at a solution
Here is my attempt at a solution
$$\nabla f(x)=f(x_{0})-\sum_{i=1}^{m}\bigg(\dfrac{a_{i}}{b_{i}-a_{i}^{T}x}\bigg)(x-x_{0})-\dfrac{1}{2}(x-x_{0})^{T}\sum_{i=1}^{m}\bigg(\dfrac{a_{i}^{T}a_{i}}{(b_{i}-a_{i}^{T}x)^{2}}\bigg)(x-x_{0})$$

So, my problem is i've never really encountered calculating gradients and hessians for vector valued functions of only one variable (x, in this case). Am I way out to lunch?

Thanks.

2. Sep 25, 2015

### Greg Bernhardt

Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

3. Sep 26, 2015

### HallsofIvy

Staff Emeritus
Perhaps it would help if you were to say exactly what the "log barrier function" is!

4. Sep 26, 2015

### FOIWATER

oh I typed it wrong

$$f(x)=-\sum_{i=1}^{m}log(b_{i}-a_{i}^{T}x)$$

But I know now that what I have above is not correct. I think the gradient might be right actually, but the hessian is certainly not.