I am a newbie to SE. I am a computer science undergrad pursuing my masters in Data Science.
I am trying to prove the following result.
For $i=1,\ldots,m$, let $\alpha_i\in\mathbb{R}^n$ and $c_i>0$. Put $f=\sum c_c \exp((\alpha_i,x))$. Show that $\log(f_i)$ is convex. Hint: compute the second derivative and write the numerator as a sum of exponentials with coefficients. Use the inequality between arithmetic mean and geometric mean to show that the coefficients are positive.
Proof. (My Attempt)
I think that the exponent involves an inner product. So, I might write
$\begin{align} \log f_i(x)&=\log(\sum ce^{\alpha^T_{ik}\boldsymbol{x}})\\ \partial(\log f_i(x))&=\frac{ce^{\alpha^T_{ik}\boldsymbol{x}}}{\sum ce^{\alpha^T_{ik}\boldsymbol{x}}}\alpha^T_{ik} \end{align}$
But, I am not sure if my first order derivative is correct and how to proceed ahead. Any inputs, tips leading to the solving the problem would help.
Also, this happens to be my first exposure to applied linear algebra and numerical optimization. I find very hard understanding what the problem asks for and find it difficult to follow my class lectures. The recommended text is https://web.stanford.edu/~boyd/cvxbook/.
What are some books/material I can refer, to get to grips with the absolute basics of numerical optimization? And what are its pre-requisites? I would like to work towards successfully completing my course, making the best out of the time. Sorry, if this has already been asked in another thread, but I felt my situation is unique.
Kind Regards, Nasreen