Linear regression model is linear in inputs or linear in terms of its optimization function that is minimized to learn the weights? In various literature i have seen both being claimed and now i am confused which one is true ?
The one that says linear in terms of weights claims that one can simply add higher order terms for the inputs and still learn non linear models while optimization problem remains linear in terms of weights.
There are other that claim it is linear in terms of inputs.
Can someone clarify on this ?
Following is what is given in Elements of Statistical Learning :

 A: This is quite an old post so maybe you already settled this problem. However, I also got stuck on the same problem and I think I understand it now. My inuition had always been that the model linear in the input, for the reason that we often think of the inputs being transformed to produce an output that we can then calculate an error of and train the model. So when I first saw "linear in the weights" this didn't make sense at all. However, one person pointed out that from the perspective of the learning algorithm itself, the data (i.e. the X_j in your equation) are fixed and cannot change. The only relevant variables are the weights, which we train. Therefore, the input data somehow play the role of coefficients, and the weights as variables.
A: Recall that linearity is defined as following. Function $f$ is linear if 
$$f(\alpha x+\beta y)=\alpha f(x)+\beta f(y),$$
where $f$, $\alpha,\beta$ and $x,y$ are such that this equation makes sense. 
The linear regression model is defined as
$$y_i=\sum_{i=1}^p\alpha_i x_i+\varepsilon_i,$$
which we can write as 
$$y_i=f(x_1,...,x_p)+\varepsilon_i=g(x_1,...,x_p,\varepsilon_i).$$
Both $f$ and $g$ are linear functions. 
Now the optimisation function for the linear regression is 
$$c(\alpha_1,...,\alpha_p)=\sum_{i=1}^n(y_i-f(x_1,...,x_p))^2.$$
It is not a linear function. 
Hope this clarifies things a bit.
