How would I write the following as a standard form LP? Minimizing $\sum_{i=1}^n x_i + c\max(a_i-x_i)$ for $a_i \ge 0$ and what is the optimal value for when $c=n$
How to express minimize $\frac{1}{2} ||x-y||_2^2$ subj to $y \in K \text{ a closed convex cone}$ as a standard form quadratic program? Why is the solution unique?
For 1, I want something of form min $<c,x>$ sub to $Ax=b$ but how do I take care of the max in my case? For 2, the standard form QP is min $x^TQx+c^Tx$ sub to $Ax=b$. So is it just $\frac{1}{2} (x-y)^T(x-y)$ what are the constraints then? Any help is greatly appreciated!
(1) : Basically the trick will be to get the $\max$ out of the objective function. Otherwise it's linear, right? So what you do is to add a new variable. Let $t = \max(a_i - x_i)$, or in the form of constraints, $t \geq a_i - x_i$ for all $i$. When you make any of these constraints tight, $t$ will be the maximum of all the $a_i-x_i$.
(2) : Hint: $\|x-y\|_2^2 = \|x\|_2^2+\|y\|_2^2-2\langle x,y\rangle$. Your variable is going to be in the form $\begin{pmatrix}x\\y\end{pmatrix}$, and you can use the fact that $\langle x,y\rangle = x^TIy$. I'll leave the rest to you.
As for $y\in K$, this is standard notation for QPs. Unless you have more specifics about how $K$ is constructed, you can't really convert it any further.