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Not necessarily as written in the generality you want. Suppose that we are in $\mathbb R^1$, there is no linear constraint, and $Q(y)$ is some positive function of $y$. Then $v(y)\equiv 0$, which is c …

Anyway, If you know a 5 line proof for the first inequality please share it with us OK, here goes. Assume that $\inf_P f=-1$ and that it is (nearly) attained at the origin. Let $K=\{x\in P: f(x)\le …

In other words, you are asking if the scalar product with every vertex is maximized by the vertex itself (we can certainly create $r$ to make any vertex we want the maximizer of the scalar product wit …

I would try to approach your original problem a bit differently. Note that $|x-a|\le \frac 12(r|x-a|^2+r^{-1})$ and the equality is attained for $r=|x-a|^{-1}$. Thus, $$ \min_x[|x-a|+\langle b,x\rang …

For what it is worth, I played a bit with the non-constrained optimization of this type and noticed a strange thing. Suppose that you want to minimize $\sum_i a_i\max (F_i(x),0)^2$. Choose any $y$. In …