Does convex functios $f$ always achieves a minimum on closed convex sets whenever the $f$ have an unconstrained minimum?
Solution 1:
In general no. Consider the counter-example in which $\Omega$ is the (unbounded) first-quadrant hyperbolic region $xy\geq 1$ and let $f(x,y)=|y|$, which is convex and has its minimum attained on the $x$ axis. The restriction of $f$ to $\Omega$ has no attained minimum.