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I have a convex function $f: \mathbb{R}^{n}\rightarrow \mathbb{R}$ that achieves minimum at some point of $\mathbb{R}^{n}$. I want to know if the problem \begin{equation*} \begin{array}{c c} \text{minimize}_{x\in\mathbb{R}^n} & f(x) \\ \text{subject to} & \begin{aligned} x \in \Omega,\\ \end{aligned} \end{array} \end{equation*} with $\Omega$ closed and convex, have a solution. Does it have?

R. W. Prado
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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.

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  • And If $\Omega$ is a vector space? – R. W. Prado Jan 06 '22 at 22:07
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    What if the function $f$ is strictly convex? – Hosein Rahnama Dec 09 '23 at 14:42
  • When a convex function is strictly convex, it is coercive. When a function is coercive, such a function achieves minimum over any closed domain, and not only convex domain, see https://math.stackexchange.com/questions/2240269/coercive-continuous-function-on-a-closed-subset-has-a-global-minimum-proof – R. W. Prado Dec 10 '23 at 22:13