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I'm stuck on a problem for my optimization class.

Minimize $f(x)$ st $x \in \Omega$ where $\Omega=\{x\in \mathbb{R^2}:x_1^2+x_2^2\geq 1\}$ and $ f(x)=x_2$.

a) Find all points that satisfy FONC

b) Which points from (a) satisfy SONC

c) Which points from (a) are local minimizers

I have $d^T\nabla f=\begin{bmatrix} 0 & d_2\end{bmatrix}^T$, so any value st $d_2\geq 0$ satisfies $d^T\nabla f\geq0$. I'm stuck from here.

I found an answer online that goes $l(x,\mu)=\nabla f(x)+\mu\Omega(x) \implies 0=\begin{bmatrix}1 & 0\end{bmatrix}^T+\begin{bmatrix}2\mu_1x_1 \\ 2\mu_1x_2\end{bmatrix} \implies 2\mu_1x_1=-1$ and $2\mu_1x_2=0$ therefore $x_1=0,x_2=0$. This answer doesn't make any sense to me as the gradient seems backwards and they got $x_1=0$ when its equation is equal to $-1$.

If anyone can explain, that would be really appreciated.

Albibi
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    What are $d$, $l$ and $\mu$? In any way, are you sure there isn't a typo in the problem? As stated, the target function $f$ is not bound below, since any vector $(0, -M)$ satisfies the constraints $\Omega$ if $M \geq 1$. – lisyarus Oct 17 '23 at 23:00
  • Quoting some online solution that you didn't understand without giving any links really isn't helpful. We no way of determining whether the problem is in that solution or your understanding. Also, throwing around undefined symbols and abbreviations makes your post understandable to only those few who have seen the exact notations and abbreviations used. – Paul Sinclair Oct 18 '23 at 17:58
  • @lisyarus $d$ is what we use for a vector to describe feasible directions. $l$ and $\mu$ I do not know what they are as they are on the online answer, and that is why I don't understand the solution. The online solution is from Chegg, and I was hoping people would understand what it meant because I've never seen those symbols used in solutions. The problem itself is written exactly the same as in my textbook (I made one typo when I originally wrote the problem where instead of $x_1^2+x_2^2\geq 1$ I wrote $x_1^2+x_2^2\leq 1$, but fixed it within 5 minutes). – Albibi Oct 18 '23 at 18:33

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