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I wondered about this question for sometime.

Definition of Stationarity
(P)
$\mbox{min} f(x)$
s.t
$x\in C$
Let $f$ be $C^1$ function over a closed and convex set $C$ . then $x^*$ is called a stationary point of (P) if $\nabla f(x^*)(x-x^*)\geq0$ for any $x\in C$

Assume that we know the stationary condition for some two sets $A$ ,$B$ can we say something about the stationary condition for $C=A\cap B$?
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i.e if we look at $\mathbb{R}_n^+\cap B[0,r]$ can we get the conditions in easier way than checking again by definition?

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