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Let $X$ be a convex set. Let $f:\mathbb{R}^n \to \mathbb{R}$ a $C^1$ function. The idea is to minimize f subject to $x \in X$.

Let $x^* \in X$ be a point such that $$\nabla f(x^*)^T (x-x^*) \geq 0$$ for all $x \in X$. Then can I conclude that $x^*$ is a local minimum? If not, is there an obvious counterexample?

root
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1 Answers1

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Here is a simple counter example in one dimension:

$X = [0, 2]$, $f(x) = -(x-1)^2$, $\nabla f(x) = -2(x-1)$

$\nabla f(x^*)^T(x-x^*) = -2(x^*-1)(x-x^*)$

Let $x^* = 1$. Then clearly the condition is zero for any choice of $x \in X$.

However $x^*$ is not a local minimum, it is a local maximum.

Titan
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