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The following is from Prof. Jahn's book " Intro. to the theory of nonlinear optimization" about tangent cone:

enter image description here

After definition, he gave an example:

enter image description here

My question is: Does the tangent cone include S? (In the def., it says tangent cone is a set of all tangent vectors to S at bar(x))

In the figure, T(S, bar(x)) should not include the region S?

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First of all, let's clear something up. Note that the tangent cone is a set of vectors at a point, which is not exactly the set of points in the highlighted region (the two sets contain elements of different "data types"). The question that you seem to be thinking of, then, is whether the tangent cone at a point contains vectors that point from $\bar x$ into $S$.

Note that if your point $\bar x$ happens to be $\bar 0$, then you can validly think of the vectors emerging from $\bar x$ as points whose coordinates are given by their components, hence the first picture.

In general, the tangent cone (at a point $\bar x \in S$) may or may not contain vectors that, when drawn emerging from $\bar x$, pass through $S$. In your particular example, the tangent cone does contain such vectors. In general, as long as $S$ is convex, you can think of the tangent cone as comprising (of the closure) of "all the directions that take you through $S$".

As an example of when the tangent cone doesn't necessarily contain vectors pointing into the space, take your region to be the border of the set $S$ that is drawn.

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  • $\begingroup$ I have a question about tangent cones here: math.stackexchange.com/questions/2502215/… and was wondering if you could help me with it. You've answered questions for me before, and I always find what you have to say extremely beneficial. Thank you. $\endgroup$ – ALannister Nov 4 '17 at 17:05
  • $\begingroup$ there is a 100 point bounty on it. $\endgroup$ – ALannister Nov 4 '17 at 23:51
  • $\begingroup$ @DaenerysDracarys I don't remember answering any of your questions before; perhaps you changed your username. In any case, I'll definitely take a look. For what it's worth I'm not particularly knowledgeable when it comes to tangent cones or optimization for that matter; I just answered this question by looking carefully at the definition. $\endgroup$ – Omnomnomnom Nov 5 '17 at 13:51
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$T(S, \bar{x}) + \bar{x}$ includes $S$ if $S$ is convex. Note that it is also a cone, i.e. if $y \in T(S, \bar{x})$, then $\lambda y \in T(S, \bar{x})$ for any real $\lambda \geq 0$.

To see the first assertion, for any $x \in S$ take a sequence $(x_n)$ of points on the segment $\{(1-\alpha) x + \alpha\bar{x}: \alpha \in [0, 1)\}$ that converges to $\bar{x}$. Since $S$ is convex, this segment is contained in $S$. If $x_n = (1-\alpha_n) x + \alpha_n\bar{x}$, take $\lambda_n = 1/(1-\alpha_n)$, and you have $\lambda_n(x_n - \bar{x}) = x - \bar{x}$ for all $n$.

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