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In this lecture a tangent cone is defined as the closure of the feasible directions.

Definition 9. (Tangent Cone) Let $C ⊆\mathbb R^n$ be a nonempty set, and let $x ∈ C$. Then the tangent cone of $C$ at $x$, denoted by $TC(x)$, is defined as follows: $TC(x) = \mathrm{closure}(FC(x))$.

I do not have good visual as to what it looks like. What would be an example of tangent cone? Also what would be some application of tangent cone?

user26857
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Olórin
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1 Answers1

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The tangent cone is largely a theoretical construct used to prove other stuff, like coming up with constraint qualifications for KKT conditions if you've heard of those. It's not easy to construct in general. Consider the set $-x \leq 0, -y \leq 0, y \leq x^3, y \geq x^5$ and the cone of feasible directions at the origin. It is empty. The tangent cone, however, is the x-axis. Try graphing it and you might understand why.

Loosely speaking, the tangent direction at the origin is the x-axis, but the feasible region near the origin is "too thin" to allow for a feasible direction. There is no $\epsilon \in \mathbb{R}$ small enough to accomodate a step in ANY direction that will give you feasibility. But there is still a direction that is "pointing towards feasibility" - the tangent direction.

Edit: Regarding the visual, it would graph exactly like the cone of feasible directions. The difference is infinitesimal, litterally :) It's just that theoretically ever-so-important boundary that differs.

Benjamin Lindqvist
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  • Thank you for the answer which I just found on MSE (not my question, but recently I've been interested in this). I've a dumb question: I can clearly see that visually, the tangent cone is the x-axis in your example, but how come it's non-empty, being the closure of an empty set, namely the set of all feasible directions? – Mathguest Nov 07 '24 at 19:46
  • @LearningMath It is because they are not using the same definition of a tangent cone as in the original question. Consider the definition in these slides. It should be evident from this definition that the accepted answer occurs. As you can see in these slides the definitions are equivalent if we work with convex sets. – bassen Feb 13 '25 at 14:13