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I am aware of the definition of a cone set $C$. $x_1, x_2 \in C$ and $\theta_1, \theta_2 \ge 0$

$$\theta_1x_1+\theta_2x_2 \in C$$

However, what is a first-order and second-order cone?

Specifically, relating to second order cone programming, why do we call the constraint

$$ \lVert Ax+b \rVert_2 \le c^Tx+d$$

a second order constraint and why do we call it a cone constraint?

Rufus
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  • This is a second order cone constraint as it contains quadratic terms in x from the euclidian $||.||_{2}$ norm. – fGDu94 Nov 25 '19 at 03:20
  • @GeorgeDewhirst I also don't fully understand why it's called a cone constraint in the first place, would you care to elaborate on that as well? – Rufus Nov 25 '19 at 04:04
  • Because whenever the $x$ satisfies this inequality it will belong to a cone – fGDu94 Nov 25 '19 at 04:09
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    Actually, @GeorgeDewhirst that is only true if $b=0$, $d=0$. In general, the set of $x$ that satisfies the constraint is not a cone. The reason this is a conic constraint is that its of the form $(Ax+b,c^Tx+d)\in C$, where $C$ is the cone ${(y,z),|,|y|_2\leq z}$. – Michael Grant Nov 27 '19 at 02:50
  • fair play....... – fGDu94 Nov 27 '19 at 02:53
  • @MichaelGrant Could you please give an example of an x that satisfies the constraint but is not in the cone? Also, could you elaborate on what you say "it is only true if $b=0, d=0$?

    Is there a decent textbook that you can recommend with a good chapter on cones?

     – Frank Jan 17 '20 at 02:48

1 Answers1

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It's called the second order cone because it's defined by a quadratic equation rather than a linear one. This naming is consistent with a quadratic polynomials being of degree two while a linear polynomial is of degree one.

CyclotomicField
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  • I also don't fully understand why it's called a cone constraint in the first place, would you care to elaborate on that as well? – Rufus Nov 25 '19 at 04:01
  • @Rufus Linear cones and quadratic cones are both bundle of lines connecting points on the interior to a special convex subset of the cone. For a typical quadratic cone that's the single point at the "apex" of the cone. Informally linear cones are similar but have hyper-plane boundaries instead of hyper-circles. – CyclotomicField Nov 25 '19 at 04:17