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I'm reading texts about optimization when given a set of polynomial constraints, like $f_1=\cdots=f_m=0$ and $q_1\ge 0,\ldots,q_\ell\ge 0$, over the reals.

I have encountered statements about "the cone" of a polynomial or a set of polynomials.

For example, sometimes the cone is defined to be the minimal set $C$ that contains a given collection $f_1,...,f_m$ of polynomials, such that if $g,h\in C$ then $g\cdot h\in C$, and $ag+bh\in C$, for $a,b$ positive scalars in $\mathbb{R}$. Here is an example use of the term cone that is undefined: here

Question: What is a cone (in the context of polynomial equations, optimization and SDP)? What is $\operatorname{cone}(f)$ for $f$ a polynomial? And what is $\operatorname{cone}(S)$ for $S$ a set of polynomials?

Jack
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  • Please cite your source. – Brian Borchers Dec 19 '17 at 23:57
  • It's likely that the author means "the conical hull of the set $S$", which is the set of all points $x$ such $x$ can be written as a linear combination of $k$ points $x_{1}$, $x_{2}$, $ldots$, $x_{k}$ $S$ with nonnegative coefficients in the linear combination. – Brian Borchers Dec 20 '17 at 00:11
  • There is no single source. It's multiple sources. – Jack Dec 20 '17 at 12:44
  • I don't think this is what they mean. I added an example. – Jack Dec 20 '17 at 12:45
  • @Jack Your example agrees with Brian's comment. Pretty sure the author is talking about the conical hull, i.e. the set of all conical combinations of the given vectors (which are polynomials in your case) – M. Winter Dec 20 '17 at 16:41
  • @M.Winter, hmm...maybe. But then, how come we can multiply by polynomials and not just by scalars? – Jack Dec 21 '17 at 00:04
  • @Jack You are right! This $g\cdot h\in C$ is strange. It would be very helpful to know the source of this example! Is there a chance you can give it? – M. Winter Dec 21 '17 at 01:30
  • Here, www.mit.edu/~parrilo/ecc03_course/06_positivstellensatz.pdf He uses cone, but don't define it, since these are slides. He uses cone in two different settings, one I think as you did, and another as something about polynomial multiplication... – Jack Dec 21 '17 at 15:46
  • The definition of a cone in polynomial optimization is somewhat more general than the more commonly used (in other areas of convex optimization) definition of a cone. Parillo's notes are pretty clear on this unusual definition: http://www.mit.edu/~parrilo/ecc03_course/03_algebra_and_duality.pdf – Brian Borchers Dec 21 '17 at 18:37
  • Now, given the definitions in part 3 of Parillo's notes, what is your question? – Brian Borchers Dec 21 '17 at 18:38
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    well, the question is whether there is a connection to other "cones" as you defined it? What is the meaning of such a cone? I.e., why is it called a cone? – Jack Dec 21 '17 at 22:26

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