I am studying convex optimisation, I keep seeing this phrase without any further explanation: "Proper cones can be used to generalise the idea of an inequality."
Could someone elaborate please?
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See Rockafellar, "Convex Analysis" or Boyd & Vandenberghe, "Convex Optimization" for example. – copper.hat Mar 18 '13 at 23:47
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The inequality $x\ge y$ can be written as $x-y\in [0,\infty)$. Here the set $[0,\infty)$ is a (proper) cone; it's closed under addition and multiplication by nonnegative scalars. Generalizing this, we can write $x\ge_C y$ if $x-y\in C$, where $C$ is a cone in some real vector space.
A simple example is $C=\{\vec x \in\mathbb R^n:x_i\ge 0 \ \forall i\}$, for which $\ge_C$ is the componentwise comparison of vectors. A less simple, but important example arises when $C$ is the cone of positive semidefinite matrices, considered as a subset of $\mathbb R^{n\times n}$.
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