Proper cone

Summary

A proper cone in $\R^n$ is a closed convex cone with nonempty interior that contains no line. Proper cones define the generalized inequalities $x \preceq_K y \iff y - x \in K$ used in conic and semidefinite optimisation. The nonnegative orthant, the second-order cone and the cone of positive semidefinite matrices are the standard examples. The dual cone of a proper cone is again proper.