In contrast to symmetric cone optimization, there has no unified framework for non-symmetric cone optimization. One main reason is that the structure of various non-symmetric cone differs case by case. Especially, their boundary conditions are usually mysterious. In this paper, we provide characterizations of boundary conditions on some non-symmetric cones, including p-order cone, ellipsoidal cone, power cone and general closed convex cone. These results will be key bricks for further investigations on non-symmetric cone optimization accordingly.
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