We present an infinite family of linear valid inequalities for a mixed-integer conic program, and prove that these inequalities describe the convex hull of the feasible set when this set is bounded and described by integral data. The main element of our proof is to establish a new strong superadditive dual for mixed-integer conic programming that, unlike the existing dual from literature, is much cleaner to describe since it does not include directional derivative constraints, and becomes a finite-dimensional problem when the input data is integral.
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