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We consider the problem of minimizing a convex function over a subset of R^nthat is not necessarily convex (minimization of a convex function over theinteger points in a polytope is a special case). We define a family of dualsfor this problem and show that, under some natural conditions, strong dualityholds for a dual problem in this family that is more restrictive thanpreviously considered duals.

Optimality certificates for convex minimization and Helly numbers