Technical Note—Conic Mixed-Binary Sets: Convex Hull Characterizations and Applications

We consider a general conic mixed-binary set where each homogeneous conic constraint j involves an affine function of independent continuous variables and an epigraph variable associated with a nonnegative function, f j , of common binary variables. Sets of this form naturally arise as substructures in a number of applications, including mean-risk optimization, chance-constrained problems, portfolio optimization, lot sizing and scheduling, fractional programming, variants of the best subset selection problem, a class of sparse semidefinite programs, and distributionally robust chance-constrained programs. We give a convex hull description of this set that relies on simultaneous characterization of the epigraphs of f j ’s, which is easy to do when all functions f j ’s are submodular. Our result unifies and generalizes an existing result in two important directions. First, it considers multiple general convex cone constraints instead of a single second-order cone type constraint. Second, it takes arbitrary nonnegative functions instead of a specific submodular function obtained from the square root of an affine function. We close by demonstrating the applicability of our results in the context of a number of problem classes.