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

Fatma Kilinç-Karzan, Simge Küçükyavuz, Dabeen Lee, Soroosh Shafieezadeh-Abadeh

Abstract: 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, fj, 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 fj’s, which is easy to do when all functions fj’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. Funding: The research is supported, in part, by ONR [Grants N00014-19-1-2321 and N00014-22-1-2602], AFOSR [Grant FA9550-22-1-0365], the Institute for Basic Science [IBS-R029-C1, Y2], the FOUR Brain Korea 21 Program [NRF-5199990113928], the National Research Foundation of Korea [NRF-2022M3J6A1063021], the KAIST Starting Fund [KAIST-G04220016], NSF [Grant CMMI 1454548], and Early Postdoc Mobility Fellowship SNSF [Grant P2ELP2_195149].