Conic Mixed-Binary Sets: Convex Hull Characterizations and Applications

TL;DR

This paper provides a unified convex hull characterization for a broad class of conic mixed-binary sets, generalizing previous results and enabling improved formulations for various optimization problems.

Contribution

It introduces a convex hull description for conic mixed-binary sets involving multiple convex cones and arbitrary nonnegative functions, extending existing submodular-based results.

Findings

Unified convex hull description for conic mixed-binary sets.

Applicable to diverse optimization problems like portfolio and scheduling.

Generalizes previous results to multiple cones and arbitrary functions.

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, $f_j$, 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 \emph{multiple general convex cone} constraints instead of a single second-order cone type constraint. Second, it takes \emph{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.