Mixed-Integer Programming Approaches to Generalized Submodular Optimization and its Applications

Submodularity is an important concept in integer and combinatorial optimization. A classical submodular set function models the utility of selecting homogenous items from a single ground set, and such selections can be represented by binary variables. In practice, many problem contexts involve choosing heterogenous items from more than one ground set or selecting multiple copies … Read more

On Constrained Mixed-Integer DR-Submodular Minimization

DR-submodular functions encompass a broad class of functions which are generally non-convex and non-concave. We study the problem of minimizing any DR-submodular function, with continuous and general integer variables, under box constraints and possibly additional monotonicity constraints. We propose valid linear inequalities for the epigraph of any DR-submodular function under the constraints. We further provide … Read more

Strong valid inequalities for a class of concave submodular minimization problems under cardinality constraints

We study the polyhedral convex hull structure of a mixed-integer set which arises in a class of cardinality-constrained concave submodular minimization problems. This class of problems has an objective function in the form of $f(a^\top x)$, where $f$ is a univariate concave function, $a$ is a non-negative vector, and $x$ is a binary vector of … Read more

An Exact Cutting Plane Method for hBcsubmodular Function Maximization

A natural and important generalization of submodularity—$k$-submodularity—applies to set functions with $k$ arguments and appears in a broad range of applications, such as infrastructure design, machine learning, and healthcare. In this paper, we study maximization problems with $k$-submodular objective functions. We propose valid linear inequalities, namely the $k$-submodular inequalities, for the hypograph of any $k$-submodular … Read more

A Polyhedral Approach to Bisubmodular Function Minimization

We consider minimization problems with bisubmodular objective functions. We propose a class of valid inequalities, which we call the poly-bimatroid inequalities and prove that these inequalities, along with trivial bound constraints, fully describe the convex hull of the epigraph of a bisubmodular function. We develop a cutting plane algorithm for general bisubmodular minimization problems using … Read more