Convexification of classes of mixed-integer sets with L♮-convexity
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Qimeng Yu
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