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 class cardinality-constrained concave submodular minimization problems. This problems has an objective function form $$f(a^\top x)$$ , where f is univariate function, non-negative vector, and x binary vector appropriate dimension. Such frequently appear applications that involve risk-aversion or economies scale. propose three classes strong valid linear inequalities for this specify their facet conditions when two distinct values. show how to use these obtain general contains multiple further provide complete description values cardinality constraint upper bound two. Our computational experiments on mean-risk optimization problem demonstrate effectiveness proposed branch-and-cut framework.