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 appropriate dimension. Such minimization problems frequently appear in applications that involve risk-aversion or economies of scale. We propose three classes of strong valid linear inequalities for this convex hull and specify their facet conditions when $a$ has two distinct values. We show how to use these inequalities to obtain valid inequalities for general $a$ that contains multiple values. We further provide a complete linear convex hull description for this mixed-integer set when $a$ contains two distinct values and the cardinality constraint upper bound is two. Our computational experiments on the mean-risk optimization problem demonstrate the effectiveness of the proposed inequalities in a branch-and-cut framework.

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