Polyhedral results for a class of cardinality constrained submodular minimization problems

Motivated by concave cost combinatorial optimization problems, we study the following mixed integer nonlinear set: P={(w,x)∈R×{0,1}n:w≥f(a′x),e′x≤k}P={(w,x)∈R×{0,1}n:w≥f(a′x),e′x≤k} where f:R→Rf:R→R is a concave function, nn and kk are positive integers, a∈Rna∈Rn is a nonnegative vector, e∈Rne∈Rn is a vector of ones, and x′yx′y denotes the scalar product of vectors xx and yy of same dimension. A standard linearization approach for PP is to exploit the fact that f(a′x)f(a′x) is submodular with respect to the binary vector xx. We extend this approach to take the cardinality constraint e′x≤ke′x≤k into account and provide a full description of the convex hull of PP when the vector aa has identical components. We also develop a family of facet-defining inequalities when the vector aa has nonidentical components. Computational results using the proposed inequalities in a branch-and-cut framework to solve mean-risk knapsack problems show significant decrease in both time and the number of nodes over standard methods.