a r t i c l e i n f o Article history: Available online xxxx MSC: 90C11 90C30 a b s t r a c tMotivated by concave cost combinatorial optimization problems, we study the following mixed integer nonlinear set:is a concave function, n and k are positive integers, a ∈ R n is a nonnegative vector, e ∈ R n is a vector of ones, and x ′ y denotes the scalar product of vectors x and y of same dimension. A standard linearization approach for P is to exploit the fact that f (a ′ x) is submodular with respect to the binary vector x. We extend this approach to take the cardinality constraint e ′ x ≤ k into account and provide a full description of the convex hull of P when the vector a has identical components. We also develop a family of facet-defining inequalities when the vector a 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.

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The expected utility knapsack problem is to pick a set of items with random values so as to maximize the expected utility of the total value of the items picked subject to a knapsack constraint. We devise an approximation algorithm for this problem by combining sample average approximation and greedy submodular maximization. Our main result is an algorithm that maximizes an increasing submodular function over a knapsack constraint with an approximation ratio better than the well known (1-1/e) factor.

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