Maximizing a Class of Utility Functions Over the Vertices of a Polytope

Given a polytope X, a monotone concave univariate function g, and two vectors c and d, we consider the discrete optimization problem of finding a vertex of X that maximizes the utility function c′x+g(d′x). This problem has numerous applications in combinatorial optimization with a probabilistic objective, including estimation of project duration with stochastic times, in reliability models and in multinomial logit models. We show that the problem is NP -hard for any strictly concave function g even for simple polytopes, such as the uniform matroid, assignment and path polytopes; and propose a 1/2-approximation algorithm for it. We discuss improvements for special cases where g is the square root, log utility, negative exponential utility and multinomial logit probability function. In particular, for the square root function, the approximation ratio is 4/5. Although the worst case bounds are tight, computational experiments with an implementation using Lagrangian relaxations indicate that the suggested approach finds solutions within 1-2% optimality gap for most of the instances very quickly.