Distributionally robust optimization through the lens of submodularity

Distributionally robust optimization is used to solve decision making problems under adversarial uncertainty where the distribution of the uncertainty is itself ambiguous. In this paper, we identify a class of these instances that is solvable in polynomial time by viewing it through the lens of submodularity. We show that the sharpest upper bound on the … Read more

Submodularity, pairwise independence and correlation gap

In this paper, we provide a characterization of the expected value of monotone submodular set functions with $n$ pairwise independent random inputs. Inspired by the notion of “correlation gap”, we study the ratio of the maximum expected value of a function with arbitrary dependence among the random inputs with given marginal probabilities to the maximum … Read more

Tight Probability Bounds with Pairwise Independence

\(\) While useful probability bounds for \(n\) pairwise independent Bernoulli random variables adding up to at least an integer \(k\) have been proposed in the literature, none of these bounds are tight in general. In this paper, we provide several results in this direction. Firstly, when \(k = 1\), the tightest upper bound on the … Read more

Extremal Probability Bounds in Combinatorial Optimization

In this paper, we compute the tightest possible bounds on the probability that the optimal value of a combinatorial optimization problem in maximization form with a random objective exceeds a given number, assuming only knowledge of the marginal distributions of the objective coefficient vector. The bounds are “extremal” since they are valid across all joint … Read more

Robust Conic Satisficing

In practical optimization problems, we typically model uncertainty as a random variable though its true probability distribution is unobservable to the decision maker. Historical data provides some information of this distribution that we can use to approximately quantify the risk of an evaluation function that depends on both our decision and the uncertainty. This empirical … Read more