A Short Proof of Strassen’s Theorem Using Convex Analysis

We give a simple proof of Strassen’s theorem on stochastic dominance using linear programming duality, without requiring measure-theoretic arguments. The result extends to generalized inequalities using conic optimization duality and provides an additional, intuitive optimization formulation for stochastic dominance. Citation Northwestern Univ., Aug., 2013 Article Download View A Short Proof of Strassen's Theorem Using Convex … Read more

Decision Making under Uncertainty when Preference Information is Incomplete

We consider the problem of optimal decision making under uncertainty but assume that the decision maker’s utility function is not completely known. Instead, we consider all the utilities that meet some criteria, such as preferring certain lotteries over certain other lotteries and being risk averse, s-shaped, or prudent. This extends the notion of stochastic dominance. … Read more

Models and Formulations for Multivariate Dominance Constrained Stochastic Programs

Dentcheva and Ruszczynski recently proposed using a stochastic dominance constraint to specify risk preferences in a stochastic program. Such a constraint requires the random outcome resulting from one’s decision to stochastically dominate a given random comparator. These ideas have been extended to problems with multiple random outcomes, using the notion of positive linear stochastic dominance. … Read more

Sample Complexity of Smooth Stochastic Optimization

Let $N(\epsilon,\delta)$ be the number of samples needed when solving a stochastic program such that the objective function evaluated at the sample optimizer is within $\epsilon$ of the true optimum with probability $1-\delta$. Previous results are of the form $N(\epsilon,\delta)=O(\epsilon^{-2}\log \delta^{-1})$. However, a smooth objective function is often locally quadratic at an interior optimum. For … Read more