We propose a randomized method for solving linear programs with a large number of columns but a relatively small number of constraints. Since enumerating all the columns is usually unrealistic, such linear programs are commonly solved by column generation, which is often still computationally challenging due to the intractability of the subproblem in many applications. … Read more
In this paper we show that when a multistage stochastic problem with stage-wise independent realizations has only RHS uncertainties, solving one tree provides a valid lower bound for all trees with the same number of scenarios per stage without any additional computational effort. The only change to the traditional algorithm is the way cuts are … Read more
Distributionally robust chance-constrained programs (DR-CCP) over Wasserstein ambiguity sets exhibit attractive out-of-sample performance and admit big-$M$-based mixed-integer programming (MIP) reformulations with conic constraints. However, the resulting formulations often suffer from scalability issues as sample size increases. To address this shortcoming, we derive stronger formulations that scale well with respect to the sample size. Our focus … Read more
When considering the design of electricity-intensive industrial processes, a challenge is that future electricity prices are highly uncertain. Design decisions made before construction can affect operations decades into the future. We thus explore whether including electricity price uncertainty into the design process affects design decisions. We apply stochastic optimization to the design and operations of … Read more
Spectral risk measure (SRM) is a weighted average of value at risk (VaR) where the weighting function (also known as risk spectrum or distortion function) characterizes the decision maker’s risk attitude. In this paper, we consider the case where the decision maker’s risk spectrum is ambiguous and introduce a robust SRM model based on the … Read more
In this paper, we consider the rate of convergence with sample average approximation (SAA) under heavy tailed distributions and quantify it under both independent identically distributed (iid) sampling and non-iid sampling. We rst derive the polynomial rate of convergence for random variable under iid sampling. Then, the uniform polynomial rates of convergence for both random … Read more
Structured non-convex learning problems, for which critical points have favorable statistical properties, arise frequently in statistical machine learning. Algorithmic convergence and statistical estimation rates are well-understood for such problems. However, quantifying the uncertainty associated with the underlying training algorithm is not well-studied in the non-convex setting. In order to address this short-coming, in this work, … Read more
We propose a new universal framework for multi-stage decision making under limited information availability. It is developed as part of a larger research project which aims at providing analytical methods to compare and evaluate different models and algorithms for multi-stage decision making. In our setting, we have an open time horizon and limited information about … Read more
We consider an n-player non-cooperative game where each player has expected value payoff function and her strategy set is defined by a joint chance constraint. The random constraint vectors are independent. We propose a subset of probability distributions from elliptical family of distributions. We consider the case when the probability distribution of each random constraint … Read more
We consider a two player zero-sum game with stochastic linear constraints. The probability distributions of the vectors associated with the constraints are partially known. The available information with respect to the distribution is based mainly on the two first moments. In this vein, we formulate the stochastic linear constraints as distributionally robust chance constraints. We … Read more