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Distributionally robust optimization (DRO) has been introduced for solving stochastic programs in which the distribution of the random variables is unknown and must be estimated by samples from that distribution. A key element of DRO is the construction of the ambiguity set, which is a set of distributions that contains the true distribution with a high probability. Assuming that the true distribution has a probability density function, we propose a class of ambiguity sets based on confidence bands of the true density function. As examples, we consider the shape-restricted confidence bands and the confidence bands constructed with a kernel density estimation technique. The former allows us to incorporate the prior knowledge of the shape of the underlying density function (e.g., unimodality and monotonicity), and the latter enables us to handle multidimensional cases. Furthermore, we establish the convergence of the optimal value of DRO to that of the underlying stochastic program as the sample size increases. The DRO with our ambiguity set involves functional decision variables and infinitely many constraints. To address this challenge, we apply duality theory to reformulate the DRO to a finite-dimensional stochastic program, which is amenable to a stochastic subgradient scheme as a solution method. 

The performance of clustering depends on an appropriately defined similarity between two items. When the similarity is measured based on human perception, human workers are often employed to estimate a similarity score between items in order to support clustering, leading to a procedure called crowdsourced clustering. Assuming a monetary reward is paid to a worker for each similarity score and assuming the similarities between pairs and workers' reliability have a large diversity, when the budget is limited, it is critical to wisely assign pairs of items to different workers to optimize the clustering result. We model this budget allocation problem as a Markov decision process where item pairs are dynamically assigned to workers based on the historical similarity scores they provided. We propose an optimistic knowledge gradient policy where the assignment of items in each stage is based on the minimum-weight K-cut defined on a similarity graph. We provide simulation studies and real data analysis to demonstrate the performance of the proposed method.