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We study decision rule approximations for generic multistage robust linear optimization problems. We examine linear decision rules for the case when the objective coefficients, the recourse matrices, and the right-hand sides are uncertain, and we explore quadratic decision rules for the case when only the right-hand sides are uncertain. The resulting optimization problems are NP hard but amenable to copositive programming reformulations that give rise to tight, tractable semidefinite programming solution approaches. We further enhance these approximations through new piecewise decision rule schemes. Finally, we prove that our proposed approximations are tighter than the state-of-the-art schemes and demonstrate their superiority through numerical experiments. 

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.