Lectures on Stochastic Programming - Modeling and Theory

By Alexander Shapiro, Darinka Dentcheva †. 2009

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Data-driven risk-averse stochastic optimization with Wasserstein metric (2018, methodology): “To solve the problem, the sample average approximation method [21] is usually applied in which the random parameters are captured by a number of scenarios sampled from the true distribution.”
Kullback-Leibler divergence constrained distributionally robust optimization (2012, methodology): “Therefore, the convergence of the optimal value and the set of optimal solutions of the SAA, i.e., Problem (17), to those of the true problem, i.e., Problem (16), can be guaranteed; see, e.g., Theorem 5.3 of Shapiro et al. (2009) and the followed discussions.”
A unified framework for stochastic optimization (2019, background): “It is also known as the sample average approximation, introduced in Kleywegt, Shapiro, and Homem-de Mello (2002) with a nice summary in Shapiro et al. (2014) .”
Statistical estimation of composite risk functionals and risk optimization problems (2015, methodology): “However, the riskmeasure for p = 1 has an equivalent min–max formulation, for which a Central Limit Theorem has already been derived in the literature (see, Shapiro 2008; Shapiro et al. 2009, Section 6.5).”
A Sampling-and-Discarding Approach to Chance-Constrained Optimization: Feasibility and Optimality (2011, background): “…to spread in the context of stochastic optimization due to the increase of computing capabilities, see e.g. [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35], and sample-based optimization has also been used in connection of various application domains, see e.g. [36] [37] [38] [39] [40].”
A Guide to Sample Average Approximation (2015, methodology): “When the sample functions are convex, the set of regularity conditions in Theorem 9 can be relaxed by using the theory of epi-convergence [Shapiro et al., 2009, Theorem 5.4].”
Distributionally Robust Stochastic Optimization with Wasserstein Distance (2016, background): “In such settings, the decision making problems can be formulated as stochastic optimization problems as follows: We refer to Shapiro et al. [40] for a thorough study of stochastic optimization.”
Wasserstein Distributional Robustness and Regularization in Statistical Learning (2017, methodology): “It then follows from the disintegration theorem (see, e.g., Theorem Using interchangeability principle (see, e.g, Theorem 7.80 in Shapiro et al. (2014)), we exchange the inﬁmum and integration: inf Combining the equations above yields inf By the assumption that Z is a linear space, we have that for…”
Robust risk measurement and model risk (2014, unknown): “Shapiro et al. (2009). Similar estimators are available in the α-divergence framework.”
Optimal stochastic extragradient schemes for pseudomonotone stochastic variational inequality problems and their variants (2014, unknown): “[19]), it can be seen that based on the optimal initial steplength, we have that E[‖xK − x∗‖]2 ≤ 1 K max ( 2(B2+ν2) σ2 , ‖x0 − x∗‖2 ) .”
The Big Data Newsvendor: Practical Insights from Machine Learning (2013, methodology): “This approach is called the Sample Average Approximation (SAA) approach in stochastic optimization [for an excellent general reference, see Shapiro et al. (2009)].”
Recovering Best Statistical Guarantees via the Empirical Divergence-Based Distributionally Robust Optimization (2016, background): “SAA entails replacing the unknown expectation Z 0 ( x ) with the sample average (1 /n ) n i =1 h ( x ; ξ i ), leading to ˆ h ( x ) := 1 The issue with naively using SAA in this setting is that a solution feasible according to (2) may be mistakeably infeasible for (1).”