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1. Exploring the Effect of Clustering Algorithms on Sample Average Approximation

   Jacobson, Daniel; Hassan, Menna; Dong, Zhijie S. (January 2021, 2021 Institute of Industrial and Systems Engineers (IISE) Annual Conference & Expo)

   null (Ed.)

   Stochastic Programming (SP) is used in disaster management, supply chain design, and other complex problems. Many of the real-world problems that SP is applied to produce large-size models. It is important but challenging that they are optimized quickly and efficiently. Existing optimization algorithms are limited in capability of solving these larger problems. Sample Average Approximation (SAA) method is a common approach for solving large scale SP problems by using the Monte Carlo simulation. This paper focuses on applying clustering algorithms to the data before the random sample is selected for the SAA algorithm. Once clustered, a sample is randomly selected from each of the clusters instead of from the entire dataset. This project looks to analyze five clustering techniques compared to each other and compared to the original SAA algorithm in order to see if clustering improves both the speed and the optimal solution of the SAA method for solving stochastic optimization problems. 

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2. New Sample Complexity Bounds for Sample Average Approximation in Heavy-Tailed Stochastic Programming

   Liu, Hongcheng; Tong, Jindong (July 2024, Proceedings of the 41st International Conference on Machine Learning)

   This paper studies sample average approximation (SAA) and its simple regularized variation in solving convex or strongly convex stochastic programming problems. Under heavy-tailed assumptions and comparable regularity conditions as in the typical SAA literature, we show — perhaps for the first time — that the sample complexity can be completely free from any complexity measure (e.g., logarithm of the covering number) of the feasible region. As a result, our new bounds can be more advantageous than the state-of-the-art in terms of the dependence on the problem dimensionality. 

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3. A Decision Rule Approach for Two-Stage Data-Driven Distributionally Robust Optimization Problems with Random Recourse

   https://doi.org/10.1287/ijoc.2021.0306  

   Fan, Xiangyi; Hanasusanto, Grani A. (November 2023, INFORMS Journal on Computing)

   We study two-stage stochastic optimization problems with random recourse, where the coefficients of the adaptive decisions involve uncertain parameters. To deal with the infinite-dimensional recourse decisions, we propose a scalable approximation scheme via piecewise linear and piecewise quadratic decision rules. We develop a data-driven distributionally robust framework with two layers of robustness to address distributional uncertainty. We also establish out-of-sample performance guarantees for the proposed scheme. Applying known ideas, the resulting optimization problem can be reformulated as an exact copositive program that admits semidefinite programming approximations. We design an iterative decomposition algorithm, which converges under some regularity conditions, to reduce the runtime needed to solve this program. Through numerical examples for various known operations management applications, we demonstrate that our method produces significantly better solutions than the traditional sample-average approximation scheme especially when the data are limited. For the problem instances for which only the recourse cost coefficients are random, our method exhibits slightly inferior out-of-sample performance but shorter runtimes compared with a competing approach. 

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4. Data Pooling in Stochastic Optimization

   https://doi.org/10.1287/mnsc.2020.3933  

   Gupta, Vishal; Kallus, Nathan (January 2021, Management Science)

   null (Ed.)

   Managing large-scale systems often involves simultaneously solving thousands of unrelated stochastic optimization problems, each with limited data. Intuition suggests that one can decouple these unrelated problems and solve them separately without loss of generality. We propose a novel data-pooling algorithm called Shrunken-SAA that disproves this intuition. In particular, we prove that combining data across problems can outperform decoupling, even when there is no a priori structure linking the problems and data are drawn independently. Our approach does not require strong distributional assumptions and applies to constrained, possibly nonconvex, nonsmooth optimization problems such as vehicle-routing, economic lot-sizing, or facility location. We compare and contrast our results to a similar phenomenon in statistics (Stein’s phenomenon), highlighting unique features that arise in the optimization setting that are not present in estimation. We further prove that, as the number of problems grows large, Shrunken-SAA learns if pooling can improve upon decoupling and the optimal amount to pool, even if the average amount of data per problem is fixed and bounded. Importantly, we highlight a simple intuition based on stability that highlights when and why data pooling offers a benefit, elucidating this perhaps surprising phenomenon. This intuition further suggests that data pooling offers the most benefits when there are many problems, each of which has a small amount of relevant data. Finally, we demonstrate the practical benefits of data pooling using real data from a chain of retail drug stores in the context of inventory management. This paper was accepted by Chung Piaw Teo, Special Issue on Data-Driven Prescriptive Analytics. 

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5. Two-stage linear decision rules for multi-stage stochastic programming

   https://doi.org/10.1007/s10107-018-1339-4  

   Bodur, Merve; Luedtke, James R. (October 2018, Mathematical Programming)

   Multi-stage stochastic linear programs (MSLPs) are notoriously hard to solve in general. Linear decision rules (LDRs) yield an approximation of an MSLP by restricting the decisions at each stage to be an affine function of the observed uncertain parameters. Finding an optimal LDR is a static optimization problem that provides an upper bound on the optimal value of the MSLP, and, under certain assumptions, can be formulated as an explicit linear program. Similarly, as proposed by Kuhn et al. (Math Program 130(1):177–209, 2011) a lower bound for an MSLP can be obtained by restricting decisions in the dual of the MSLP to follow an LDR. We propose a new approximation approach for MSLPs, two-stage LDRs. The idea is to require only the state variables in an MSLP to follow an LDR, which is sufficient to obtain an approximation of an MSLP that is a two-stage stochastic linear program (2SLP). We similarly propose to apply LDR only to a subset of the variables in the dual of the MSLP, which yields a 2SLP approximation of the dual that provides a lower bound on the optimal value of the MSLP. Although solving the corresponding 2SLP approximations exactly is intractable in general, we investigate how approximate solution approaches that have been developed for solving 2SLP can be applied to solve these approximation problems, and derive statistical upper and lower bounds on the optimal value of the MSLP. In addition to potentially yielding better policies and bounds, this approach requires many fewer assumptions than are required to obtain an explicit reformulation when using the standard static LDR approach. A computational study on two example problems demonstrates that using a two-stage LDR can yield significantly better primal policies and modestly better dual policies than using policies based on a static LDR. 

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