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1. 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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2. Stage-discharge relationship in tidal channels: Tidal stage-discharge relationship

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   Kearney, William S.; Mariotti, Giulio; Deegan, Linda A.; Fagherazzi, Sergio (April 2017, Limnology and Oceanography: Methods)
3. How intra-stage and inter-stage competition affect overcompensation in density and hydra effects in single-species, stage-structured models

   https://doi.org/10.1007/s12080-020-00488-1  

   Sorenson, Darian K.; Cortez, Michael H. (March 2021, Theoretical Ecology)

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4. Paleofloods stage a comeback

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   St. George, Scott; Hefner, Amanda M.; Avila, Judith (December 2020, Nature Geoscience)

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5. Multi-stage Relational Programming

   https://doi.org/10.1145/3729314  

   Ballantyne, Michael; Sanna, Rafaello; Hemann, Jason; Byrd, William E; Amin, Nada (June 2025, Proceedings of the ACM on Programming Languages)

   We transport multi-stage programming from functional to relational programming, with novel constructs to give programmers control over staging and non-determinism. We stage interpreters written as relations, in which the programs under interpretation can contain holes representing unknown expressions or values. By compiling the known parts without interpretive overhead and deferring interpretation to run time only for the unknown parts, we compound the benefits of staging (e.g., turning interpreters into compilers) and relational interpretation (e.g., turning functions into relations and synthesizing from sketches). We extend miniKanren with staging constructs and apply the resulting multi-stage language to relational interpreters for subsets of Racket and miniKanren as well as a relational recognizer for context-free grammars. We demonstrate significant performance gains across multiple synthesis problems, systematically comparing unstaged and staged computation, as well as indicatively comparing with an existing hand-tuned relational interpreter. 

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