• Corpus ID: 224814186

Tropical Dynamic Programming for Lipschitz Multistage Stochastic Programming.

@article{Akian2020TropicalDP,
  title={Tropical Dynamic Programming for Lipschitz Multistage Stochastic Programming.},
  author={Marianne Akian and Jean-Philippe Chancelier and Benoit Tran},
  journal={arXiv: Optimization and Control},
  year={2020}
}
We present an algorithm called Tropical Dynamic Programming (TDP) which builds upper and lower approximations of the Bellman value functions in risk-neutral Multistage Stochastic Programming (MSP), with independent noises of finite supports. To tackle the curse of dimensionality, popular parametric variants of Approximate Dynamic Programming approximate the Bellman value function as linear combinations of basis functions. Here, Tropical Dynamic Programming builds upper (resp. lower… 

Figures and Tables from this paper

References

SHOWING 1-10 OF 26 REFERENCES

A stochastic algorithm for deterministic multistage optimization problems

TLDR
A common framework is built for both the SDDP and a discrete time version of Zheng Qu's algorithm to solve deterministic multistage optimization problems and generates monotone approximations of the value functions as a pointwise supremum, or infimum, of basic functions which are randomly selected.

Exact Converging Bounds for Stochastic Dual Dynamic Programming via Fenchel Duality

TLDR
This paper presents a dual SDDP algorithm that yields a converging exact upper bound for the optimal value of the optimization problem and shows how to compute an alternative control policy based on an inner approximation of Bellman value functions instead of the outer approximation given by the standardSDDP algorithm.

A deterministic algorithm for solving stochastic minimax dynamic programmes

TLDR
This paper presents an algorithm for solving stochastic minimax dynamic programmes where state and action sets are convex and compact, and applies the theory developed in this paper to multistage risk-averse optimisation.

Stochastic Lipschitz dynamic programming

We propose a new algorithm for solving multistage stochastic mixed integer linear programming (MILP) problems with complete continuous recourse. In a similar way to cutting plane methods, we

Stochastic dual dynamic integer programming

TLDR
An extension to SDDP—called stochastic dual dynamic integer programming (SDDiP)—for solving MSIP problems with binary state variables is proposed and it is shown that, under fairly reasonable assumptions, an MSIP problem with general state variables can be approximated by one withbinary state variables to desired precision with only a modest increase in problem size.

Approximate dynamic programming: solving the curses of dimensionality

TLDR
This book provides detailed coverage of modelling decision processes under uncertainty, robustness, designing and estimating value function approximations, choosing effective step-size rules, and convergence issues and is an excellent textbook for advanced undergraduate and beginning graduate students.

On Solving Multistage Stochastic Programs with Coherent Risk Measures

TLDR
A general computational approach based on dynamic programming is derived that can be shown to converge to an optimal policy by computing an inner approximation to future cost functions and an outer approximation that delivers a lower bound.

A max-plus based randomized algorithm for solving a class of HJB PDEs

  • Zheng Qu
  • Computer Science
    53rd IEEE Conference on Decision and Control
  • 2014
TLDR
A new max-plus based randomized algorithm to solve the same class of infinite horizon optimal control problems, but with a major difference that, instead of adding a large number of functions and then pruning the less useful ones, the new algorithm finds in cheap computation time, useful quadratic functions and adds only those functions to the set of basis functions.

SDDP for some interstage dependent risk-averse problems and application to hydro-thermal planning

  • V. Guigues
  • Computer Science
    Comput. Optim. Appl.
  • 2014
TLDR
The analysis includes some enhancements of this algorithm such as the definition of a state vector of minimal size, the computation of feasibility cuts without the assumption of relatively complete recourse, as well as efficient formulas for sharing optimality and feasibility cuts between nodes of the same stage.