# Difference between revisions of "Adaptive robust optimization"

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== Introduction == | == Introduction == | ||

− | Traditionally, robust optimization has solved problems based on static decisions which are predetermined by the decision makers. Once the decisions were made, the problem was solved and whenever a new uncertainty was realized, the uncertainty was incorporated to the original problem and the entire problem was solved again to account for the uncertainty.[1] | + | Traditionally, robust optimization has solved problems based on static decisions which are predetermined by the decision makers. Once the decisions were made, the problem was solved and whenever a new uncertainty was realized, the uncertainty was incorporated to the original problem and the entire problem was solved again to account for the uncertainty.[1] Generally, robust optimization problem is formulated as follows. |

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Unfortunately, most robust optimization problems for real life applications require multiple stages to account for uncertainties and traditional static robust has shown limitations. In order to improve the pre-existing technique, adaptive robust optimization was studied and advances in the field was made to address the problems which could not be easily handled with previous methods.[2] <br> | Unfortunately, most robust optimization problems for real life applications require multiple stages to account for uncertainties and traditional static robust has shown limitations. In order to improve the pre-existing technique, adaptive robust optimization was studied and advances in the field was made to address the problems which could not be easily handled with previous methods.[2] <br> | ||

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[[File:wsaroeq3.png|center]] | [[File:wsaroeq3.png|center]] | ||

Then, the second subset of possible scenarios are used to calculate the regrets at each iteration for all scenarios which were previously ruled out in the possible scenario categorization. If the optimal objective value obtained from running the first subset is greater than or equal to the calculated regret from optimality, the optimal condition is met and the algorithm terminates. If not, previously not included scenarios would be included and then the iteration will repeat until the optimum solution is found. | Then, the second subset of possible scenarios are used to calculate the regrets at each iteration for all scenarios which were previously ruled out in the possible scenario categorization. If the optimal objective value obtained from running the first subset is greater than or equal to the calculated regret from optimality, the optimal condition is met and the algorithm terminates. If not, previously not included scenarios would be included and then the iteration will repeat until the optimum solution is found. | ||

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== Example == | == Example == |

## Revision as of 16:53, 30 May 2015

Author: Woo Soo Choe (ChE 345 Spring 2015)

Steward: Dajun Yue, Fengqi You

## Contents |

## Introduction

Traditionally, robust optimization has solved problems based on static decisions which are predetermined by the decision makers. Once the decisions were made, the problem was solved and whenever a new uncertainty was realized, the uncertainty was incorporated to the original problem and the entire problem was solved again to account for the uncertainty.[1] Generally, robust optimization problem is formulated as follows.

## Example

As an example problem which implements the previously covered methodology a warehouse and resource application problem is used[2].

Also, the data of the problem are given as such[2].

Furthermore, the variables of the problem are defined as follows[2]

In this example, the problem is seen in multiple different approaches. In general, the location of the factories and the warehouses and the maximum capacities are the first-stage decision variables because the first-stage decisions must be made before the uncertainties are taken into account.

## Results

When the example problem is solved using the algorithm posed in methodology section yields the following result[2]. [[File:

## Conclusion

Consequently, the studied algorithm may not always guarantee the reduction of computing power or computing time, but by reducing the number of possible solutions to run, it provides a way for an adaptive robust optimization to cope with a highly complicated problem. Also, this approach has a relevance to the technique's learned in optimization classes. By controlling the scope of the different scenarios to try, further advance in adaptive robust optimization may be possible by making improvement in the already computationally taxing nature of adaptive robust optimization.

## References

1. Virginie Gabrel, *Recent Advance in Robust Optimization: An Overview*, Universite Paris Dauphine, 2013

2. Tiravat Assavapokee, *Scenario Relaxation Algorithm for Finite Scenario-Based min–max Regret and Min–Max Relative Regret Robust Optimization*

3. Kouvelis P, Yu G, *Robust Discrete Optimization and Its Application*, Dordrecht, 1997

4. Ben-Tal A, Nemirovski A, *Robust Convex Optimization* Mathematical Methods of Operations Research, 1998