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Online stochastic optimization under time constraints

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  • Pascal Hentenryck
  • Russell Bent
  • Eli Upfal

Abstract

This paper considers online stochastic combinatorial optimization problems where uncertainties, i.e., which requests come and when, are characterized by distributions that can be sampled and where time constraints severely limit the number of offline optimizations which can be performed at decision time and/or in between decisions. It proposes online stochastic algorithms that combine the frameworks of online and stochastic optimization. Online stochastic algorithms differ from traditional a priori methods such as stochastic programming and Markov Decision Processes by focusing on the instance data that is revealed over time. The paper proposes three main algorithms: expectation E, consensus C, and regret R. They all make online decisions by approximating, for each decision, the solution to a multi-stage stochastic program using an exterior sampling method and a polynomial number of samples. The algorithms were evaluated experimentally and theoretically. The experimental results were obtained on three applications of different nature: packet scheduling, multiple vehicle routing with time windows, and multiple vehicle dispatching. The theoretical results show that, under assumptions which seem to hold on these, and other, applications, algorithm E has an expected constant loss compared to the offline optimal solution. Algorithm R reduces the number of optimizations by a factor |R|, where R is the number of requests, and has an expected ρ(1+o(1)) loss when the regret gives a ρ-approximation to the offline problem. Copyright Springer Science+Business Media, LLC 2010

Suggested Citation

  • Pascal Hentenryck & Russell Bent & Eli Upfal, 2010. "Online stochastic optimization under time constraints," Annals of Operations Research, Springer, vol. 177(1), pages 151-183, June.
  • Handle: RePEc:spr:annopr:v:177:y:2010:i:1:p:151-183:10.1007/s10479-009-0605-5
    DOI: 10.1007/s10479-009-0605-5
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    References listed on IDEAS

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    1. Rüdiger Schultz, 1993. "Continuity Properties of Expectation Functions in Stochastic Integer Programming," Mathematics of Operations Research, INFORMS, vol. 18(3), pages 578-589, August.
    2. Marius M. Solomon, 1987. "Algorithms for the Vehicle Routing and Scheduling Problems with Time Window Constraints," Operations Research, INFORMS, vol. 35(2), pages 254-265, April.
    3. Russell Bent & Pascal Van Hentenryck, 2004. "A Two-Stage Hybrid Local Search for the Vehicle Routing Problem with Time Windows," Transportation Science, INFORMS, vol. 38(4), pages 515-530, November.
    4. A Larsen & O Madsen & M Solomon, 2002. "Partially dynamic vehicle routing—models and algorithms," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 53(6), pages 637-646, June.
    5. Russell W. Bent & Pascal Van Hentenryck, 2004. "Scenario-Based Planning for Partially Dynamic Vehicle Routing with Stochastic Customers," Operations Research, INFORMS, vol. 52(6), pages 977-987, December.
    6. Werner Römisch & Rüdiger Schultz, 1993. "Stability of Solutions for Stochastic Programs with Complete Recourse," Mathematics of Operations Research, INFORMS, vol. 18(3), pages 590-609, August.
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