Technical Note—On the Strength of Relaxations of Weakly Coupled Stochastic Dynamic Programs
Many sequential decision problems have a weakly coupled structure in that a set of linking constraints couples an otherwise independent collection of subproblems. This structure arises in a wide variety of applications, such as network revenue management, online advertising, assortment planning, int...
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Veröffentlicht in: | Operations research 2023-11, Vol.71 (6), p.2374-2389 |
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Zusammenfassung: | Many sequential decision problems have a weakly coupled structure in that a set of linking constraints couples an otherwise independent collection of subproblems. This structure arises in a wide variety of applications, such as network revenue management, online advertising, assortment planning, interactive marketing, optimization of power systems, and multilocation inventory management to name only a few. Such problems can be modeled as dynamic programs but are quite difficult to solve. Two widely studied approximation methods are approximate linear programs, which involve finding a best approximation of total value that is additive across the subsystems, and Lagrangian relaxations, which involve relaxing the linking constraints. It is well known that both of these approaches provide upper bounds to the optimal value, and the approximate linear programming approach is a better bound but also, more difficult to compute. In this paper, we provide a detailed theoretical analysis of these two approximations and show that, under fairly broad conditions, these two approximations lead to upper bounds that are very close and often identical. Our theory suggests that, between these two approximations, Lagrangian relaxations should usually be the preferred choice for researchers studying applications involving weakly coupled dynamic programs.
Many stochastic dynamic programs (DPs) have a weakly coupled structure in that a set of linking constraints in each period couples an otherwise independent collection of subproblems. Two widely studied approximations of such problems are approximate linear programs (ALPs), which involve optimizing value function approximations that additively separate across subproblems, and Lagrangian relaxations, which involve relaxing the linking constraints. It is well known that both of these approximations provide upper bounds on the optimal value function in all states and that the ALP provides a tighter upper bound in the initial state. The purpose of this short paper is to provide theoretical justification for the fact that these upper bounds are often close if not identical. We show that (i) for any weakly coupled DP, the difference between these two upper bounds—the
relaxation gap
—is bounded from above in terms of the integrality gap of the separation problems associated with the ALP. (ii) If subproblem rewards are uniformly bounded and some broadly applicable conditions on the linking constraints hold, the relaxation gap is bounde |
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ISSN: | 0030-364X 1526-5463 |
DOI: | 10.1287/opre.2022.2287 |