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Nonanticipative duality, relaxations, and formulations for chance-constrained stochastic programs

Abstract

We propose two new Lagrangian dual problems for chance-constrained stochastic programs based on relaxing nonanticipativity constraints. We compare the strength of the proposed dual bounds and demonstrate that they are superior to the bound obtained from the continuous relaxation of a standard mixed-integer programming (MIP) formulation. For a given dual solution, the associated Lagrangian relaxation bounds can be calculated by solving a set of single scenario subproblems and then solving a single knapsack problem. We also derive two new primal MIP formulations and demonstrate that for chance-constrained linear programs, the continuous relaxations of these formulations yield bounds equal to the proposed dual bounds. We propose a new heuristic method and two new exact algorithms based on these duals and formulations. The first exact algorithm applies to chance-constrained binary programs, and uses either of the proposed dual bounds in concert with cuts that eliminate solutions found by the subproblems. The second exact method is a branch-and-cut algorithm for solving either of the primal formulations. Our computational results indicate that the proposed dual bounds and heuristic solutions can be obtained efficiently, and the gaps between the best dual bounds and the heuristic solutions are small.

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Acknowledgments

S. Ahmed was supported in part by the National Science Foundation Grant 1331426 and the Office of Naval Research Grant N00014-15-1-2078. J. Luedtke was partly supported by NSF Grant CMMI-0952907 and by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program under contract number DE-AC02-06CH11357. W. Xie was supported in part by the National Science Foundation Grant 1129871 and a fellowship from the Algorithms and Randomness Center at Georgia Tech. The authors acknowledge the valuable comments from the editors and three anonymous reviewers.

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Correspondence to Yongjia Song.

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Ahmed, S., Luedtke, J., Song, Y. et al. Nonanticipative duality, relaxations, and formulations for chance-constrained stochastic programs. Math. Program. 162, 51–81 (2017). https://doi.org/10.1007/s10107-016-1029-z

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Mathematics Subject Classification

  • 90C11
  • 90C15
  • 90C57
  • 49M29