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

• 成果类型: Article
• 署名作者: Ahmed, Shabbir; Luedtke, James; Song, Yongjia; Xie, Weijun
• 署名单位: University System of Georgia; Georgia Institute of Technology; University of Wisconsin System; University of Wisconsin Madison; University of Wisconsin System; University of Wisconsin Madison; Virginia Commonwealth University
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-016-1029-z
• 发表日期: 2017
• 页码: 51-81
• 关键词: bundle methods; DECOMPOSITION; optimization; equivalents; gaps

摘要

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.