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        "description": "The partitioned subproblem is linear; the remaining master problem may still be difficult. The computational argument assumes the stated boundedness/continuity and correctly solved master/subproblems. Finite polyhedral generators support feasibility-ray and optimality-point cuts, not polynomial runtime for every mixed problem. A big-M variant requires a sufficiently large/adapted value or a separate phase. Reported test cases are not an independently reproduced performance study.",
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      "statement": "# Benders: separating a hard decision from a linear subproblem\n\nBenders partitions a mixed-variable optimization problem into a master problem and a linear subproblem, generating only the dual constraints needed to identify an optimum or detect infeasibility or unboundedness, under explicit solution and compactness assumptions.\n\nA large optimization problem may combine difficult choices with a simpler linear calculation. Benders separates the two. The master proposes a choice; the linear subproblem checks what that choice makes possible. A successful check sends back a bound on the objective. An impossible choice sends back a constraint explaining why it must be excluded. These constraints accumulate, so the master learns from earlier proposals without constructing every possible constraint in advance. The method also keeps upper and lower bounds, showing how much uncertainty remains. It does not make the difficult master disappear: that problem must still be solved correctly. The original paper gives a finite theoretical procedure and small computer experiments, while stating that useful general estimates of the number of steps were not yet available.\n\n## Scientific scope\n\nThe partitioned subproblem is linear; the remaining master problem may still be difficult. The computational argument assumes the stated boundedness/continuity and correctly solved master/subproblems. Finite polyhedral generators support feasibility-ray and optimality-point cuts, not polynomial runtime for every mixed problem. A big-M variant requires a sufficiently large/adapted value or a separate phase. Reported test cases are not an independently reproduced performance study.\n\n## People and systems\n\nJ. F. Benders\n\n\n\n## Original sources\n\n- [J. F. Benders, Partitioning procedures for solving mixed-variables programming problems; exact1962 original](https://www.im-uff.mat.br/puc-rio/disciplinas/2006.1/soe/arquivos/benders-numerische-mathematik-1962.pdf)\n- [Publisher-deposited identity for the exact1962 Benders partitioning paper](https://api.crossref.org/works/10.1007%2FBF01386316)\n",
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    "scope": "Benders partitions a mixed-variable optimization problem into a master problem and a linear subproblem, generating only the dual constraints needed to identify an optimum or detect infeasibility or unboundedness, under explicit solution and compactness assumptions. The partitioned subproblem is linear; the remaining master problem may still be difficult. The computational argument assumes the stated boundedness/continuity and correctly solved master/subproblems. Finite polyhedral generators support feasibility-ray and optimality-point cuts, not polynomial runtime for every mixed problem. A big-M variant requires a sufficiently large/adapted value or a separate phase. Reported test cases are not an independently reproduced performance study.",
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        "title": "J. F. Benders, Partitioning procedures for solving mixed-variables programming problems; exact 1962 original",
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