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Benders decomposition (or Benders' decomposition) is a technique in mathematical programming that allows the solution of very large linear programming problems that have a special block structure. This block structure often occurs in applications such as stochastic programming as the uncertainty
Branch and price is a branch and bound method in which at each node of the search tree, columns may be added to the linear programming relaxation (LP relaxation). At the start of the algorithm, sets of columns are excluded from the LP relaxation in order to reduce the computational and memory requirements and then columns are added back to the LP relaxation as needed.
The following is the skeleton of a generic branch and bound algorithm for minimizing an arbitrary objective function f. [3] To obtain an actual algorithm from this, one requires a bounding function bound, that computes lower bounds of f on nodes of the search tree, as well as a problem-specific branching rule.
A variation on that is MI for free negative, giving an upper bound of 0 but no lower bound. Bound type PL is for a free positive from zero to plus infinity, but as this is the normal default, it is seldom used. Additionally, in some modifications of the mps file format there exists bound types for use in MIP models. Such as BV for binary, being ...
The cost of the solution produced by the algorithm is within 3/2 of the optimum. To prove this, let C be the optimal traveling salesman tour. Removing an edge from C produces a spanning tree, which must have weight at least that of the minimum spanning tree, implying that w(T) ≤ w(C) - lower bound to the cost of the optimal solution.
Such algorithms are called output-sensitive algorithms. They may be asymptotically more efficient than Θ(n log n) algorithms in cases when h = o(n). The lower bound on worst-case running time of output-sensitive convex hull algorithms was established to be Ω(n log h) in the planar case. [1]
This method [6] runs a branch-and-bound algorithm on problems, where is the number of variables. Each such problem is the subproblem obtained by dropping a sequence of variables x 1 , … , x i {\displaystyle x_{1},\ldots ,x_{i}} from the original problem, along with the constraints containing them.
The optimal algorithm is by Andris Ambainis. [7] Yaoyun Shi first proved a tight lower bound when the size of the range is sufficiently large. [8] Ambainis [9] and Kutin [10] independently (and via different proofs) extended his work to obtain the lower bound for all functions.