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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.
In Boolean algebra, Petrick's method [1] (also known as Petrick function [2] or branch-and-bound method) is a technique described by Stanley R. Petrick (1931–2006) [3] [4] in 1956 [5] [6] for determining all minimum sum-of-products solutions from a prime implicant chart. [7]
Branch and cut [1] is a method of combinatorial optimization for solving integer linear programs (ILPs), that is, linear programming (LP) problems where some or all the unknowns are restricted to integer values. [2] Branch and cut involves running a branch and bound algorithm and using cutting planes to tighten
In general, the number of possible patterns grows exponentially as a function of m, the number of orders. As the number of orders increases, it may therefore become impractical to enumerate the possible cutting patterns. An alternative approach uses delayed column-generation. This method solves the cutting-stock problem by starting with just a ...
An integer programming problem is a mathematical optimization or feasibility program in which some or all of the variables are restricted to be integers.In many settings the term refers to integer linear programming (ILP), in which the objective function and the constraints (other than the integer constraints) are linear.
For each clause c in C, let S + c and S − c denote the sets of variables which are not negated in c, and those that are negated in c, respectively. The variables y x of the ILP will correspond to the variables of the formula F, whereas the variables z c will correspond to the clauses. The ILP is as follows:
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.
Because the feasible space only shrinks as information is added, the objective value for the master function provides a lower bound on the objective function of the overall problem. Benders Decomposition is applicable to problems with a largely block-diagonal structure.