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Branch and price: Integer programming with column generation; Decomposition techniques for MILP: Lagrangian relaxation; Genetic algorithms; Integer linear complementary problem; Integer programming; Integer programming: Algebraic methods; Integer programming: Branch and cut algorithms; Integer programming: Cutting plane algorithms; Integer programming duality; Integer programming: Lagrangian relaxation; LCP: Pardalos–Rosen mixed integer formulation, Linear programming; Interior point methods; Mixed integer classification problems; Multi-objective integer linear programming; Multi-objective mixed integer programming; Multiparametric mixed integer linear programming; Parametric mixed integer nonlinear optimization; Set covering, packing and partitioning problems; Simplicial pivoting algorithms for integer programming; Stochastic integer programming: Continuity, stability, rates of convergence; Stochastic integer programs; Time-dependent traveling salesman problem INTEGER PROGRAMMING: BRANCH AND BOUND METHODS

branch and bound

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Encyclopedia of Optimization

Overview

An integer programming problem (IP) is an optimization problem in which some or all of the variables are restricted to take on only integer values. The exposition presented here will focus on the case in which the objective and constraints of the optimization problem are defined via linear functions. In addition, for simplicity, it will be assumed that all of the variables are restricted to be nonnegative integer valued. Thus, the mathematical formulation of the problem under consideration can be stated as:

where A ∈ R m × n, b ∈ R m and c ∈ R n. For notational convenience, let S denote the constraint set of problem (IP); i.e.,

The classical approach to solving integer programs is branch and bound [39]. The branch and bound method is based on the idea of iteratively partitioning the set S(branching) to form subproblems of the original integer program. Each subproblem is solved — either exactly or approximately — to obtain an upper bound on the subproblem objective value. The...

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Authors and Affiliations

  1. Industrial and Systems Engin. Georgia Inst. Technol., Atlanta, GA, USA

    Eva K. Lee

  2. Math. Sci. Rensselaer Polytechnic Inst., Troy, NY, USA

    John E. Mitchell

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  1. Eva K. Lee
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  2. John E. Mitchell
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Lee, E.K., Mitchell, J.E. (2001). Branch and price: Integer programming with column generation; Decomposition techniques for MILP: Lagrangian relaxation; Genetic algorithms; Integer linear complementary problem; Integer programming; Integer programming: Algebraic methods; Integer programming: Branch and cut algorithms; Integer programming: Cutting plane algorithms; Integer programming duality; Integer programming: Lagrangian relaxation; LCP: Pardalos–Rosen mixed integer formulation, Linear programming; Interior point methods; Mixed integer classification problems; Multi-objective integer linear programming; Multi-objective mixed integer programming; Multiparametric mixed integer linear programming; Parametric mixed integer nonlinear optimization; Set covering, packing and partitioning problems; Simplicial pivoting algorithms for integer programming; Stochastic integer programming: Continuity, stability, rates of convergence; Stochastic integer programs; Time-dependent traveling salesman problem INTEGER PROGRAMMING: BRANCH AND BOUND METHODS . In: Encyclopedia of Optimization. Springer, Boston, MA. https://doi.org/10.1007/0-306-48332-7_214

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  • DOI: https://doi.org/10.1007/0-306-48332-7_214

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-0-7923-6932-5

  • Online ISBN: 978-0-306-48332-5

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