On obtaining the ‘best’ multipliers for a lagrangean relaxation for integer programming

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Abstract

In a workshop on Integer Programming held in Germany in 1975, Geoffrion presented as an open important area for research, “What are the Cost Effective Ways to Determine the Best Multipliers for Generalized Lagrangean Relaxation?” In this paper, a method for computing the Lagrangean multipliers is presented. The method is applicable for a relatively wide class of Integer Programming problems. Four examples are presented in which, by applying this approach, it was possible to directly compute the multipliers. First, the general approach is outlined, and then the examples are presented. It is hoped that this approach and those specific examples will be of use in developing more general methods for computing the optimal Lagrangean multipliers and reduce the solution time of Integer Programming problems that fall within this class.

Preliminary computational tests have been performed on a variety of interval-bounded Zero-one Knapsack problems, using a Branch and Bound algorithm and a Lagrangean Relaxation that applied a method similar to the one presented in this research to obtain the best multipliers. The computational tests were very favorable.

References (23)

  • U. Akinc et al.

    An Efficient Branch and Bound Algorithm for the Capacitated Warehouse Location Problem

    (1974)
  • V. Balachandran

    An Integer Generalized Transportation Model for Optimal Job Assignment in Computer Networks

    Ops Res.

    (1976)
  • R. Brooks et al.

    Finding Everett's Lagrange Multipliers by Linear Programming

    Ops Res.

    (1966)
  • A.O. Demaio et al.

    An All Zero-One Algorithm for a Certain Class of Transportation Problems

    Ops Res.

    (1971)
  • H. Everett

    Generalized Lagrange Multiplier Method for Solving Problems of Optimum Allocation of Resources

    Ops Res.

    (1963)
  • B. Gavish et al.

    Municipal Bond Coupon Schedules with Limitations on the Number of Different Coupon Values

    (1977)
  • A.M. Geoffrion

    How Can Specialized Discrete and Convex Optimization Methods be Married

  • A.M. Geoffrion

    Lagrangean Relaxation for Integer Programming

    Math. Programming Study

    (1974)
  • A.M. Geoffrion et al.

    Multicommodity Distribution System Design by Benders Decomposition

    Mgt Sci.

    (1974)
  • C. Gotlieb

    The Construction of Class-Teacher Timetables

  • M. Held et al.

    Validation of Subgradient Optimization

    Math. Prog.

    (1974)
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    Bezalel Gavish is an Assistant Professor of Computers and Information Systems and Operations Research at the Graduate School of Management, University of Rochester. He received his MSc. and Ph.D. from the Department of Industrial Engineering and Management Science at the Technion Israel Institute of Technology. He had previously been a department head for Systems Analysis and Scientific Programming (1969–1973), and a staff member of the IBM Israel Scientific Center (1973–1976). His publications have appeared in Management Science, IEEE Trans. and other professional journals.

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