Excerpt from Generalized Lagrange Multipliers: In Integer Programming

Several authors have proposed generalized Lagrangian methods for finding good or Optimal solutions to integer programming problems. The capital budgeting problem of Lorie and Savage essentially the 0-1 multi-dimensional Knapsack problem, has received particular attention in this regard. In Nemhauser and Ullman prove the somewhat negative result that the approach of Everett [4] applied to the capital budgeting problem by Kaplan in [8] can yield an optimal solution only if there is an Optimal linear programming solution that is integer. In this paper, we use group theory to reformulate the integer programming problem, thereby obtaining a Lagrangian problem which appears to Offer greater combinatorial resolution than previous methods. Conversely, the usefulness Of the group theoretic approach is enhanced by the Lagrangian problem.

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