Cutting and Surrogate Constraint Analysis for Improved Multidimensional Knapsack Solutions

We use surrogate analysis and constraint pairing in multidimensional knapsack problems to fix some variables to zero and to separate the rest into two groups – those that tend to be zero and those that tend to be one, in an optimal integer solution. Using an initial feasible integer solution, we generate logic cuts based on our analysis before solving the problem with branch and bound. Computational testing, including the set of problems in the OR-library and our own set of difficult problems, shows our approach helps to solve difficult problems in a reasonable amount of time and, in most cases, with a fewer number of nodes in the search tree than leading commercial software.     

Classical Cuts for Mixed-Integer Programming and Branch-and-Cut

We review classical valid linear inequalities for mixed-integer programming, i.e., Gomory's fractional and mixed-integer cuts, and discuss their use in branch-and-cut. In particular, a generalization of the recent mixed-integer rounding (MIR) inequality and a sufficient condition for the global validity of classical cuts after branching has occurred are derived. Work supported in part by a grant from the Office of Naval Research (N00014-96-0327). Reprinted from Math. Meth. Oper. Res. (2001) 53, 173–203.