Closing the Duality Gap on Integer Linear Programming

Dual formulations for optimization problems provide a new way to view the same problem, revealing its structure and bounding optimal solutions. A strong dual formulation constructs a problem that can be solved in place of or alongside the primal and converges to the same optimal objective function value. In this paper we examine the general dual framework for nonlinear programs suggested by Everett and developed by Gould. This is leveraged to provide a novel strong dual formulation for general and 0–1 integer linear programming with qualitatively different structure from other known strong dual formulations. In particular, the dual problem optimizes three variables subject to a bounded, exponential number of constraints. The zero duality gap and small domain may prove useful in developing new techniques for solving integer linear programs.