Valid integer polytope (VIP) penalties for branch-and-bound enumeration

We introduce new penalties, called valid integer polytope (VIP) penalties, that tighten the bound of an integer-linear program during branch-and-bound enumeration. Early commercial codes for branch and bound commonly employed penalties developed from the dual simplicial lower bound on the cost of restricting fractional integer variables to proximate integral values. VIP penalties extend and tighten these for ubiquitous k-pack, k-partition, and k-cover constraints. In real-world problems, VIP penalties occasionally tighten the bound by more than an order of magnitude, but they usually offer small bound improvement. Their ease of implementation, speed of execution, and occasional, overwhelming success make them an attractive addition during branch-and-bound enumeration.