Intersection Cuts - A New Type of Cutting Planes for Integer Programming

This paper proposes a new class of cutting planes for integer programming. A typical member of the class is generated as follows. Let X be the feasible set, and x the optimal (noninteger) solution to the linear program associated with an integer program in n-space. Consider a unit hypercube containing x, whose vertices are integer, and the hypersphere circumscribing the cube. This hypersphere is intersected in n independent points by the n halflines originating at x and containing the n edges of X adjacent to x (if x is degenerate, X is replaced by X′ ⊃ X having exactly n edges adjacent to x). The hyperplane through these n points of intersection defines a valid cut, the (spherical) intersection cut. The paper gives a simple formula for finding the equation of the hyperplane, discusses some ways of strengthening the cut, proposes an algorithm, and gives a finiteness proof. A straightforward extension of these geometric ideas yields an analogous (cylindrical) intersection cut for the mixed-integer ca...