Intersection Cuts Research Articles

The SIGMAP-UW nonlinear programming symposium

BALAS, GON: INTERSECTION CUTS FOR NONCONVEX PROGRAMMING. MONDAY, P.M.: A new class of intersection cuts is discussed, which is applicable to a large variety of nonconvex programming problems, like linearly constrained quadratic programs, separable programs, linear programs with disjunctive or other logical constraints, and of course integer programs. These cutting planes subsume or dominate many of those proposed earlier, and are easier to generate. When used in a branch and bound context, they yield stronger bounds than the ones currently in use.

Ranking the facets of the octahedron

This paper describes a procedure for ranking the facets of the n-dimensional (regular) octahedron in the order in which they are intersected by a halfline. The problem discussed here arise, in the context of integer programming via convex analysis (or intersection cuts).

Integer programming and convex analysis: Intersection cuts from outer polars

Recently a new, geometrically motivated approach was proposed [1] for integer programming, based on generating intersection cuts from some convex setS whose interior contains the linear programming optimum $$\bar x$$ but no feasible integer point. Larger sets tend to produce stronger cuts, and in this paper convex analysis is used to construct sets as large as possible within the above requirements. Information is generated from all problem constraints within a unit cubeK containing $$\bar x$$ The key concept is that of outer polars, viewed as maximal convex extensions of the ballB circumscribingK, relative to the problem constraints. The outer polarF * of the feasible setF overB is shown to be a convex set whose boundary contains all feasible vertices ofK, and whose interior contains no feasible integer point. The existence of a dual correspondence betweenF andF *, and the fact that polarity is inclusion-reversing, leads to a dualization of operations onF. As one possible procedure based on this approach, we construct a generalized intersection cut, that can be strengthened whenever some vertex ofF is cut off. This makes it possible to fruitfully combine intersection cuts with implicit enumeration or branch-and-bound. While valid for arbitrary integer programs, the theory developed here is relevant primarily to (pure or mixed-integer) 0–1 problems. Other topics discussed include: generalized polars, intersection cuts from related problems, connections with asymptotic theory.

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 x̄ the optimal (noninteger) solution to the linear program associated with an integer program in n-space. Consider a unit hypercube containing x̄, whose vertices are integer, and the hypersphere circumscribing the cube. This hypersphere is intersected in n independent points by the n halflines originating at x̄ and containing the n edges of X adjacent to x̄ (if x̄ is degenerate, X is replaced by X′ ⊃ X having exactly n edges adjacent to x̄). 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 case. A discussion of relations to other work (Young, Gomory) is followed by the introduction of some additional intersection cuts obtained from hypersurfaces other than the sphere or the cylinder. The message of this paper lies, not so much in the particular cuts that it proposes, as in the basically new approach to integer programming that these cuts typify. This approach is geometrically motivated and uses the “local” properties (in the integer-programming sense) of the feasible set.

An Intersection Cut from the Dual of the Unit Hypercube

This note extends the work of Balas (and the related work of Young) on intersection cuts for integer programming. Replacing the Euclidean hypersphere used by Balas with a convex polyhedron dual to the unit hypercube, we obtain a stronger intersection cut.