Integrability of vector and multivector fields associated with interior point methods for linear programming

In the feasible region of a linear programming problem, a number of ‘desirably good’ directions have been defined in connexion with various interior point methods. Each of them determines a contravariant vector field in the region whose only stable critical point is the optimum point. Some interior point methods incorporate a two- or higher-dimensional search, which naturally leads to the introduction of the corresponding contravariant multivector field. The paper investigates the integrability of those multivector fields, i.e., whether a contravariant p-vector field is Xp-forming, is enveloped by a family of Xq’s (q>p) or envelops a family of Xq’s (q<p) (in J.A. Schouten’s terminology), where Xq is a q-dimensional manifold. Immediate consequences of known facts are: (1)The directions hitherto proposed are X1-forming with the optimum point of the linear programming problem as the stable accumulation point, and (2)there is an X2-forming contravariant bivector field for which the center path is the critical submanifold. Most of the meaningful p-vector fields with p≥3 are not Xp-forming in general, though they envelop that bivector field. This observation will add another circumstantial evidence that the bivector field has a kind of invariant significance in the geometry of interior point methods for linear programming. For a kind of appendix, it it noted that, if there are several objectives, i.e., in the case of multiobjective linear programs, extension to higher dimensions is easily obtained.