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 us to the introduction of the corresponding contravariant multivector field. We investigate the integrability of those multivector fields, i.e., whether a contravariant p-vector field is X(p)-forming, is enveloped by a family of X(q)'s (q > p) or envelops a family of X(q)'s (q < p) (in J.A. Schouten's terminology), where X(q) 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 greater-than-or-equal-to 3 are not X(p)-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 is noted that, if we have several objectives, i.e., in the case of multiobjective linear programming extension to higher dimensions is easily obtained.