A Theorem of the Alternative for Multihomogeneous Functions and Its Relationship to Diagonal Scaling of Matrices

We prove that, given a multihomogeneous function satisfying some initial conditions, either it has a certain nonnegative zero over a given subspace, or an associated logarithmic barrier function has a constrained stationary point. Under convexity precisely one of these conditions is satisfied. The main ingredients of the proof are the derivation of significant properties of the constrained stationary points of the logarithmic barrier function, and their relationship to corresponding points of an associated Karmarkar potential function. Corollaries of the theorem include a duality for Karmarkar's canonical linear program, which happens to be a stronger version of Gordan's theorem; a more general duality than an existing one concerning the diagonal scaling of symmetric matrices, also shown to be a stronger version of Gordan's theorem; and a diagonal scalability result for a class of multihomogeneous polynomials which is more general than a previously known result on the scalability of positive multidimensional matrices. We also give an algorithmic proof of the theorem of the alternative through a projective algorithm which is a generalization of a modified Karmarkar linear programming algorithm, and in the context of nonnegative matrix scaling becomes a variant of the well-known RAS algorithm.