Generalization of the Theorems of Alternatives in Semidefinite Programming

This paper deals with the generalization of the theorems of alternatives in semidefinite programming proposed by Balakrishnan and Vandenberghe. In their original form, the theorems assume that the domain of the linear mapping be a finite-dimensional Hilbert space. We show that the validity of the basic theorems does not rely on the finite-dimensional assumption, and the derived theorems can also be appropriately generalized. The Moore-Penrose inverse plays a crucial role in the generalization.