Separation by Hyperplanes: A Linear Semi-Infinite Programming Approach

Abstract

In this paper we analyze the separability and the strong separability of two given sets in a real normed space by means of a topological hyperplane. The existence of such a separating hyperplane is characterized by the negativity of the optimal value of some related (infinite dimensional) linear optimization problems. In the finite dimensional case, such hyperplane can be effectively calculated by means of well-known linear semi-infinite optimization methods. If the sets to be separated are compact, and they are contained in a separable normed space, a conceptual cutting plane algorithm for their strong separation is proposed. This algorithm solves a semi-infinite programming problem (in the sense that it has finitely many constraints) at each iteration, and its convergence to an optimal solution, providing the desired separating hyperplane, is detailedly studied. Finally, the strong separation of finite sets in the Hadamard space is also approached, and a grid discretization method is proposed in this case.