Cone algorithm: an extension of the perceptron algorithm

The perceptron convergence theorem played an important role in the early development of machine learning. Mathematically, the perceptron learning algorithm is an iterative procedure for finding a separating hyperplane for a finite set of linearly separable vectors, or equivalently, for finding a separating hyperplane for a finite set of linearly contained vectors. In this paper, the author shows that the perceptron algorithm can be extended to a more general algorithm, called the cone algorithm, for finding a covering cone for a finite set of linearly contained vectors. A proof of the convergence of the cone algorithm is given. The relationship between the cone algorithm and other related algorithms is discussed. The equivalence of the problem of finding a covering cone for a set of linearly contained vectors and the problem of finding a solution cone for a system of homogeneous linear inequalities is established