Multicategory discrimination via linear programming

A single linear program is proposed for discriminating between the elements of κ disjoint point sets in the n-dimensional real space Rn . When the conical hulls of the κ sets are (κ−1)-point disjoint in Rn +1, a κ-piece piecewise-linear surface generated by the linear program completely separates the κ sets. This improves on a previous linear programming approach which required that each set be linearly separable from the remaining κ−1 sets. When the conical hulls of the κ sets are not (κ;−1)-point disjoint, the proposed linear program generates an error-minimizing piecewise-linear separator for the κ Sets. For this case it is shown that the null solution is never a unique solver of the linear program and occurs only under the rather rare condition when the mean of each point set equals the mean of the means of the other κ−l sets. This makes the proposed linear computational programming formulation useful for approximately discriminating between κ sets that are not piecewise-linear separable. Computationa...