Misclassiication Minimization

The problem of minimizing the number of misclassiied points by a plane, attempting to separate two point sets with intersecting convex hulls in n-dimensional real space, is formulated as a linear program with equilibrium constraints (LPEC). This general LPEC can be converted to an exact penalty problem with a quadratic objective and linear constraints. A Frank-Wolfe-type algorithm is proposed for the penalty problem that terminates at a stationary point or a global solution. Novel aspects of the approach include: (i) A linear complementarity formulation of the step function that \counts" misclassiications, (ii) Exact penalty formulation without boundedness, nondegeneracy or constraint qualiication assumptions, (iii) An exact solution extraction from the sequence of minimizers of the penalty function for a nite value of the penalty parameter for the general LPEC and an explicitly exact solution for the LPEC with uncoupled constraints, and (iv) A parametric quadratic programming formulation of the LPEC associated with the misclassiication minimization problem.