On Finding the Maximum Feasible Subsystem of a System of Linear Inequalities

Abstract

Some methods for finding the maximum feasible subsystems of systems of linear inequalities are considered. The problem of finding the most accurate algorithm in a parametric family of linear classification algorithms is one of the most important problems in machine learning. In order to solve this discrete optimization problem, an exact (combinatorial) algorithm, its approximations (relaxation and greedy combinatorial descent algorithms), and the approximation algorithm are given. The latter consists in replacing the original discrete optimization problem with a nonlinear programming problem by changing from linear inequalities to their sigmoid functions. The initial results of their comparison are presented.