A maximum feasible subset algorithm with application to radiation therapy

Consider a set of linear one sided or two sided inequality constraints on a real vector X. The problem of interest is selection of X so as to maximize the number of constraints that are simultaneously satisfied, or equivalently, combinatorial selection of a maximum cardinality subset of feasible inequalities. Special classes of this problem are of interest in a variety of areas such as pattern recognition, machine learning, operations research, and medical treatment planning. This problem is generally solvable in exponential time. A heuristic polynomial time algorithm is presented in this paper. The algorithm relies on an iterative constraint removal procedure where constraints are eliminated from a set proposed by solutions to minmax linear programs. The method is illustrated by a simulated example of a linear system with double sided bounds and a case from the area of radiation therapy.