Linear Superiorization for Infeasible Linear Programming

TL;DR

This paper introduces Linear Superiorization (LinSup), a method that guides feasibility-seeking algorithms towards solutions with lower objective function values even when the linear constraints are infeasible, demonstrated through experimental results.

Contribution

The paper proposes LinSup, a novel approach that steers feasibility-seeking processes towards lower objective values in infeasible linear programming problems.

Findings

LinSup converges to a point minimizing constraint violation.

LinSup tends to reduce the objective function value.

Experimental results illustrate LinSup's behavior on infeasible LPs.

Abstract

Linear superiorization (abbreviated: LinSup) considers linear programming (LP) problems wherein the constraints as well as the objective function are linear. It allows to steer the iterates of a feasibility-seeking iterative process toward feasible points that have lower (not necessarily minimal) values of the objective function than points that would have been reached by the same feasiblity-seeking iterative process without superiorization. Using a feasibility-seeking iterative process that converges even if the linear feasible set is empty, LinSup generates an iterative sequence that converges to a point that minimizes a proximity function which measures the linear constraints violation. In addition, due to LinSup's repeated objective function reduction steps such a point will most probably have a reduced objective function value. We present an exploratory experimental result that illustrates the behavior of LinSup on an infeasible LP problem.