A Potential Reduction Algorithm with User-Specified Phase I–Phase II Balance for Solving a Linear Program from an Infeasible Warm Start

Abstract

This paper develops a potential reduction algorithm for solving a linear programming problem directly from a “warm start” initial point that is neither feasible nor optimal. The algorithm is an “interior point” variety that seeks to reduce a single potential function which simultaneously coerces feasibility improvement (Phase I) and objective value improvement (Phase II). The key feature of the algorithm is the ability to specify beforehand the desired balance between infeasibility and nonoptimality in the following sense. Given a prespecified balancing parameter $\beta > 0$, the algorithm maintains the following Phase I–Phase II “$\beta $-balancing constraint” throughout \[ \left( c^T x - z^* \right) < \beta \xi^T x, \] where $c^T x$ is the objective function, $z^ * $ is the (unknown) optimal objective value of the linear program, and $\xi^T x$ measures the infeasibility of the current iterate x. This balancing constraint can be used to either emphasize rapid attainment of feasibility (set $\beta $ large) at the possible expense of good objective function values or to emphasize rapid attainment of good objective values (set $\beta$ small) at the possible expense of a lower infeasibility gap. The algorithm seeks to minimize the feasibility gap while maintaining the $\beta $-balancing condition, thus solving the original linear program as a consequence. The algorithm exhibits the following advantageous features: (i) the iterate solutions monotonically decrease the infeasibility measure, (ii) the iterate solutions satisfy the $\beta $-balancing constraint, (iii) the iterate solutions achieve constant improvement in both Phase I and Phase II in $O( n t)$ iterations, (iv) there is always a possibility of finite termination of the Phase I problem, and (v) the algorithm is amenable to acceleration via linesearch of the potential function.