Following a “Balanced” Trajectory from an Infeasible Point to an Optimal Linear Programming Solution with a Polynomial-Time Algorithm

Abstract

This paper is concerned with the problem of following a trajectory from an infeasible starting point directly to an optimal solution of the linear programming problem. A class of trajectories for the problem is defined, based on the notion of a β-balanced solution to the problem. Given a prespecified positive balancing constant β, an infeasible solution x is said to be β-balanced if the optimal value gap is less than or equal to β times the infeasibility gap. Mathematically, this can be written as cTx − z* ≤ βξTx, where the linear form ξTx is the Phase I objective function and z* is the optimal objective value of the linear program. The concept of a β-balanced solution is used to define a class of trajectories from an infeasible point to an optimal solution of a given linear program. Each trajectory has the property that all points on or near the trajectory (in a suitable metric) are β-balanced. The main thrust of the paper is the development of an algorithm that traces a given β-balanced trajectory from a starting point near the trajectory to an optimal solution to the given linear programming problem in polynomial time. More specifically, the algorithm allows for fixed improvement in the bound on the Phase I and the Phase II objectives in O(n) Newton steps.