A Polynomial Predictor-Corrector Trust-Region Algorithm for Linear Programming

In this paper we present a scaling-invariant, interior-point, predictor-corrector type algorithm for linear programming (LP) whose iteration-complexity is polynomially bounded by the dimension and the logarithm of a certain condition number of the LP constraint matrix. At the predictor stage, the algorithm either takes the step along the standard affine scaling (AS) direction or a new trust-region type direction, whose construction depends on a scaling-invariant bipartition of the variables determined by the AS direction. This contrasts with the layered least squares direction introduced in S. Vavasis and Y. Ye [Math. Program., 74 (1996), pp. 79–120], whose construction depends on multiple-layered partitions of the variables that are not scaling-invariant. Moreover, it is shown that the overall arithmetic complexity of the algorithm (weakly) depends on the right-hand side and the cost of the LP in view of the work involved in the computation of the trust region steps.