On the complexity of following the central path of linear programs by linear extrapolation II

Abstract

A class of algorithms is proposed for solving linear programming problems (withm inequality constraints) by following the central path using linear extrapolation with a special adaptive choice of steplengths. The latter is based on explicit results concerning the convergence behaviour of Newton's method to compute points on the central pathx(r), r>0, and this allows to estimate the complexity, i.e. the total numberN = N(R, δ) of steps needed to go from an initial pointx(R) to a final pointx(δ), R>δ>0, by an integral of the local “weighted curvature” of the (primal—dual) path. Here, the central curve is parametrized with the logarithmic penalty parameterr↓0. It is shown that for large classes of problems the complexity integral, i.e. the number of stepsN, is not greater than constm ^α log(R/δ), whereα < 1/2 e.g.α = 1/4 orα = 3/8 (note thatα = 1/2 gives the complexity of zero order methods). We also provide a lower bound for the complexity showing that for some problems the above estimation can hold only forα ⩾ 1/3.

As a byproduct, many analytical and structural properties of the primal—dual central path are obtained: there are, for instance, close relations between the weighted curvature and the logarithmic derivatives of the slack variables; the dependence of these quantities on the parameterr is described. Also, related results hold for a family of weighted trajectories, into which the central path can be embedded.