Rudiments of an average case complexity theory for piecewise-linear path following algorithms

Abstract

We examine the efficiency of PL path following algorithms in followingF ^-1_T (0), whereF _T is the PL approximation, induced by the simplicial triangulationT, to a mapf:ℝⁿ→ℝ^n-1. In particular, we consider the problem of determining an upper bound on the expected number of pivots made per unit length off ⁻¹(0) that is approximated. We show that if the sizes of the simplices ofT are “sufficiently small”, where “sufficiently small” is an explicitly given quantity dependent on measurements of how “nice”f is, then the average directional density ofT, as introduced by Todd, really does give a good approximation to the expected number of pivots made, confirming what researchers have believed on intuitive grounds for a decade. Because what constitutes “sufficiently small” is a precisely given quantity, i.e., non-asymptotic, we are able to provide some rigorous justification for the claim that the expected number of pivots grows only polynomially inn, the number of variables.

Several other issues are also examined.