  This paper revisits the fundamental equations for the solution of the
frictionless unilateral normal contact problem between a rough rigid surface
and a linear elastic half-plane using the boundary element method (BEM). After
recasting the resulting Linear Complementarity Problem (LCP) as a convex
quadratic program (QP) with nonnegative constraints, different optimization
algorithms are compared for its solution: (i) a Greedy method, based on
different solvers for the unconstrained linear system (Conjugate Gradient CG,
Gauss-Seidel, Cholesky factorization), (ii) a constrained CG algorithm, (iii)
the Alternating Direction Method of Multipliers (ADMM), and ($iv$) the
Non-Negative Least Squares (NNLS) algorithm, possibly warm-started by
accelerated gradient projection steps or taking advantage of a loading history.
The latter method is two orders of magnitude faster than the Greedy CG method
and one order of magnitude faster than the constrained CG algorithm. Finally,
we propose another type of warm start based on a refined criterion for the
identification of the initial trial contact domain that can be used in
conjunction with all the previous optimization algorithms. This method, called
Cascade Multi-Resolution (CMR), takes advantage of physical considerations
regarding the scaling of the contact predictions by changing the surface
resolution. The method is very efficient and accurate when applied to real or
numerically generated rough surfaces, provided that their power spectral
density function is of power-law type, as in case of self-similar fractal
surfaces.
