Mesh-dependent iteration count growth in primal-dual active set strategies

Abstract

Primal-dual active set strategies (PDAS) are popular iterative solvers for mixed complementarity problems such as constrained optimization problems with pointwise inequality constraints. Examples include the reduced-space active set algorithm vinewtonrsls found in PETSc. When applied to discretized infinite-dimensional problems, PDAS exhibit local superlinear convergence thanks to their equivalence to a semismooth Newton method (SSN). However, for many problem classes the number of iterations, to reach convergence, grows without bound under mesh refinement. In this paper we numerically study PDAS iteration counts on uniformly refined meshes for obstacle problems, Signorini problems, and related models. As the mesh size tends to zero, PDAS applied to Signorini-type problems lose their local superlinear convergence, resulting in linear growth of the iteration count (adding some iterations with each refinement). For obstacle problems, PDAS stagnates, leading to exponential iteration growth (asymptotically doubling with each refinement). We explain these phenomena by (i) proving that, for obstacle problems, nodal degrees of freedom only peel away from the obstacle layer-by-layer during the deactivation phase, (ii) deriving a general global convergence rate for PDAS that depends on the magnitude of dual feasibility violation, and (iii) demonstrating why, in the infinite-dimensional setting, this leads to a well-defined solver, but without local superlinear convergence, for some problems yet divergence for others.