  In this paper, we build on recent work using a mathematical programming
approach for incremental state update in analysis of non-linear mechanics
models. In particular, we consider quasi-static analysis of continuum problems
in the linearized kinematics regime, with non-linear material models described
using convex energy functions. We find in this case that the classical
displacement-based nested approach for incremental state update can be
reformulated as solving a reduced dual optimization problem. This reformulation
provides insights into the working of the algorithm, and eliminates the need
for some heuristics. An important purpose of this paper is to further
illustrate the unifying nature of the mathematical programming approach. We
therefore present relationships with several of these types of algorithms
recently presented in the literature for incremental state update.
