  The topology optimization community has regularly employed nonlinear
programming (NLP) algorithms from the operations research community. However,
these algorithms are implemented in the real vector space $\mathbb{R}^n$
instead of the proper function space where the design variable resides. In this
article, we show how the volume fraction variable discretization on non-uniform
meshes affects the convergence of $\mathbb{R}^n$ based NLP algorithms. We do so
by first summarizing the functional analysis tools necessary to understand why
convergence is affected by the mesh. Namely, the distinction between derivative
definitions and the role of the mesh-dependent inner product within the NLP
algorithm. These tools are then used to make the Globally Convergent Method of
Moving Asymptotes (GCMMA), a popular NLP algorithm in the topology optimization
community, converge in a mesh independent fashion when starting from the same
initial design. We then benchmark our algorithms with three common problems in
topology optimization.
